Octave 3.8, jcobi/3

Percentage Accurate: 94.9% → 99.8%

Time: 19.7s

Alternatives: 22

Speedup: 2.1×

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: 94.9% 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, 1.3× speedup

Math

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

FPCore

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

C

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

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 = beta + (alpha + 2.0d0)
    code = (((beta + 1.0d0) * ((1.0d0 + alpha) / t_0)) / t_0) / (1.0d0 + (2.0d0 + (beta + alpha)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.4%

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

   1. div-inv 96.4%

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

   2. +-commutative 96.4%

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

   3. associate-+l+ 96.4%

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

   4. *-commutative 96.4%

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

   5. metadata-eval 96.4%

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

   6. +-commutative 96.4%

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

   7. metadata-eval 96.4%

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

   8. +-commutative 96.4%

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

3. Applied egg-rr 96.4%

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

   1. associate-*l/ 96.4%

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

   2. associate-*r/ 96.4%

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

   3. *-rgt-identity 96.4%

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

   4. associate-+r+ 96.4%

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

   5. *-rgt-identity 96.4%

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

   6. +-commutative 96.4%

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

   7. distribute-rgt1-in 96.4%

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

   8. distribute-lft-in 96.4%

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

   9. +-commutative 96.4%

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

   10. +-commutative 96.4%

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

   11. *-commutative 96.4%

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

   12. +-commutative 96.4%

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

   13. +-commutative 96.4%

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

   14. associate-*r/ 99.8%

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

   15. +-commutative 99.8%

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

   16. +-commutative 99.8%

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

   17. associate-+r+ 99.8%

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

   18. +-commutative 99.8%

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

   19. associate-+r+ 99.8%

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

   20. +-commutative 99.8%

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

5. Simplified 99.8%

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

6. Final simplification 99.8%

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

Alternative 2: 99.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 2 \cdot 10^{+93}:\\ \;\;\;\;\left(1 + \alpha\right) \cdot \frac{\frac{\beta + 1}{t_0}}{t_0 \cdot \left(\alpha + \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{t_0} \cdot \left(1 + \frac{-1 - \alpha}{\beta}\right)}{\beta + \left(\alpha + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 2e+93)
     (* (+ 1.0 alpha) (/ (/ (+ beta 1.0) t_0) (* t_0 (+ alpha (+ beta 3.0)))))
     (/
      (* (/ (+ 1.0 alpha) t_0) (+ 1.0 (/ (- -1.0 alpha) beta)))
      (+ beta (+ alpha 3.0))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 2e+93) {
		tmp = (1.0 + alpha) * (((beta + 1.0) / t_0) / (t_0 * (alpha + (beta + 3.0))));
	} else {
		tmp = (((1.0 + alpha) / t_0) * (1.0 + ((-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) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    if (beta <= 2d+93) then
        tmp = (1.0d0 + alpha) * (((beta + 1.0d0) / t_0) / (t_0 * (alpha + (beta + 3.0d0))))
    else
        tmp = (((1.0d0 + alpha) / t_0) * (1.0d0 + (((-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 t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 2e+93) {
		tmp = (1.0 + alpha) * (((beta + 1.0) / t_0) / (t_0 * (alpha + (beta + 3.0))));
	} else {
		tmp = (((1.0 + alpha) / t_0) * (1.0 + ((-1.0 - alpha) / beta))) / (beta + (alpha + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 2e+93:
		tmp = (1.0 + alpha) * (((beta + 1.0) / t_0) / (t_0 * (alpha + (beta + 3.0))))
	else:
		tmp = (((1.0 + alpha) / t_0) * (1.0 + ((-1.0 - alpha) / beta))) / (beta + (alpha + 3.0))
	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 <= 2e+93)
		tmp = Float64(Float64(1.0 + alpha) * Float64(Float64(Float64(beta + 1.0) / t_0) / Float64(t_0 * Float64(alpha + Float64(beta + 3.0)))));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + alpha) / t_0) * Float64(1.0 + 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)
	t_0 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 2e+93)
		tmp = (1.0 + alpha) * (((beta + 1.0) / t_0) / (t_0 * (alpha + (beta + 3.0))));
	else
		tmp = (((1.0 + alpha) / t_0) * (1.0 + ((-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_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2e+93], N[(N[(1.0 + alpha), $MachinePrecision] * N[(N[(N[(beta + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 * N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(1.0 + N[(N[(-1.0 - alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.00000000000000009e93

   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/ 99.1%

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

      2. associate-+l+ 99.1%

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

      3. +-commutative 99.1%

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

      4. associate-+r+ 99.1%

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

      5. associate-+l+ 99.1%

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

      6. distribute-rgt1-in 99.1%

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

      7. *-rgt-identity 99.1%

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

      8. distribute-lft-out 99.1%

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

      9. +-commutative 99.1%

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

      10. associate-*l/ 99.1%

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

      11. *-commutative 99.1%

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

      12. associate-*r/ 96.5%

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

   3. Simplified 96.5%

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

   if 2.00000000000000009e93 < beta

   1. Initial program 83.9%

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

      1. div-inv 83.9%

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

      2. +-commutative 83.9%

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

      3. associate-+l+ 83.9%

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

      4. *-commutative 83.9%

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

      5. metadata-eval 83.9%

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

      6. +-commutative 83.9%

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

      7. metadata-eval 83.9%

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

      8. +-commutative 83.9%

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

   3. Applied egg-rr 83.9%

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

      1. associate-*l/ 83.9%

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

      2. associate-*r/ 83.9%

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

      3. *-rgt-identity 83.9%

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

      4. associate-+r+ 83.9%

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

      5. *-rgt-identity 83.9%

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

      6. +-commutative 83.9%

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

      7. distribute-rgt1-in 83.9%

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

      8. distribute-lft-in 83.9%

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

      9. +-commutative 83.9%

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

      10. +-commutative 83.9%

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

      11. *-commutative 83.9%

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

      12. +-commutative 83.9%

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

      13. +-commutative 83.9%

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

      14. associate-*r/ 99.9%

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

      15. +-commutative 99.9%

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

      16. +-commutative 99.9%

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

      17. associate-+r+ 99.9%

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

      18. +-commutative 99.9%

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

      19. associate-+r+ 99.9%

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

      20. +-commutative 99.9%

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

   5. Simplified 99.9%

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

      1. div-inv 99.8%

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

      2. associate-/l* 98.8%

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

      3. associate-+r+ 98.8%

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

      4. +-commutative 98.8%

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

      5. associate-+r+ 98.8%

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

      6. +-commutative 98.8%

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

      7. metadata-eval 98.8%

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

      8. associate-+l+ 98.8%

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

      9. +-commutative 98.8%

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

      10. metadata-eval 98.8%

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

   7. Applied egg-rr 98.8%

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

      1. associate-*r/ 98.9%

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

      2. *-rgt-identity 98.9%

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

      3. associate-/r/ 99.9%

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

      4. +-commutative 99.9%

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

      5. +-commutative 99.9%

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

      6. +-commutative 99.9%

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

      7. associate-+l+ 99.9%

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

   9. Simplified 99.9%

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

   10. Taylor expanded in beta around inf 87.9%

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

       1. mul-1-neg 87.9%

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

       2. unsub-neg 87.9%

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

   12. Simplified 87.9%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2 \cdot 10^{+93}:\\ \;\;\;\;\left(1 + \alpha\right) \cdot \frac{\frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \left(1 + \frac{-1 - \alpha}{\beta}\right)}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]

Alternative 3: 98.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.8:\\ \;\;\;\;\frac{1}{\alpha + 3} \cdot \frac{1 + \alpha}{4 + \alpha \cdot \left(\alpha + 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \left(1 + \frac{-1 - \alpha}{\beta}\right)}{\beta + \left(\alpha + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.8)
   (* (/ 1.0 (+ alpha 3.0)) (/ (+ 1.0 alpha) (+ 4.0 (* alpha (+ alpha 4.0)))))
   (/
    (*
     (/ (+ 1.0 alpha) (+ alpha (+ beta 2.0)))
     (+ 1.0 (/ (- -1.0 alpha) beta)))
    (+ beta (+ alpha 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.8) {
		tmp = (1.0 / (alpha + 3.0)) * ((1.0 + alpha) / (4.0 + (alpha * (alpha + 4.0))));
	} else {
		tmp = (((1.0 + alpha) / (alpha + (beta + 2.0))) * (1.0 + ((-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 <= 3.8d0) then
        tmp = (1.0d0 / (alpha + 3.0d0)) * ((1.0d0 + alpha) / (4.0d0 + (alpha * (alpha + 4.0d0))))
    else
        tmp = (((1.0d0 + alpha) / (alpha + (beta + 2.0d0))) * (1.0d0 + (((-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 <= 3.8) {
		tmp = (1.0 / (alpha + 3.0)) * ((1.0 + alpha) / (4.0 + (alpha * (alpha + 4.0))));
	} else {
		tmp = (((1.0 + alpha) / (alpha + (beta + 2.0))) * (1.0 + ((-1.0 - alpha) / beta))) / (beta + (alpha + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 3.8:
		tmp = (1.0 / (alpha + 3.0)) * ((1.0 + alpha) / (4.0 + (alpha * (alpha + 4.0))))
	else:
		tmp = (((1.0 + alpha) / (alpha + (beta + 2.0))) * (1.0 + ((-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 <= 3.8)
		tmp = Float64(Float64(1.0 / Float64(alpha + 3.0)) * Float64(Float64(1.0 + alpha) / Float64(4.0 + Float64(alpha * Float64(alpha + 4.0)))));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(beta + 2.0))) * Float64(1.0 + 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 <= 3.8)
		tmp = (1.0 / (alpha + 3.0)) * ((1.0 + alpha) / (4.0 + (alpha * (alpha + 4.0))));
	else
		tmp = (((1.0 + alpha) / (alpha + (beta + 2.0))) * (1.0 + ((-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, 3.8], N[(N[(1.0 / N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + alpha), $MachinePrecision] / N[(4.0 + N[(alpha * N[(alpha + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(-1.0 - alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.7999999999999998

   1. Initial program 99.8%

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

      1. associate-/l/ 99.5%

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

      2. associate-/l/ 96.5%

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

      3. associate-+l+ 96.5%

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

      4. +-commutative 96.5%

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

      5. associate-+r+ 96.5%

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

      6. associate-+l+ 96.5%

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

      7. distribute-rgt1-in 96.5%

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

      8. *-rgt-identity 96.5%

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

      9. distribute-lft-out 96.5%

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

      10. +-commutative 96.5%

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

      11. times-frac 99.5%

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

   3. Simplified 99.5%

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

      1. distribute-lft-in 99.5%

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

      2. associate-+r+ 99.5%

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

      3. +-commutative 99.5%

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

      4. associate-+r+ 99.5%

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

      5. +-commutative 99.5%

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

   5. Applied egg-rr 99.5%

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

   6. Taylor expanded in beta around 0 99.3%

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

   7. Taylor expanded in alpha around 0 99.4%

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

      1. unpow2 99.4%

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

      2. distribute-rgt-out 99.4%

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

   9. Simplified 99.4%

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

   10. Taylor expanded in beta around 0 99.0%

       \[\leadsto \color{blue}{\frac{1}{3 + \alpha}} \cdot \frac{1 + \alpha}{4 + \alpha \cdot \left(\alpha + 4\right)} \]

   if 3.7999999999999998 < beta

   1. Initial program 88.0%

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

      1. div-inv 88.1%

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

      2. +-commutative 88.1%

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

      3. associate-+l+ 88.1%

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

      4. *-commutative 88.1%

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

      5. metadata-eval 88.1%

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

      6. +-commutative 88.1%

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

      7. metadata-eval 88.1%

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

      8. +-commutative 88.1%

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

   3. Applied egg-rr 88.1%

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

      1. associate-*l/ 88.1%

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

      2. associate-*r/ 88.0%

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

      3. *-rgt-identity 88.0%

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

      4. associate-+r+ 88.0%

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

      5. *-rgt-identity 88.0%

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

      6. +-commutative 88.0%

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

      7. distribute-rgt1-in 88.0%

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

      8. distribute-lft-in 88.0%

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

      9. +-commutative 88.0%

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

      10. +-commutative 88.0%

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

      11. *-commutative 88.0%

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

      12. +-commutative 88.0%

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

      13. +-commutative 88.0%

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

      14. associate-*r/ 99.8%

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

      15. +-commutative 99.8%

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

      16. +-commutative 99.8%

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

      17. associate-+r+ 99.8%

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

      18. +-commutative 99.8%

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

      19. associate-+r+ 99.8%

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

      20. +-commutative 99.8%

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

   5. Simplified 99.8%

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

      1. div-inv 99.7%

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

      2. associate-/l* 99.0%

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

      3. associate-+r+ 99.0%

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

      4. +-commutative 99.0%

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

      5. associate-+r+ 99.0%

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

      6. +-commutative 99.0%

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

      7. metadata-eval 99.0%

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

      8. associate-+l+ 99.0%

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

      9. +-commutative 99.0%

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

      10. metadata-eval 99.0%

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

   7. Applied egg-rr 99.0%

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

      1. associate-*r/ 99.1%

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

      2. *-rgt-identity 99.1%

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

      3. associate-/r/ 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. +-commutative 99.8%

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

      7. associate-+l+ 99.8%

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

   9. Simplified 99.8%

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

   10. Taylor expanded in beta around inf 82.9%

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

       1. mul-1-neg 82.9%

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

       2. unsub-neg 82.9%

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

   12. Simplified 82.9%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.8:\\ \;\;\;\;\frac{1}{\alpha + 3} \cdot \frac{1 + \alpha}{4 + \alpha \cdot \left(\alpha + 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \left(1 + \frac{-1 - \alpha}{\beta}\right)}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]

Alternative 4: 99.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 96.4%

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

   1. div-inv 96.4%

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

   2. +-commutative 96.4%

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

   3. associate-+l+ 96.4%

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

   4. *-commutative 96.4%

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

   5. metadata-eval 96.4%

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

   6. +-commutative 96.4%

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

   7. metadata-eval 96.4%

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

   8. +-commutative 96.4%

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

3. Applied egg-rr 96.4%

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

   1. associate-*l/ 96.4%

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

   2. associate-*r/ 96.4%

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

   3. *-rgt-identity 96.4%

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

   4. associate-+r+ 96.4%

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

   5. *-rgt-identity 96.4%

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

   6. +-commutative 96.4%

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

   7. distribute-rgt1-in 96.4%

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

   8. distribute-lft-in 96.4%

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

   9. +-commutative 96.4%

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

   10. +-commutative 96.4%

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

   11. *-commutative 96.4%

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

   12. +-commutative 96.4%

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

   13. +-commutative 96.4%

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

   14. associate-*r/ 99.8%

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

   15. +-commutative 99.8%

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

   16. +-commutative 99.8%

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

   17. associate-+r+ 99.8%

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

   18. +-commutative 99.8%

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

   19. associate-+r+ 99.8%

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

   20. +-commutative 99.8%

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

5. Simplified 99.8%

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

   1. div-inv 99.7%

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

   2. associate-/l* 99.5%

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

   3. associate-+r+ 99.5%

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

   4. +-commutative 99.5%

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

   5. associate-+r+ 99.5%

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

   6. +-commutative 99.5%

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

   7. metadata-eval 99.5%

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

   8. associate-+l+ 99.5%

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

   9. +-commutative 99.5%

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

   10. metadata-eval 99.5%

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

7. Applied egg-rr 99.5%

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

   1. associate-*r/ 99.6%

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

   2. *-rgt-identity 99.6%

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

   3. associate-/r/ 99.8%

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

   4. +-commutative 99.8%

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

   5. +-commutative 99.8%

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

   6. +-commutative 99.8%

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

   7. associate-+l+ 99.8%

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

9. Simplified 99.8%

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

10. Final simplification 99.8%

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

Alternative 5: 99.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.4%

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

   1. div-inv 96.4%

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

   2. +-commutative 96.4%

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

   3. associate-+l+ 96.4%

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

   4. *-commutative 96.4%

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

   5. metadata-eval 96.4%

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

   6. +-commutative 96.4%

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

   7. metadata-eval 96.4%

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

   8. +-commutative 96.4%

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

3. Applied egg-rr 96.4%

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

   1. associate-*l/ 96.4%

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

   2. associate-*r/ 96.4%

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

   3. *-rgt-identity 96.4%

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

   4. associate-+r+ 96.4%

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

   5. *-rgt-identity 96.4%

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

   6. +-commutative 96.4%

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

   7. distribute-rgt1-in 96.4%

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

   8. distribute-lft-in 96.4%

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

   9. +-commutative 96.4%

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

   10. +-commutative 96.4%

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

   11. *-commutative 96.4%

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

   12. +-commutative 96.4%

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

   13. +-commutative 96.4%

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

   14. associate-*r/ 99.8%

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

   15. +-commutative 99.8%

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

   16. +-commutative 99.8%

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

   17. associate-+r+ 99.8%

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

   18. +-commutative 99.8%

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

   19. associate-+r+ 99.8%

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

   20. +-commutative 99.8%

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

5. Simplified 99.8%

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

   1. div-inv 99.7%

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

   2. associate-/l* 99.5%

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

   3. associate-+r+ 99.5%

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

   4. +-commutative 99.5%

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

   5. associate-+r+ 99.5%

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

   6. +-commutative 99.5%

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

   7. metadata-eval 99.5%

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

   8. associate-+l+ 99.5%

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

   9. +-commutative 99.5%

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

   10. metadata-eval 99.5%

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

7. Applied egg-rr 99.5%

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

   1. associate-*r/ 99.6%

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

   2. *-rgt-identity 99.6%

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

   3. associate-/r/ 99.8%

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

   4. +-commutative 99.8%

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

   5. +-commutative 99.8%

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

   6. +-commutative 99.8%

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

   7. associate-+l+ 99.8%

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

9. Simplified 99.8%

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

    1. associate-*l/ 99.8%

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

    2. +-commutative 99.8%

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

    3. +-commutative 99.8%

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

11. Applied egg-rr 99.8%

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

12. Final simplification 99.8%

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

Alternative 6: 98.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 9 \cdot 10^{+32}:\\ \;\;\;\;\frac{\left(\beta + 1\right) \cdot \left(1 + \alpha\right)}{t_0 \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{t_0}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 9e+32)
     (/ (* (+ beta 1.0) (+ 1.0 alpha)) (* t_0 (* (+ beta 2.0) (+ beta 3.0))))
     (/ (/ (+ 1.0 alpha) t_0) (+ beta (+ alpha 3.0))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 9e+32) {
		tmp = ((beta + 1.0) * (1.0 + alpha)) / (t_0 * ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / t_0) / (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) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    if (beta <= 9d+32) then
        tmp = ((beta + 1.0d0) * (1.0d0 + alpha)) / (t_0 * ((beta + 2.0d0) * (beta + 3.0d0)))
    else
        tmp = ((1.0d0 + alpha) / t_0) / (beta + (alpha + 3.0d0))
    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 <= 9e+32) {
		tmp = ((beta + 1.0) * (1.0 + alpha)) / (t_0 * ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / t_0) / (beta + (alpha + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 9e+32:
		tmp = ((beta + 1.0) * (1.0 + alpha)) / (t_0 * ((beta + 2.0) * (beta + 3.0)))
	else:
		tmp = ((1.0 + alpha) / t_0) / (beta + (alpha + 3.0))
	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 <= 9e+32)
		tmp = Float64(Float64(Float64(beta + 1.0) * Float64(1.0 + alpha)) / Float64(t_0 * Float64(Float64(beta + 2.0) * Float64(beta + 3.0))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / t_0) / Float64(beta + Float64(alpha + 3.0)));
	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 <= 9e+32)
		tmp = ((beta + 1.0) * (1.0 + alpha)) / (t_0 * ((beta + 2.0) * (beta + 3.0)));
	else
		tmp = ((1.0 + alpha) / t_0) / (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_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 9e+32], N[(N[(N[(beta + 1.0), $MachinePrecision] * N[(1.0 + alpha), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 9.0000000000000007e32

   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/ 99.6%

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

      2. associate-/r* 96.1%

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

      3. associate-+l+ 96.1%

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

      4. +-commutative 96.1%

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

      5. associate-+r+ 96.1%

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

      6. associate-+l+ 96.1%

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

      7. distribute-rgt1-in 96.1%

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

      8. *-rgt-identity 96.1%

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

      9. distribute-lft-out 96.1%

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

      10. *-commutative 96.1%

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

      11. metadata-eval 96.1%

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

      12. associate-+l+ 96.1%

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

      13. +-commutative 96.1%

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

   3. Simplified 96.1%

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

   4. Taylor expanded in alpha around 0 65.7%

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

   if 9.0000000000000007e32 < beta

   1. Initial program 87.0%

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

      1. div-inv 87.0%

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

      2. +-commutative 87.0%

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

      3. associate-+l+ 87.0%

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

      4. *-commutative 87.0%

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

      5. metadata-eval 87.0%

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

      6. +-commutative 87.0%

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

      7. metadata-eval 87.0%

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

      8. +-commutative 87.0%

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

   3. Applied egg-rr 87.0%

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

      1. associate-*l/ 87.0%

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

      2. associate-*r/ 87.0%

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

      3. *-rgt-identity 87.0%

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

      4. associate-+r+ 87.0%

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

      5. *-rgt-identity 87.0%

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

      6. +-commutative 87.0%

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

      7. distribute-rgt1-in 87.0%

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

      8. distribute-lft-in 87.0%

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

      9. +-commutative 87.0%

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

      10. +-commutative 87.0%

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

      11. *-commutative 87.0%

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

      12. +-commutative 87.0%

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

      13. +-commutative 87.0%

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

      14. associate-*r/ 99.8%

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

      15. +-commutative 99.8%

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

      16. +-commutative 99.8%

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

      17. associate-+r+ 99.8%

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

      18. +-commutative 99.8%

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

      19. associate-+r+ 99.8%

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

      20. +-commutative 99.8%

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

   5. Simplified 99.8%

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

      1. div-inv 99.7%

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

      2. associate-/l* 98.9%

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

      3. associate-+r+ 98.9%

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

      4. +-commutative 98.9%

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

      5. associate-+r+ 98.9%

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

      6. +-commutative 98.9%

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

      7. metadata-eval 98.9%

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

      8. associate-+l+ 98.9%

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

      9. +-commutative 98.9%

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

      10. metadata-eval 98.9%

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

   7. Applied egg-rr 98.9%

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

      1. associate-*r/ 99.0%

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

      2. *-rgt-identity 99.0%

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

      3. associate-/r/ 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. +-commutative 99.8%

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

      7. associate-+l+ 99.8%

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

   9. Simplified 99.8%

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

       1. associate-*l/ 99.8%

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

       2. +-commutative 99.8%

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

       3. +-commutative 99.8%

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

   11. Applied egg-rr 99.8%

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

   12. Taylor expanded in beta around inf 86.2%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 9 \cdot 10^{+32}:\\ \;\;\;\;\frac{\left(\beta + 1\right) \cdot \left(1 + \alpha\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]

Alternative 7: 98.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 9 \cdot 10^{+32}:\\ \;\;\;\;\frac{\left(\beta + 1\right) \cdot \left(1 + \alpha\right)}{t_0 \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + 1}{t_0} \cdot \frac{1 + \alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 9e+32)
     (/ (* (+ beta 1.0) (+ 1.0 alpha)) (* t_0 (* (+ beta 2.0) (+ beta 3.0))))
     (/
      (* (/ (+ beta 1.0) t_0) (/ (+ 1.0 alpha) beta))
      (+ beta (+ alpha 3.0))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 9e+32) {
		tmp = ((beta + 1.0) * (1.0 + alpha)) / (t_0 * ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = (((beta + 1.0) / t_0) * ((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) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    if (beta <= 9d+32) then
        tmp = ((beta + 1.0d0) * (1.0d0 + alpha)) / (t_0 * ((beta + 2.0d0) * (beta + 3.0d0)))
    else
        tmp = (((beta + 1.0d0) / t_0) * ((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 t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 9e+32) {
		tmp = ((beta + 1.0) * (1.0 + alpha)) / (t_0 * ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = (((beta + 1.0) / t_0) * ((1.0 + alpha) / beta)) / (beta + (alpha + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 9e+32:
		tmp = ((beta + 1.0) * (1.0 + alpha)) / (t_0 * ((beta + 2.0) * (beta + 3.0)))
	else:
		tmp = (((beta + 1.0) / t_0) * ((1.0 + alpha) / beta)) / (beta + (alpha + 3.0))
	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 <= 9e+32)
		tmp = Float64(Float64(Float64(beta + 1.0) * Float64(1.0 + alpha)) / Float64(t_0 * Float64(Float64(beta + 2.0) * Float64(beta + 3.0))));
	else
		tmp = Float64(Float64(Float64(Float64(beta + 1.0) / t_0) * 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)
	t_0 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 9e+32)
		tmp = ((beta + 1.0) * (1.0 + alpha)) / (t_0 * ((beta + 2.0) * (beta + 3.0)));
	else
		tmp = (((beta + 1.0) / t_0) * ((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_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 9e+32], N[(N[(N[(beta + 1.0), $MachinePrecision] * N[(1.0 + alpha), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(beta + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 9.0000000000000007e32

   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/ 99.6%

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

      2. associate-/r* 96.1%

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

      3. associate-+l+ 96.1%

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

      4. +-commutative 96.1%

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

      5. associate-+r+ 96.1%

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

      6. associate-+l+ 96.1%

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

      7. distribute-rgt1-in 96.1%

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

      8. *-rgt-identity 96.1%

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

      9. distribute-lft-out 96.1%

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

      10. *-commutative 96.1%

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

      11. metadata-eval 96.1%

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

      12. associate-+l+ 96.1%

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

      13. +-commutative 96.1%

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

   3. Simplified 96.1%

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

   4. Taylor expanded in alpha around 0 65.7%

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

   if 9.0000000000000007e32 < beta

   1. Initial program 87.0%

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

      1. div-inv 87.0%

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

      2. +-commutative 87.0%

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

      3. associate-+l+ 87.0%

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

      4. *-commutative 87.0%

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

      5. metadata-eval 87.0%

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

      6. +-commutative 87.0%

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

      7. metadata-eval 87.0%

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

      8. +-commutative 87.0%

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

   3. Applied egg-rr 87.0%

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

      1. associate-*l/ 87.0%

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

      2. associate-*r/ 87.0%

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

      3. *-rgt-identity 87.0%

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

      4. associate-+r+ 87.0%

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

      5. *-rgt-identity 87.0%

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

      6. +-commutative 87.0%

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

      7. distribute-rgt1-in 87.0%

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

      8. distribute-lft-in 87.0%

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

      9. +-commutative 87.0%

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

      10. +-commutative 87.0%

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

      11. *-commutative 87.0%

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

      12. +-commutative 87.0%

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

      13. +-commutative 87.0%

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

      14. associate-*r/ 99.8%

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

      15. +-commutative 99.8%

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

      16. +-commutative 99.8%

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

      17. associate-+r+ 99.8%

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

      18. +-commutative 99.8%

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

      19. associate-+r+ 99.8%

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

      20. +-commutative 99.8%

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

   5. Simplified 99.8%

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

      1. div-inv 99.7%

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

      2. associate-/l* 98.9%

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

      3. associate-+r+ 98.9%

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

      4. +-commutative 98.9%

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

      5. associate-+r+ 98.9%

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

      6. +-commutative 98.9%

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

      7. metadata-eval 98.9%

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

      8. associate-+l+ 98.9%

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

      9. +-commutative 98.9%

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

      10. metadata-eval 98.9%

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

   7. Applied egg-rr 98.9%

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

      1. associate-*r/ 99.0%

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

      2. *-rgt-identity 99.0%

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

      3. associate-/r/ 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. +-commutative 99.8%

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

      7. associate-+l+ 99.8%

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

   9. Simplified 99.8%

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

   10. Taylor expanded in beta around inf 86.2%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 9 \cdot 10^{+32}:\\ \;\;\;\;\frac{\left(\beta + 1\right) \cdot \left(1 + \alpha\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]

Alternative 8: 98.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \beta + \left(\alpha + 3\right)\\ t_1 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 1.42 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{\left(\beta + 1\right) \cdot \frac{1}{\beta + 2}}{t_1}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + 1}{t_1} \cdot \frac{1 + \alpha}{\beta}}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ beta (+ alpha 3.0))) (t_1 (+ alpha (+ beta 2.0))))
   (if (<= beta 1.42e+16)
     (/ (/ (* (+ beta 1.0) (/ 1.0 (+ beta 2.0))) t_1) t_0)
     (/ (* (/ (+ beta 1.0) t_1) (/ (+ 1.0 alpha) beta)) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = beta + (alpha + 3.0);
	double t_1 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 1.42e+16) {
		tmp = (((beta + 1.0) * (1.0 / (beta + 2.0))) / t_1) / t_0;
	} else {
		tmp = (((beta + 1.0) / t_1) * ((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) :: t_1
    real(8) :: tmp
    t_0 = beta + (alpha + 3.0d0)
    t_1 = alpha + (beta + 2.0d0)
    if (beta <= 1.42d+16) then
        tmp = (((beta + 1.0d0) * (1.0d0 / (beta + 2.0d0))) / t_1) / t_0
    else
        tmp = (((beta + 1.0d0) / t_1) * ((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 = beta + (alpha + 3.0);
	double t_1 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 1.42e+16) {
		tmp = (((beta + 1.0) * (1.0 / (beta + 2.0))) / t_1) / t_0;
	} else {
		tmp = (((beta + 1.0) / t_1) * ((1.0 + alpha) / beta)) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = beta + (alpha + 3.0)
	t_1 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 1.42e+16:
		tmp = (((beta + 1.0) * (1.0 / (beta + 2.0))) / t_1) / t_0
	else:
		tmp = (((beta + 1.0) / t_1) * ((1.0 + alpha) / beta)) / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(beta + Float64(alpha + 3.0))
	t_1 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 1.42e+16)
		tmp = Float64(Float64(Float64(Float64(beta + 1.0) * Float64(1.0 / Float64(beta + 2.0))) / t_1) / t_0);
	else
		tmp = Float64(Float64(Float64(Float64(beta + 1.0) / t_1) * 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 = beta + (alpha + 3.0);
	t_1 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 1.42e+16)
		tmp = (((beta + 1.0) * (1.0 / (beta + 2.0))) / t_1) / t_0;
	else
		tmp = (((beta + 1.0) / t_1) * ((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[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1.42e+16], N[(N[(N[(N[(beta + 1.0), $MachinePrecision] * N[(1.0 / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(N[(beta + 1.0), $MachinePrecision] / t$95$1), $MachinePrecision] * N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.42e16

   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. div-inv 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+l+ 99.8%

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

      4. *-commutative 99.8%

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

      5. metadata-eval 99.8%

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

      6. +-commutative 99.8%

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

      7. metadata-eval 99.8%

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

      8. +-commutative 99.8%

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

   3. Applied egg-rr 99.8%

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

      1. associate-*l/ 99.8%

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

      2. associate-*r/ 99.8%

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

      3. *-rgt-identity 99.8%

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

      4. associate-+r+ 99.8%

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

      5. *-rgt-identity 99.8%

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

      6. +-commutative 99.8%

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

      7. distribute-rgt1-in 99.8%

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

      8. distribute-lft-in 99.8%

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

      9. +-commutative 99.8%

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

      10. +-commutative 99.8%

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

      11. *-commutative 99.8%

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

      12. +-commutative 99.8%

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

      13. +-commutative 99.8%

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

      14. associate-*r/ 99.8%

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

      15. +-commutative 99.8%

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

      16. +-commutative 99.8%

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

      17. associate-+r+ 99.8%

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

      18. +-commutative 99.8%

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

      19. associate-+r+ 99.8%

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

      20. +-commutative 99.8%

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

   5. Simplified 99.8%

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

      1. div-inv 99.8%

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

      2. associate-/l* 99.8%

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

      3. associate-+r+ 99.8%

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

      4. +-commutative 99.8%

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

      5. associate-+r+ 99.8%

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

      6. +-commutative 99.8%

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

      7. metadata-eval 99.8%

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

      8. associate-+l+ 99.8%

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

      9. +-commutative 99.8%

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

      10. metadata-eval 99.8%

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

   7. Applied egg-rr 99.8%

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

      1. associate-*r/ 99.8%

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

      2. *-rgt-identity 99.8%

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

      3. associate-/r/ 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. +-commutative 99.8%

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

      7. associate-+l+ 99.8%

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

   9. Simplified 99.8%

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

       1. associate-*l/ 99.8%

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

       2. +-commutative 99.8%

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

       3. +-commutative 99.8%

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

   11. Applied egg-rr 99.8%

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

   12. Taylor expanded in alpha around 0 86.2%

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

   if 1.42e16 < beta

   1. Initial program 87.7%

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

      1. div-inv 87.7%

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

      2. +-commutative 87.7%

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

      3. associate-+l+ 87.7%

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

      4. *-commutative 87.7%

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

      5. metadata-eval 87.7%

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

      6. +-commutative 87.7%

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

      7. metadata-eval 87.7%

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

      8. +-commutative 87.7%

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

   3. Applied egg-rr 87.7%

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

      1. associate-*l/ 87.7%

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

      2. associate-*r/ 87.7%

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

      3. *-rgt-identity 87.7%

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

      4. associate-+r+ 87.7%

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

      5. *-rgt-identity 87.7%

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

      6. +-commutative 87.7%

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

      7. distribute-rgt1-in 87.7%

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

      8. distribute-lft-in 87.7%

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

      9. +-commutative 87.7%

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

      10. +-commutative 87.7%

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

      11. *-commutative 87.7%

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

      12. +-commutative 87.7%

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

      13. +-commutative 87.7%

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

      14. associate-*r/ 99.8%

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

      15. +-commutative 99.8%

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

      16. +-commutative 99.8%

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

      17. associate-+r+ 99.8%

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

      18. +-commutative 99.8%

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

      19. associate-+r+ 99.8%

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

      20. +-commutative 99.8%

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

   5. Simplified 99.8%

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

      1. div-inv 99.7%

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

      2. associate-/l* 99.0%

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

      3. associate-+r+ 99.0%

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

      4. +-commutative 99.0%

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

      5. associate-+r+ 99.0%

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

      6. +-commutative 99.0%

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

      7. metadata-eval 99.0%

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

      8. associate-+l+ 99.0%

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

      9. +-commutative 99.0%

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

      10. metadata-eval 99.0%

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

   7. Applied egg-rr 99.0%

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

      1. associate-*r/ 99.1%

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

      2. *-rgt-identity 99.1%

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

      3. associate-/r/ 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. +-commutative 99.8%

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

      7. associate-+l+ 99.8%

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

   9. Simplified 99.8%

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

   10. Taylor expanded in beta around inf 84.4%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.42 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{\left(\beta + 1\right) \cdot \frac{1}{\beta + 2}}{\alpha + \left(\beta + 2\right)}}{\beta + \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]

Alternative 9: 98.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 9 \cdot 10^{+32}:\\ \;\;\;\;\frac{\left(\beta + 1\right) \cdot \left(1 + \alpha\right)}{\left(\beta + 2\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 9e+32)
   (/
    (* (+ beta 1.0) (+ 1.0 alpha))
    (* (+ beta 2.0) (* (+ beta 2.0) (+ beta 3.0))))
   (/ (/ (+ 1.0 alpha) (+ alpha (+ beta 2.0))) (+ beta (+ alpha 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 9e+32) {
		tmp = ((beta + 1.0) * (1.0 + alpha)) / ((beta + 2.0) * ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) / (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 <= 9d+32) then
        tmp = ((beta + 1.0d0) * (1.0d0 + alpha)) / ((beta + 2.0d0) * ((beta + 2.0d0) * (beta + 3.0d0)))
    else
        tmp = ((1.0d0 + alpha) / (alpha + (beta + 2.0d0))) / (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 <= 9e+32) {
		tmp = ((beta + 1.0) * (1.0 + alpha)) / ((beta + 2.0) * ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) / (beta + (alpha + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 9e+32:
		tmp = ((beta + 1.0) * (1.0 + alpha)) / ((beta + 2.0) * ((beta + 2.0) * (beta + 3.0)))
	else:
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) / (beta + (alpha + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 9e+32)
		tmp = Float64(Float64(Float64(beta + 1.0) * Float64(1.0 + alpha)) / Float64(Float64(beta + 2.0) * Float64(Float64(beta + 2.0) * Float64(beta + 3.0))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(beta + 2.0))) / 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 <= 9e+32)
		tmp = ((beta + 1.0) * (1.0 + alpha)) / ((beta + 2.0) * ((beta + 2.0) * (beta + 3.0)));
	else
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) / (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, 9e+32], N[(N[(N[(beta + 1.0), $MachinePrecision] * N[(1.0 + alpha), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 9.0000000000000007e32

   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/ 99.6%

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

      2. associate-/r* 96.1%

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

      3. associate-+l+ 96.1%

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

      4. +-commutative 96.1%

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

      5. associate-+r+ 96.1%

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

      6. associate-+l+ 96.1%

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

      7. distribute-rgt1-in 96.1%

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

      8. *-rgt-identity 96.1%

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

      9. distribute-lft-out 96.1%

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

      10. *-commutative 96.1%

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

      11. metadata-eval 96.1%

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

      12. associate-+l+ 96.1%

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

      13. +-commutative 96.1%

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

   3. Simplified 96.1%

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

   4. Taylor expanded in alpha around 0 65.7%

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

   5. Taylor expanded in alpha around 0 65.2%

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

   if 9.0000000000000007e32 < beta

   1. Initial program 87.0%

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

      1. div-inv 87.0%

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

      2. +-commutative 87.0%

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

      3. associate-+l+ 87.0%

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

      4. *-commutative 87.0%

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

      5. metadata-eval 87.0%

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

      6. +-commutative 87.0%

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

      7. metadata-eval 87.0%

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

      8. +-commutative 87.0%

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

   3. Applied egg-rr 87.0%

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

      1. associate-*l/ 87.0%

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

      2. associate-*r/ 87.0%

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

      3. *-rgt-identity 87.0%

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

      4. associate-+r+ 87.0%

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

      5. *-rgt-identity 87.0%

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

      6. +-commutative 87.0%

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

      7. distribute-rgt1-in 87.0%

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

      8. distribute-lft-in 87.0%

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

      9. +-commutative 87.0%

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

      10. +-commutative 87.0%

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

      11. *-commutative 87.0%

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

      12. +-commutative 87.0%

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

      13. +-commutative 87.0%

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

      14. associate-*r/ 99.8%

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

      15. +-commutative 99.8%

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

      16. +-commutative 99.8%

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

      17. associate-+r+ 99.8%

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

      18. +-commutative 99.8%

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

      19. associate-+r+ 99.8%

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

      20. +-commutative 99.8%

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

   5. Simplified 99.8%

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

      1. div-inv 99.7%

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

      2. associate-/l* 98.9%

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

      3. associate-+r+ 98.9%

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

      4. +-commutative 98.9%

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

      5. associate-+r+ 98.9%

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

      6. +-commutative 98.9%

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

      7. metadata-eval 98.9%

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

      8. associate-+l+ 98.9%

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

      9. +-commutative 98.9%

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

      10. metadata-eval 98.9%

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

   7. Applied egg-rr 98.9%

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

      1. associate-*r/ 99.0%

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

      2. *-rgt-identity 99.0%

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

      3. associate-/r/ 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. +-commutative 99.8%

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

      7. associate-+l+ 99.8%

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

   9. Simplified 99.8%

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

       1. associate-*l/ 99.8%

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

       2. +-commutative 99.8%

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

       3. +-commutative 99.8%

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

   11. Applied egg-rr 99.8%

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

   12. Taylor expanded in beta around inf 86.2%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 9 \cdot 10^{+32}:\\ \;\;\;\;\frac{\left(\beta + 1\right) \cdot \left(1 + \alpha\right)}{\left(\beta + 2\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]

Alternative 10: 97.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.55:\\ \;\;\;\;\frac{1}{\alpha + 3} \cdot \frac{1 + \alpha}{4 + \alpha \cdot \left(\alpha + 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.55)
   (* (/ 1.0 (+ alpha 3.0)) (/ (+ 1.0 alpha) (+ 4.0 (* alpha (+ alpha 4.0)))))
   (/ (/ (+ 1.0 alpha) (+ alpha (+ beta 2.0))) (+ beta (+ alpha 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.55) {
		tmp = (1.0 / (alpha + 3.0)) * ((1.0 + alpha) / (4.0 + (alpha * (alpha + 4.0))));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) / (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 <= 1.55d0) then
        tmp = (1.0d0 / (alpha + 3.0d0)) * ((1.0d0 + alpha) / (4.0d0 + (alpha * (alpha + 4.0d0))))
    else
        tmp = ((1.0d0 + alpha) / (alpha + (beta + 2.0d0))) / (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 <= 1.55) {
		tmp = (1.0 / (alpha + 3.0)) * ((1.0 + alpha) / (4.0 + (alpha * (alpha + 4.0))));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) / (beta + (alpha + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.55:
		tmp = (1.0 / (alpha + 3.0)) * ((1.0 + alpha) / (4.0 + (alpha * (alpha + 4.0))))
	else:
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) / (beta + (alpha + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.55)
		tmp = Float64(Float64(1.0 / Float64(alpha + 3.0)) * Float64(Float64(1.0 + alpha) / Float64(4.0 + Float64(alpha * Float64(alpha + 4.0)))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(beta + 2.0))) / 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 <= 1.55)
		tmp = (1.0 / (alpha + 3.0)) * ((1.0 + alpha) / (4.0 + (alpha * (alpha + 4.0))));
	else
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) / (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, 1.55], N[(N[(1.0 / N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + alpha), $MachinePrecision] / N[(4.0 + N[(alpha * N[(alpha + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.55000000000000004

   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%

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

      2. associate-/l/ 96.5%

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

      3. associate-+l+ 96.5%

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

      4. +-commutative 96.5%

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

      5. associate-+r+ 96.5%

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

      6. associate-+l+ 96.5%

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

      7. distribute-rgt1-in 96.5%

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

      8. *-rgt-identity 96.5%

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

      9. distribute-lft-out 96.5%

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

      10. +-commutative 96.5%

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

      11. times-frac 99.5%

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

   3. Simplified 99.5%

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

      1. distribute-lft-in 99.5%

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

      2. associate-+r+ 99.5%

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

      3. +-commutative 99.5%

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

      4. associate-+r+ 99.5%

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

      5. +-commutative 99.5%

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

   5. Applied egg-rr 99.5%

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

   6. Taylor expanded in beta around 0 99.3%

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

   7. Taylor expanded in alpha around 0 99.4%

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

      1. unpow2 99.4%

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

      2. distribute-rgt-out 99.4%

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

   9. Simplified 99.4%

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

   10. Taylor expanded in beta around 0 99.0%

       \[\leadsto \color{blue}{\frac{1}{3 + \alpha}} \cdot \frac{1 + \alpha}{4 + \alpha \cdot \left(\alpha + 4\right)} \]

   if 1.55000000000000004 < beta

   1. Initial program 88.0%

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

      1. div-inv 88.1%

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

      2. +-commutative 88.1%

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

      3. associate-+l+ 88.1%

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

      4. *-commutative 88.1%

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

      5. metadata-eval 88.1%

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

      6. +-commutative 88.1%

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

      7. metadata-eval 88.1%

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

      8. +-commutative 88.1%

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

   3. Applied egg-rr 88.1%

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

      1. associate-*l/ 88.1%

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

      2. associate-*r/ 88.0%

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

      3. *-rgt-identity 88.0%

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

      4. associate-+r+ 88.0%

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

      5. *-rgt-identity 88.0%

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

      6. +-commutative 88.0%

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

      7. distribute-rgt1-in 88.0%

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

      8. distribute-lft-in 88.0%

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

      9. +-commutative 88.0%

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

      10. +-commutative 88.0%

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

      11. *-commutative 88.0%

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

      12. +-commutative 88.0%

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

      13. +-commutative 88.0%

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

      14. associate-*r/ 99.8%

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

      15. +-commutative 99.8%

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

      16. +-commutative 99.8%

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

      17. associate-+r+ 99.8%

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

      18. +-commutative 99.8%

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

      19. associate-+r+ 99.8%

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

      20. +-commutative 99.8%

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

   5. Simplified 99.8%

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

      1. div-inv 99.7%

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

      2. associate-/l* 99.0%

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

      3. associate-+r+ 99.0%

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

      4. +-commutative 99.0%

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

      5. associate-+r+ 99.0%

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

      6. +-commutative 99.0%

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

      7. metadata-eval 99.0%

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

      8. associate-+l+ 99.0%

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

      9. +-commutative 99.0%

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

      10. metadata-eval 99.0%

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

   7. Applied egg-rr 99.0%

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

      1. associate-*r/ 99.1%

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

      2. *-rgt-identity 99.1%

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

      3. associate-/r/ 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. +-commutative 99.8%

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

      7. associate-+l+ 99.8%

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

   9. Simplified 99.8%

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

       1. associate-*l/ 99.8%

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

       2. +-commutative 99.8%

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

       3. +-commutative 99.8%

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

   11. Applied egg-rr 99.8%

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

   12. Taylor expanded in beta around inf 82.8%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.55:\\ \;\;\;\;\frac{1}{\alpha + 3} \cdot \frac{1 + \alpha}{4 + \alpha \cdot \left(\alpha + 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]

Alternative 11: 94.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.8:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + 3}}{4 + \alpha \cdot \left(\alpha + 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta} + \frac{\alpha}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.8)
   (/ (/ (+ 1.0 alpha) (+ alpha 3.0)) (+ 4.0 (* alpha (+ alpha 4.0))))
   (+ (/ 1.0 (* beta beta)) (/ alpha (* beta beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.8) {
		tmp = ((1.0 + alpha) / (alpha + 3.0)) / (4.0 + (alpha * (alpha + 4.0)));
	} else {
		tmp = (1.0 / (beta * beta)) + (alpha / (beta * beta));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 4.8d0) then
        tmp = ((1.0d0 + alpha) / (alpha + 3.0d0)) / (4.0d0 + (alpha * (alpha + 4.0d0)))
    else
        tmp = (1.0d0 / (beta * beta)) + (alpha / (beta * beta))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.8) {
		tmp = ((1.0 + alpha) / (alpha + 3.0)) / (4.0 + (alpha * (alpha + 4.0)));
	} else {
		tmp = (1.0 / (beta * beta)) + (alpha / (beta * beta));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 4.8:
		tmp = ((1.0 + alpha) / (alpha + 3.0)) / (4.0 + (alpha * (alpha + 4.0)))
	else:
		tmp = (1.0 / (beta * beta)) + (alpha / (beta * beta))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.8)
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + 3.0)) / Float64(4.0 + Float64(alpha * Float64(alpha + 4.0))));
	else
		tmp = Float64(Float64(1.0 / Float64(beta * beta)) + Float64(alpha / Float64(beta * beta)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4.8)
		tmp = ((1.0 + alpha) / (alpha + 3.0)) / (4.0 + (alpha * (alpha + 4.0)));
	else
		tmp = (1.0 / (beta * beta)) + (alpha / (beta * beta));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 4.8], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision] / N[(4.0 + N[(alpha * N[(alpha + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(beta * beta), $MachinePrecision]), $MachinePrecision] + N[(alpha / N[(beta * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.79999999999999982

   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%

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

      2. associate-/l/ 96.5%

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

      3. associate-+l+ 96.5%

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

      4. +-commutative 96.5%

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

      5. associate-+r+ 96.5%

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

      6. associate-+l+ 96.5%

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

      7. distribute-rgt1-in 96.5%

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

      8. *-rgt-identity 96.5%

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

      9. distribute-lft-out 96.5%

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

      10. +-commutative 96.5%

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

      11. times-frac 99.5%

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

   3. Simplified 99.5%

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

      1. distribute-lft-in 99.5%

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

      2. associate-+r+ 99.5%

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

      3. +-commutative 99.5%

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

      4. associate-+r+ 99.5%

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

      5. +-commutative 99.5%

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

   5. Applied egg-rr 99.5%

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

   6. Taylor expanded in beta around 0 99.3%

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

   7. Taylor expanded in alpha around 0 99.4%

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

      1. unpow2 99.4%

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

      2. distribute-rgt-out 99.4%

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

   9. Simplified 99.4%

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

   10. Taylor expanded in beta around 0 96.1%

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

       1. associate-/r* 99.0%

          \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{3 + \alpha}}{4 + \left(4 + \alpha\right) \cdot \alpha}} \]

       2. +-commutative 99.0%

          \[\leadsto \frac{\frac{1 + \alpha}{\color{blue}{\alpha + 3}}}{4 + \left(4 + \alpha\right) \cdot \alpha} \]

       3. +-commutative 99.0%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + 3}}{4 + \color{blue}{\left(\alpha + 4\right)} \cdot \alpha} \]

       4. *-commutative 99.0%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + 3}}{4 + \color{blue}{\alpha \cdot \left(\alpha + 4\right)}} \]

   12. Simplified 99.0%

       \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\alpha + 3}}{4 + \alpha \cdot \left(\alpha + 4\right)}} \]

   if 4.79999999999999982 < beta

   1. Initial program 88.0%

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

      1. associate-/l/ 84.7%

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

      2. associate-+l+ 84.7%

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

      3. +-commutative 84.7%

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

      4. associate-+r+ 84.7%

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

      5. associate-+l+ 84.7%

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

      6. distribute-rgt1-in 84.7%

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

      7. *-rgt-identity 84.7%

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

      8. distribute-lft-out 84.7%

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

      9. +-commutative 84.7%

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

      10. associate-*l/ 90.6%

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

      11. *-commutative 90.6%

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

      12. associate-*r/ 88.8%

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

   3. Simplified 88.8%

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

   4. Taylor expanded in beta around inf 80.2%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 80.2%

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

   6. Simplified 80.2%

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

   7. Taylor expanded in alpha around 0 80.2%

      \[\leadsto \color{blue}{\frac{1}{{\beta}^{2}} + \frac{\alpha}{{\beta}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 80.2%

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

      2. unpow2 80.2%

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

   9. Simplified 80.2%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.8:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + 3}}{4 + \alpha \cdot \left(\alpha + 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta} + \frac{\alpha}{\beta \cdot \beta}\\ \end{array} \]

Alternative 12: 97.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.8:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + 3}}{4 + \alpha \cdot \left(\alpha + 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.8)
   (/ (/ (+ 1.0 alpha) (+ alpha 3.0)) (+ 4.0 (* alpha (+ alpha 4.0))))
   (/ (/ (+ 1.0 alpha) (+ alpha (+ beta 2.0))) (+ beta (+ alpha 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.8) {
		tmp = ((1.0 + alpha) / (alpha + 3.0)) / (4.0 + (alpha * (alpha + 4.0)));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) / (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 <= 1.8d0) then
        tmp = ((1.0d0 + alpha) / (alpha + 3.0d0)) / (4.0d0 + (alpha * (alpha + 4.0d0)))
    else
        tmp = ((1.0d0 + alpha) / (alpha + (beta + 2.0d0))) / (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 <= 1.8) {
		tmp = ((1.0 + alpha) / (alpha + 3.0)) / (4.0 + (alpha * (alpha + 4.0)));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) / (beta + (alpha + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.8:
		tmp = ((1.0 + alpha) / (alpha + 3.0)) / (4.0 + (alpha * (alpha + 4.0)))
	else:
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) / (beta + (alpha + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.8)
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + 3.0)) / Float64(4.0 + Float64(alpha * Float64(alpha + 4.0))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(beta + 2.0))) / 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 <= 1.8)
		tmp = ((1.0 + alpha) / (alpha + 3.0)) / (4.0 + (alpha * (alpha + 4.0)));
	else
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) / (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, 1.8], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision] / N[(4.0 + N[(alpha * N[(alpha + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.80000000000000004

   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%

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

      2. associate-/l/ 96.5%

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

      3. associate-+l+ 96.5%

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

      4. +-commutative 96.5%

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

      5. associate-+r+ 96.5%

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

      6. associate-+l+ 96.5%

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

      7. distribute-rgt1-in 96.5%

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

      8. *-rgt-identity 96.5%

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

      9. distribute-lft-out 96.5%

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

      10. +-commutative 96.5%

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

      11. times-frac 99.5%

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

   3. Simplified 99.5%

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

      1. distribute-lft-in 99.5%

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

      2. associate-+r+ 99.5%

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

      3. +-commutative 99.5%

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

      4. associate-+r+ 99.5%

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

      5. +-commutative 99.5%

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

   5. Applied egg-rr 99.5%

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

   6. Taylor expanded in beta around 0 99.3%

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

   7. Taylor expanded in alpha around 0 99.4%

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

      1. unpow2 99.4%

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

      2. distribute-rgt-out 99.4%

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

   9. Simplified 99.4%

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

   10. Taylor expanded in beta around 0 96.1%

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

       1. associate-/r* 99.0%

          \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{3 + \alpha}}{4 + \left(4 + \alpha\right) \cdot \alpha}} \]

       2. +-commutative 99.0%

          \[\leadsto \frac{\frac{1 + \alpha}{\color{blue}{\alpha + 3}}}{4 + \left(4 + \alpha\right) \cdot \alpha} \]

       3. +-commutative 99.0%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + 3}}{4 + \color{blue}{\left(\alpha + 4\right)} \cdot \alpha} \]

       4. *-commutative 99.0%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + 3}}{4 + \color{blue}{\alpha \cdot \left(\alpha + 4\right)}} \]

   12. Simplified 99.0%

       \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\alpha + 3}}{4 + \alpha \cdot \left(\alpha + 4\right)}} \]

   if 1.80000000000000004 < beta

   1. Initial program 88.0%

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

      1. div-inv 88.1%

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

      2. +-commutative 88.1%

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

      3. associate-+l+ 88.1%

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

      4. *-commutative 88.1%

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

      5. metadata-eval 88.1%

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

      6. +-commutative 88.1%

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

      7. metadata-eval 88.1%

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

      8. +-commutative 88.1%

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

   3. Applied egg-rr 88.1%

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

      1. associate-*l/ 88.1%

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

      2. associate-*r/ 88.0%

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

      3. *-rgt-identity 88.0%

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

      4. associate-+r+ 88.0%

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

      5. *-rgt-identity 88.0%

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

      6. +-commutative 88.0%

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

      7. distribute-rgt1-in 88.0%

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

      8. distribute-lft-in 88.0%

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

      9. +-commutative 88.0%

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

      10. +-commutative 88.0%

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

      11. *-commutative 88.0%

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

      12. +-commutative 88.0%

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

      13. +-commutative 88.0%

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

      14. associate-*r/ 99.8%

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

      15. +-commutative 99.8%

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

      16. +-commutative 99.8%

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

      17. associate-+r+ 99.8%

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

      18. +-commutative 99.8%

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

      19. associate-+r+ 99.8%

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

      20. +-commutative 99.8%

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

   5. Simplified 99.8%

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

      1. div-inv 99.7%

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

      2. associate-/l* 99.0%

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

      3. associate-+r+ 99.0%

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

      4. +-commutative 99.0%

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

      5. associate-+r+ 99.0%

         \[\leadsto \frac{\beta + 1}{\frac{\alpha + \left(\beta + 2\right)}{\frac{1 + \alpha}{\color{blue}{\left(\beta + 2\right) + \alpha}}}} \cdot \frac{1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      6. +-commutative 99.0%

         \[\leadsto \frac{\beta + 1}{\frac{\alpha + \left(\beta + 2\right)}{\frac{1 + \alpha}{\color{blue}{\alpha + \left(\beta + 2\right)}}}} \cdot \frac{1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      7. metadata-eval 99.0%

         \[\leadsto \frac{\beta + 1}{\frac{\alpha + \left(\beta + 2\right)}{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}}} \cdot \frac{1}{\left(\left(\alpha + \beta\right) + \color{blue}{2}\right) + 1} \]

      8. associate-+l+ 99.0%

         \[\leadsto \frac{\beta + 1}{\frac{\alpha + \left(\beta + 2\right)}{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}}} \cdot \frac{1}{\color{blue}{\left(\alpha + \beta\right) + \left(2 + 1\right)}} \]

      9. +-commutative 99.0%

         \[\leadsto \frac{\beta + 1}{\frac{\alpha + \left(\beta + 2\right)}{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}}} \cdot \frac{1}{\color{blue}{\left(\beta + \alpha\right)} + \left(2 + 1\right)} \]

      10. metadata-eval 99.0%

          \[\leadsto \frac{\beta + 1}{\frac{\alpha + \left(\beta + 2\right)}{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}}} \cdot \frac{1}{\left(\beta + \alpha\right) + \color{blue}{3}} \]

   7. Applied egg-rr 99.0%

      \[\leadsto \color{blue}{\frac{\beta + 1}{\frac{\alpha + \left(\beta + 2\right)}{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}}} \cdot \frac{1}{\left(\beta + \alpha\right) + 3}} \]
   8. Step-by-step derivation

      1. associate-*r/ 99.1%

         \[\leadsto \color{blue}{\frac{\frac{\beta + 1}{\frac{\alpha + \left(\beta + 2\right)}{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}}} \cdot 1}{\left(\beta + \alpha\right) + 3}} \]

      2. *-rgt-identity 99.1%

         \[\leadsto \frac{\color{blue}{\frac{\beta + 1}{\frac{\alpha + \left(\beta + 2\right)}{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}}}}}{\left(\beta + \alpha\right) + 3} \]

      3. associate-/r/ 99.8%

         \[\leadsto \frac{\color{blue}{\frac{\beta + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}}}{\left(\beta + \alpha\right) + 3} \]

      4. +-commutative 99.8%

         \[\leadsto \frac{\frac{\color{blue}{1 + \beta}}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}}{\left(\beta + \alpha\right) + 3} \]

      5. +-commutative 99.8%

         \[\leadsto \frac{\frac{1 + \beta}{\alpha + \color{blue}{\left(2 + \beta\right)}} \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}}{\left(\beta + \alpha\right) + 3} \]

      6. +-commutative 99.8%

         \[\leadsto \frac{\frac{1 + \beta}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 + \alpha}{\alpha + \color{blue}{\left(2 + \beta\right)}}}{\left(\beta + \alpha\right) + 3} \]

      7. associate-+l+ 99.8%

         \[\leadsto \frac{\frac{1 + \beta}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\color{blue}{\beta + \left(\alpha + 3\right)}} \]

   9. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{\frac{1 + \beta}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\beta + \left(\alpha + 3\right)}} \]
   10. Step-by-step derivation

       1. associate-*l/ 99.8%

          \[\leadsto \frac{\color{blue}{\frac{\left(1 + \beta\right) \cdot \frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\alpha + \left(2 + \beta\right)}}}{\beta + \left(\alpha + 3\right)} \]

       2. +-commutative 99.8%

          \[\leadsto \frac{\frac{\left(1 + \beta\right) \cdot \frac{1 + \alpha}{\alpha + \color{blue}{\left(\beta + 2\right)}}}{\alpha + \left(2 + \beta\right)}}{\beta + \left(\alpha + 3\right)} \]

       3. +-commutative 99.8%

          \[\leadsto \frac{\frac{\left(1 + \beta\right) \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}}{\alpha + \color{blue}{\left(\beta + 2\right)}}}{\beta + \left(\alpha + 3\right)} \]

   11. Applied egg-rr 99.8%

       \[\leadsto \frac{\color{blue}{\frac{\left(1 + \beta\right) \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)}}}{\beta + \left(\alpha + 3\right)} \]

   12. Taylor expanded in beta around inf 82.8%

       \[\leadsto \frac{\frac{\color{blue}{1 + \alpha}}{\alpha + \left(\beta + 2\right)}}{\beta + \left(\alpha + 3\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.8:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + 3}}{4 + \alpha \cdot \left(\alpha + 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]

Alternative 13: 94.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.5:\\ \;\;\;\;\frac{\beta + 1}{\alpha + \left(\beta + 3\right)} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.5)
   (* (/ (+ beta 1.0) (+ alpha (+ beta 3.0))) 0.25)
   (/ (+ 1.0 alpha) (* beta beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.5) {
		tmp = ((beta + 1.0) / (alpha + (beta + 3.0))) * 0.25;
	} 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 <= 2.5d0) then
        tmp = ((beta + 1.0d0) / (alpha + (beta + 3.0d0))) * 0.25d0
    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 <= 2.5) {
		tmp = ((beta + 1.0) / (alpha + (beta + 3.0))) * 0.25;
	} else {
		tmp = (1.0 + alpha) / (beta * beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.5:
		tmp = ((beta + 1.0) / (alpha + (beta + 3.0))) * 0.25
	else:
		tmp = (1.0 + alpha) / (beta * beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.5)
		tmp = Float64(Float64(Float64(beta + 1.0) / Float64(alpha + Float64(beta + 3.0))) * 0.25);
	else
		tmp = Float64(Float64(1.0 + alpha) / Float64(beta * beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.5)
		tmp = ((beta + 1.0) / (alpha + (beta + 3.0))) * 0.25;
	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, 2.5], N[(N[(N[(beta + 1.0), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision], N[(N[(1.0 + alpha), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.5

   1. Initial program 99.8%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Step-by-step derivation

      1. associate-/l/ 99.5%

         \[\leadsto \frac{\color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      2. associate-/l/ 96.5%

         \[\leadsto \color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}} \]

      3. associate-+l+ 96.5%

         \[\leadsto \frac{\color{blue}{\left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)} + 1}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      4. +-commutative 96.5%

         \[\leadsto \frac{\left(\alpha + \color{blue}{\left(\beta \cdot \alpha + \beta\right)}\right) + 1}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      5. associate-+r+ 96.5%

         \[\leadsto \frac{\color{blue}{\left(\left(\alpha + \beta \cdot \alpha\right) + \beta\right)} + 1}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      6. associate-+l+ 96.5%

         \[\leadsto \frac{\color{blue}{\left(\alpha + \beta \cdot \alpha\right) + \left(\beta + 1\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      7. distribute-rgt1-in 96.5%

         \[\leadsto \frac{\color{blue}{\left(\beta + 1\right) \cdot \alpha} + \left(\beta + 1\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      8. *-rgt-identity 96.5%

         \[\leadsto \frac{\left(\beta + 1\right) \cdot \alpha + \color{blue}{\left(\beta + 1\right) \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      9. distribute-lft-out 96.5%

         \[\leadsto \frac{\color{blue}{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      10. +-commutative 96.5%

          \[\leadsto \frac{\left(\beta + 1\right) \cdot \color{blue}{\left(1 + \alpha\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      11. times-frac 99.5%

          \[\leadsto \color{blue}{\frac{\beta + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \cdot \frac{1 + \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\frac{\beta + 1}{\alpha + \left(\beta + 3\right)} \cdot \frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}} \]
   4. Step-by-step derivation

      1. distribute-lft-in 99.5%

         \[\leadsto \frac{\beta + 1}{\alpha + \left(\beta + 3\right)} \cdot \frac{\alpha + 1}{\color{blue}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \alpha + \left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + 2\right)}} \]

      2. associate-+r+ 99.5%

         \[\leadsto \frac{\beta + 1}{\alpha + \left(\beta + 3\right)} \cdot \frac{\alpha + 1}{\color{blue}{\left(\left(\alpha + \beta\right) + 2\right)} \cdot \alpha + \left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + 2\right)} \]

      3. +-commutative 99.5%

         \[\leadsto \frac{\beta + 1}{\alpha + \left(\beta + 3\right)} \cdot \frac{\alpha + 1}{\color{blue}{\left(2 + \left(\alpha + \beta\right)\right)} \cdot \alpha + \left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + 2\right)} \]

      4. associate-+r+ 99.5%

         \[\leadsto \frac{\beta + 1}{\alpha + \left(\beta + 3\right)} \cdot \frac{\alpha + 1}{\left(2 + \left(\alpha + \beta\right)\right) \cdot \alpha + \color{blue}{\left(\left(\alpha + \beta\right) + 2\right)} \cdot \left(\beta + 2\right)} \]

      5. +-commutative 99.5%

         \[\leadsto \frac{\beta + 1}{\alpha + \left(\beta + 3\right)} \cdot \frac{\alpha + 1}{\left(2 + \left(\alpha + \beta\right)\right) \cdot \alpha + \color{blue}{\left(2 + \left(\alpha + \beta\right)\right)} \cdot \left(\beta + 2\right)} \]

   5. Applied egg-rr 99.5%

      \[\leadsto \frac{\beta + 1}{\alpha + \left(\beta + 3\right)} \cdot \frac{\alpha + 1}{\color{blue}{\left(2 + \left(\alpha + \beta\right)\right) \cdot \alpha + \left(2 + \left(\alpha + \beta\right)\right) \cdot \left(\beta + 2\right)}} \]

   6. Taylor expanded in beta around 0 99.3%

      \[\leadsto \frac{\beta + 1}{\alpha + \left(\beta + 3\right)} \cdot \color{blue}{\frac{1 + \alpha}{2 \cdot \left(2 + \alpha\right) + \left(2 + \alpha\right) \cdot \alpha}} \]

   7. Taylor expanded in alpha around 0 67.6%

      \[\leadsto \frac{\beta + 1}{\alpha + \left(\beta + 3\right)} \cdot \color{blue}{0.25} \]

   if 2.5 < beta

   1. Initial program 88.0%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Step-by-step derivation

      1. associate-/l/ 84.7%

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. associate-+l+ 84.7%

         \[\leadsto \frac{\frac{\color{blue}{\left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)} + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      3. +-commutative 84.7%

         \[\leadsto \frac{\frac{\left(\alpha + \color{blue}{\left(\beta \cdot \alpha + \beta\right)}\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      4. associate-+r+ 84.7%

         \[\leadsto \frac{\frac{\color{blue}{\left(\left(\alpha + \beta \cdot \alpha\right) + \beta\right)} + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      5. associate-+l+ 84.7%

         \[\leadsto \frac{\frac{\color{blue}{\left(\alpha + \beta \cdot \alpha\right) + \left(\beta + 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      6. distribute-rgt1-in 84.7%

         \[\leadsto \frac{\frac{\color{blue}{\left(\beta + 1\right) \cdot \alpha} + \left(\beta + 1\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      7. *-rgt-identity 84.7%

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \alpha + \color{blue}{\left(\beta + 1\right) \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      8. distribute-lft-out 84.7%

         \[\leadsto \frac{\frac{\color{blue}{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      9. +-commutative 84.7%

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \color{blue}{\left(1 + \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      10. associate-*l/ 90.6%

          \[\leadsto \frac{\color{blue}{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      11. *-commutative 90.6%

          \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right) \cdot \frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      12. associate-*r/ 88.8%

          \[\leadsto \color{blue}{\left(1 + \alpha\right) \cdot \frac{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   3. Simplified 88.8%

      \[\leadsto \color{blue}{\left(\alpha + 1\right) \cdot \frac{\frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]

   4. Taylor expanded in beta around inf 80.2%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 80.2%

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

   6. Simplified 80.2%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.5:\\ \;\;\;\;\frac{\beta + 1}{\alpha + \left(\beta + 3\right)} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \end{array} \]

Alternative 14: 94.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.5:\\ \;\;\;\;\frac{\beta + 1}{\alpha + \left(\beta + 3\right)} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta} + \frac{\alpha}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.5)
   (* (/ (+ beta 1.0) (+ alpha (+ beta 3.0))) 0.25)
   (+ (/ 1.0 (* beta beta)) (/ alpha (* beta beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.5) {
		tmp = ((beta + 1.0) / (alpha + (beta + 3.0))) * 0.25;
	} else {
		tmp = (1.0 / (beta * beta)) + (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 <= 2.5d0) then
        tmp = ((beta + 1.0d0) / (alpha + (beta + 3.0d0))) * 0.25d0
    else
        tmp = (1.0d0 / (beta * beta)) + (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 <= 2.5) {
		tmp = ((beta + 1.0) / (alpha + (beta + 3.0))) * 0.25;
	} else {
		tmp = (1.0 / (beta * beta)) + (alpha / (beta * beta));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.5:
		tmp = ((beta + 1.0) / (alpha + (beta + 3.0))) * 0.25
	else:
		tmp = (1.0 / (beta * beta)) + (alpha / (beta * beta))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.5)
		tmp = Float64(Float64(Float64(beta + 1.0) / Float64(alpha + Float64(beta + 3.0))) * 0.25);
	else
		tmp = Float64(Float64(1.0 / Float64(beta * beta)) + Float64(alpha / Float64(beta * beta)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.5)
		tmp = ((beta + 1.0) / (alpha + (beta + 3.0))) * 0.25;
	else
		tmp = (1.0 / (beta * beta)) + (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, 2.5], N[(N[(N[(beta + 1.0), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision], N[(N[(1.0 / N[(beta * beta), $MachinePrecision]), $MachinePrecision] + N[(alpha / N[(beta * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.5

   1. Initial program 99.8%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Step-by-step derivation

      1. associate-/l/ 99.5%

         \[\leadsto \frac{\color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      2. associate-/l/ 96.5%

         \[\leadsto \color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}} \]

      3. associate-+l+ 96.5%

         \[\leadsto \frac{\color{blue}{\left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)} + 1}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      4. +-commutative 96.5%

         \[\leadsto \frac{\left(\alpha + \color{blue}{\left(\beta \cdot \alpha + \beta\right)}\right) + 1}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      5. associate-+r+ 96.5%

         \[\leadsto \frac{\color{blue}{\left(\left(\alpha + \beta \cdot \alpha\right) + \beta\right)} + 1}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      6. associate-+l+ 96.5%

         \[\leadsto \frac{\color{blue}{\left(\alpha + \beta \cdot \alpha\right) + \left(\beta + 1\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      7. distribute-rgt1-in 96.5%

         \[\leadsto \frac{\color{blue}{\left(\beta + 1\right) \cdot \alpha} + \left(\beta + 1\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      8. *-rgt-identity 96.5%

         \[\leadsto \frac{\left(\beta + 1\right) \cdot \alpha + \color{blue}{\left(\beta + 1\right) \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      9. distribute-lft-out 96.5%

         \[\leadsto \frac{\color{blue}{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      10. +-commutative 96.5%

          \[\leadsto \frac{\left(\beta + 1\right) \cdot \color{blue}{\left(1 + \alpha\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      11. times-frac 99.5%

          \[\leadsto \color{blue}{\frac{\beta + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \cdot \frac{1 + \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\frac{\beta + 1}{\alpha + \left(\beta + 3\right)} \cdot \frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}} \]
   4. Step-by-step derivation

      1. distribute-lft-in 99.5%

         \[\leadsto \frac{\beta + 1}{\alpha + \left(\beta + 3\right)} \cdot \frac{\alpha + 1}{\color{blue}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \alpha + \left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + 2\right)}} \]

      2. associate-+r+ 99.5%

         \[\leadsto \frac{\beta + 1}{\alpha + \left(\beta + 3\right)} \cdot \frac{\alpha + 1}{\color{blue}{\left(\left(\alpha + \beta\right) + 2\right)} \cdot \alpha + \left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + 2\right)} \]

      3. +-commutative 99.5%

         \[\leadsto \frac{\beta + 1}{\alpha + \left(\beta + 3\right)} \cdot \frac{\alpha + 1}{\color{blue}{\left(2 + \left(\alpha + \beta\right)\right)} \cdot \alpha + \left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + 2\right)} \]

      4. associate-+r+ 99.5%

         \[\leadsto \frac{\beta + 1}{\alpha + \left(\beta + 3\right)} \cdot \frac{\alpha + 1}{\left(2 + \left(\alpha + \beta\right)\right) \cdot \alpha + \color{blue}{\left(\left(\alpha + \beta\right) + 2\right)} \cdot \left(\beta + 2\right)} \]

      5. +-commutative 99.5%

         \[\leadsto \frac{\beta + 1}{\alpha + \left(\beta + 3\right)} \cdot \frac{\alpha + 1}{\left(2 + \left(\alpha + \beta\right)\right) \cdot \alpha + \color{blue}{\left(2 + \left(\alpha + \beta\right)\right)} \cdot \left(\beta + 2\right)} \]

   5. Applied egg-rr 99.5%

      \[\leadsto \frac{\beta + 1}{\alpha + \left(\beta + 3\right)} \cdot \frac{\alpha + 1}{\color{blue}{\left(2 + \left(\alpha + \beta\right)\right) \cdot \alpha + \left(2 + \left(\alpha + \beta\right)\right) \cdot \left(\beta + 2\right)}} \]

   6. Taylor expanded in beta around 0 99.3%

      \[\leadsto \frac{\beta + 1}{\alpha + \left(\beta + 3\right)} \cdot \color{blue}{\frac{1 + \alpha}{2 \cdot \left(2 + \alpha\right) + \left(2 + \alpha\right) \cdot \alpha}} \]

   7. Taylor expanded in alpha around 0 67.6%

      \[\leadsto \frac{\beta + 1}{\alpha + \left(\beta + 3\right)} \cdot \color{blue}{0.25} \]

   if 2.5 < beta

   1. Initial program 88.0%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Step-by-step derivation

      1. associate-/l/ 84.7%

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. associate-+l+ 84.7%

         \[\leadsto \frac{\frac{\color{blue}{\left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)} + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      3. +-commutative 84.7%

         \[\leadsto \frac{\frac{\left(\alpha + \color{blue}{\left(\beta \cdot \alpha + \beta\right)}\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      4. associate-+r+ 84.7%

         \[\leadsto \frac{\frac{\color{blue}{\left(\left(\alpha + \beta \cdot \alpha\right) + \beta\right)} + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      5. associate-+l+ 84.7%

         \[\leadsto \frac{\frac{\color{blue}{\left(\alpha + \beta \cdot \alpha\right) + \left(\beta + 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      6. distribute-rgt1-in 84.7%

         \[\leadsto \frac{\frac{\color{blue}{\left(\beta + 1\right) \cdot \alpha} + \left(\beta + 1\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      7. *-rgt-identity 84.7%

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \alpha + \color{blue}{\left(\beta + 1\right) \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      8. distribute-lft-out 84.7%

         \[\leadsto \frac{\frac{\color{blue}{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      9. +-commutative 84.7%

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \color{blue}{\left(1 + \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      10. associate-*l/ 90.6%

          \[\leadsto \frac{\color{blue}{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      11. *-commutative 90.6%

          \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right) \cdot \frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      12. associate-*r/ 88.8%

          \[\leadsto \color{blue}{\left(1 + \alpha\right) \cdot \frac{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   3. Simplified 88.8%

      \[\leadsto \color{blue}{\left(\alpha + 1\right) \cdot \frac{\frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]

   4. Taylor expanded in beta around inf 80.2%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 80.2%

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

   6. Simplified 80.2%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]

   7. Taylor expanded in alpha around 0 80.2%

      \[\leadsto \color{blue}{\frac{1}{{\beta}^{2}} + \frac{\alpha}{{\beta}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 80.2%

         \[\leadsto \frac{1}{\color{blue}{\beta \cdot \beta}} + \frac{\alpha}{{\beta}^{2}} \]

      2. unpow2 80.2%

         \[\leadsto \frac{1}{\beta \cdot \beta} + \frac{\alpha}{\color{blue}{\beta \cdot \beta}} \]

   9. Simplified 80.2%

      \[\leadsto \color{blue}{\frac{1}{\beta \cdot \beta} + \frac{\alpha}{\beta \cdot \beta}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.5:\\ \;\;\;\;\frac{\beta + 1}{\alpha + \left(\beta + 3\right)} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta} + \frac{\alpha}{\beta \cdot \beta}\\ \end{array} \]

Alternative 15: 94.0% accurate, 3.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.5:\\ \;\;\;\;0.25 \cdot \frac{\beta + 1}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.5)
   (* 0.25 (/ (+ beta 1.0) (+ beta 3.0)))
   (/ (+ 1.0 alpha) (* beta beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.5) {
		tmp = 0.25 * ((beta + 1.0) / (beta + 3.0));
	} 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 <= 2.5d0) then
        tmp = 0.25d0 * ((beta + 1.0d0) / (beta + 3.0d0))
    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 <= 2.5) {
		tmp = 0.25 * ((beta + 1.0) / (beta + 3.0));
	} else {
		tmp = (1.0 + alpha) / (beta * beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.5:
		tmp = 0.25 * ((beta + 1.0) / (beta + 3.0))
	else:
		tmp = (1.0 + alpha) / (beta * beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.5)
		tmp = Float64(0.25 * Float64(Float64(beta + 1.0) / Float64(beta + 3.0)));
	else
		tmp = Float64(Float64(1.0 + alpha) / Float64(beta * beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.5)
		tmp = 0.25 * ((beta + 1.0) / (beta + 3.0));
	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, 2.5], N[(0.25 * N[(N[(beta + 1.0), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + alpha), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.5

   1. Initial program 99.8%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Step-by-step derivation

      1. associate-/l/ 99.5%

         \[\leadsto \frac{\color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      2. associate-/l/ 96.5%

         \[\leadsto \color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}} \]

      3. associate-+l+ 96.5%

         \[\leadsto \frac{\color{blue}{\left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)} + 1}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      4. +-commutative 96.5%

         \[\leadsto \frac{\left(\alpha + \color{blue}{\left(\beta \cdot \alpha + \beta\right)}\right) + 1}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      5. associate-+r+ 96.5%

         \[\leadsto \frac{\color{blue}{\left(\left(\alpha + \beta \cdot \alpha\right) + \beta\right)} + 1}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      6. associate-+l+ 96.5%

         \[\leadsto \frac{\color{blue}{\left(\alpha + \beta \cdot \alpha\right) + \left(\beta + 1\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      7. distribute-rgt1-in 96.5%

         \[\leadsto \frac{\color{blue}{\left(\beta + 1\right) \cdot \alpha} + \left(\beta + 1\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      8. *-rgt-identity 96.5%

         \[\leadsto \frac{\left(\beta + 1\right) \cdot \alpha + \color{blue}{\left(\beta + 1\right) \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      9. distribute-lft-out 96.5%

         \[\leadsto \frac{\color{blue}{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      10. +-commutative 96.5%

          \[\leadsto \frac{\left(\beta + 1\right) \cdot \color{blue}{\left(1 + \alpha\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      11. times-frac 99.5%

          \[\leadsto \color{blue}{\frac{\beta + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \cdot \frac{1 + \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\frac{\beta + 1}{\alpha + \left(\beta + 3\right)} \cdot \frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}} \]
   4. Step-by-step derivation

      1. distribute-lft-in 99.5%

         \[\leadsto \frac{\beta + 1}{\alpha + \left(\beta + 3\right)} \cdot \frac{\alpha + 1}{\color{blue}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \alpha + \left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + 2\right)}} \]

      2. associate-+r+ 99.5%

         \[\leadsto \frac{\beta + 1}{\alpha + \left(\beta + 3\right)} \cdot \frac{\alpha + 1}{\color{blue}{\left(\left(\alpha + \beta\right) + 2\right)} \cdot \alpha + \left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + 2\right)} \]

      3. +-commutative 99.5%

         \[\leadsto \frac{\beta + 1}{\alpha + \left(\beta + 3\right)} \cdot \frac{\alpha + 1}{\color{blue}{\left(2 + \left(\alpha + \beta\right)\right)} \cdot \alpha + \left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + 2\right)} \]

      4. associate-+r+ 99.5%

         \[\leadsto \frac{\beta + 1}{\alpha + \left(\beta + 3\right)} \cdot \frac{\alpha + 1}{\left(2 + \left(\alpha + \beta\right)\right) \cdot \alpha + \color{blue}{\left(\left(\alpha + \beta\right) + 2\right)} \cdot \left(\beta + 2\right)} \]

      5. +-commutative 99.5%

         \[\leadsto \frac{\beta + 1}{\alpha + \left(\beta + 3\right)} \cdot \frac{\alpha + 1}{\left(2 + \left(\alpha + \beta\right)\right) \cdot \alpha + \color{blue}{\left(2 + \left(\alpha + \beta\right)\right)} \cdot \left(\beta + 2\right)} \]

   5. Applied egg-rr 99.5%

      \[\leadsto \frac{\beta + 1}{\alpha + \left(\beta + 3\right)} \cdot \frac{\alpha + 1}{\color{blue}{\left(2 + \left(\alpha + \beta\right)\right) \cdot \alpha + \left(2 + \left(\alpha + \beta\right)\right) \cdot \left(\beta + 2\right)}} \]

   6. Taylor expanded in beta around 0 99.3%

      \[\leadsto \frac{\beta + 1}{\alpha + \left(\beta + 3\right)} \cdot \color{blue}{\frac{1 + \alpha}{2 \cdot \left(2 + \alpha\right) + \left(2 + \alpha\right) \cdot \alpha}} \]

   7. Taylor expanded in alpha around 0 99.4%

      \[\leadsto \frac{\beta + 1}{\alpha + \left(\beta + 3\right)} \cdot \frac{1 + \alpha}{\color{blue}{4 + \left({\alpha}^{2} + 4 \cdot \alpha\right)}} \]
   8. Step-by-step derivation

      1. unpow2 99.4%

         \[\leadsto \frac{\beta + 1}{\alpha + \left(\beta + 3\right)} \cdot \frac{1 + \alpha}{4 + \left(\color{blue}{\alpha \cdot \alpha} + 4 \cdot \alpha\right)} \]

      2. distribute-rgt-out 99.4%

         \[\leadsto \frac{\beta + 1}{\alpha + \left(\beta + 3\right)} \cdot \frac{1 + \alpha}{4 + \color{blue}{\alpha \cdot \left(\alpha + 4\right)}} \]

   9. Simplified 99.4%

      \[\leadsto \frac{\beta + 1}{\alpha + \left(\beta + 3\right)} \cdot \frac{1 + \alpha}{\color{blue}{4 + \alpha \cdot \left(\alpha + 4\right)}} \]

   10. Taylor expanded in alpha around 0 66.0%

       \[\leadsto \color{blue}{0.25 \cdot \frac{\beta + 1}{\beta + 3}} \]

   if 2.5 < beta

   1. Initial program 88.0%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Step-by-step derivation

      1. associate-/l/ 84.7%

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. associate-+l+ 84.7%

         \[\leadsto \frac{\frac{\color{blue}{\left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)} + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      3. +-commutative 84.7%

         \[\leadsto \frac{\frac{\left(\alpha + \color{blue}{\left(\beta \cdot \alpha + \beta\right)}\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      4. associate-+r+ 84.7%

         \[\leadsto \frac{\frac{\color{blue}{\left(\left(\alpha + \beta \cdot \alpha\right) + \beta\right)} + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      5. associate-+l+ 84.7%

         \[\leadsto \frac{\frac{\color{blue}{\left(\alpha + \beta \cdot \alpha\right) + \left(\beta + 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      6. distribute-rgt1-in 84.7%

         \[\leadsto \frac{\frac{\color{blue}{\left(\beta + 1\right) \cdot \alpha} + \left(\beta + 1\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      7. *-rgt-identity 84.7%

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \alpha + \color{blue}{\left(\beta + 1\right) \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      8. distribute-lft-out 84.7%

         \[\leadsto \frac{\frac{\color{blue}{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      9. +-commutative 84.7%

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \color{blue}{\left(1 + \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      10. associate-*l/ 90.6%

          \[\leadsto \frac{\color{blue}{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      11. *-commutative 90.6%

          \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right) \cdot \frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      12. associate-*r/ 88.8%

          \[\leadsto \color{blue}{\left(1 + \alpha\right) \cdot \frac{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   3. Simplified 88.8%

      \[\leadsto \color{blue}{\left(\alpha + 1\right) \cdot \frac{\frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]

   4. Taylor expanded in beta around inf 80.2%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 80.2%

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

   6. Simplified 80.2%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.5:\\ \;\;\;\;0.25 \cdot \frac{\beta + 1}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \end{array} \]

Alternative 16: 94.0% accurate, 3.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.7:\\ \;\;\;\;0.16666666666666666 \cdot \frac{1 + \alpha}{\alpha + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.7)
   (* 0.16666666666666666 (/ (+ 1.0 alpha) (+ alpha 2.0)))
   (/ (+ 1.0 alpha) (* beta beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.7) {
		tmp = 0.16666666666666666 * ((1.0 + alpha) / (alpha + 2.0));
	} else {
		tmp = (1.0 + alpha) / (beta * beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 3.7d0) then
        tmp = 0.16666666666666666d0 * ((1.0d0 + alpha) / (alpha + 2.0d0))
    else
        tmp = (1.0d0 + alpha) / (beta * beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.7) {
		tmp = 0.16666666666666666 * ((1.0 + alpha) / (alpha + 2.0));
	} else {
		tmp = (1.0 + alpha) / (beta * beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 3.7:
		tmp = 0.16666666666666666 * ((1.0 + alpha) / (alpha + 2.0))
	else:
		tmp = (1.0 + alpha) / (beta * beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.7)
		tmp = Float64(0.16666666666666666 * Float64(Float64(1.0 + alpha) / Float64(alpha + 2.0)));
	else
		tmp = Float64(Float64(1.0 + alpha) / Float64(beta * beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.7)
		tmp = 0.16666666666666666 * ((1.0 + alpha) / (alpha + 2.0));
	else
		tmp = (1.0 + alpha) / (beta * beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 3.7], N[(0.16666666666666666 * N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + alpha), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.7000000000000002

   1. Initial program 99.8%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Step-by-step derivation

      1. associate-/l/ 99.6%

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. associate-/r* 96.5%

         \[\leadsto \color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}} \]

      3. associate-+l+ 96.5%

         \[\leadsto \frac{\color{blue}{\left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)} + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      4. +-commutative 96.5%

         \[\leadsto \frac{\left(\alpha + \color{blue}{\left(\beta \cdot \alpha + \beta\right)}\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      5. associate-+r+ 96.5%

         \[\leadsto \frac{\color{blue}{\left(\left(\alpha + \beta \cdot \alpha\right) + \beta\right)} + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      6. associate-+l+ 96.5%

         \[\leadsto \frac{\color{blue}{\left(\alpha + \beta \cdot \alpha\right) + \left(\beta + 1\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      7. distribute-rgt1-in 96.5%

         \[\leadsto \frac{\color{blue}{\left(\beta + 1\right) \cdot \alpha} + \left(\beta + 1\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      8. *-rgt-identity 96.5%

         \[\leadsto \frac{\left(\beta + 1\right) \cdot \alpha + \color{blue}{\left(\beta + 1\right) \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      9. distribute-lft-out 96.5%

         \[\leadsto \frac{\color{blue}{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      10. *-commutative 96.5%

          \[\leadsto \frac{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)\right)}} \]

      11. metadata-eval 96.5%

          \[\leadsto \frac{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}{\left(\left(\alpha + \beta\right) + \color{blue}{2}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)\right)} \]

      12. associate-+l+ 96.5%

          \[\leadsto \frac{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}{\color{blue}{\left(\alpha + \left(\beta + 2\right)\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)\right)} \]

      13. +-commutative 96.5%

          \[\leadsto \frac{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \color{blue}{\left(1 + \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}\right)} \]

   3. Simplified 96.5%

      \[\leadsto \color{blue}{\frac{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)\right)}} \]

   4. Taylor expanded in alpha around 0 66.1%

      \[\leadsto \frac{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \color{blue}{\left(\left(\beta + 3\right) \cdot \left(\beta + 2\right)\right)}} \]

   5. Taylor expanded in beta around 0 66.0%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot \frac{1 + \alpha}{2 + \alpha}} \]
   6. Step-by-step derivation

      1. *-commutative 66.0%

         \[\leadsto \color{blue}{\frac{1 + \alpha}{2 + \alpha} \cdot 0.16666666666666666} \]

      2. +-commutative 66.0%

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\alpha + 2}} \cdot 0.16666666666666666 \]

   7. Simplified 66.0%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\alpha + 2} \cdot 0.16666666666666666} \]

   if 3.7000000000000002 < beta

   1. Initial program 88.0%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Step-by-step derivation

      1. associate-/l/ 84.7%

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. associate-+l+ 84.7%

         \[\leadsto \frac{\frac{\color{blue}{\left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)} + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      3. +-commutative 84.7%

         \[\leadsto \frac{\frac{\left(\alpha + \color{blue}{\left(\beta \cdot \alpha + \beta\right)}\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      4. associate-+r+ 84.7%

         \[\leadsto \frac{\frac{\color{blue}{\left(\left(\alpha + \beta \cdot \alpha\right) + \beta\right)} + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      5. associate-+l+ 84.7%

         \[\leadsto \frac{\frac{\color{blue}{\left(\alpha + \beta \cdot \alpha\right) + \left(\beta + 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      6. distribute-rgt1-in 84.7%

         \[\leadsto \frac{\frac{\color{blue}{\left(\beta + 1\right) \cdot \alpha} + \left(\beta + 1\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      7. *-rgt-identity 84.7%

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \alpha + \color{blue}{\left(\beta + 1\right) \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      8. distribute-lft-out 84.7%

         \[\leadsto \frac{\frac{\color{blue}{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      9. +-commutative 84.7%

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \color{blue}{\left(1 + \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      10. associate-*l/ 90.6%

          \[\leadsto \frac{\color{blue}{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      11. *-commutative 90.6%

          \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right) \cdot \frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      12. associate-*r/ 88.8%

          \[\leadsto \color{blue}{\left(1 + \alpha\right) \cdot \frac{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   3. Simplified 88.8%

      \[\leadsto \color{blue}{\left(\alpha + 1\right) \cdot \frac{\frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]

   4. Taylor expanded in beta around inf 80.2%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 80.2%

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

   6. Simplified 80.2%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.7:\\ \;\;\;\;0.16666666666666666 \cdot \frac{1 + \alpha}{\alpha + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \end{array} \]

Alternative 17: 94.0% accurate, 3.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.6:\\ \;\;\;\;\frac{\left(1 + \alpha\right) \cdot 0.16666666666666666}{\alpha + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.6)
   (/ (* (+ 1.0 alpha) 0.16666666666666666) (+ alpha 2.0))
   (/ (+ 1.0 alpha) (* beta beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.6) {
		tmp = ((1.0 + alpha) * 0.16666666666666666) / (alpha + 2.0);
	} else {
		tmp = (1.0 + alpha) / (beta * beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 3.6d0) then
        tmp = ((1.0d0 + alpha) * 0.16666666666666666d0) / (alpha + 2.0d0)
    else
        tmp = (1.0d0 + alpha) / (beta * beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.6) {
		tmp = ((1.0 + alpha) * 0.16666666666666666) / (alpha + 2.0);
	} else {
		tmp = (1.0 + alpha) / (beta * beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 3.6:
		tmp = ((1.0 + alpha) * 0.16666666666666666) / (alpha + 2.0)
	else:
		tmp = (1.0 + alpha) / (beta * beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.6)
		tmp = Float64(Float64(Float64(1.0 + alpha) * 0.16666666666666666) / Float64(alpha + 2.0));
	else
		tmp = Float64(Float64(1.0 + alpha) / Float64(beta * beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.6)
		tmp = ((1.0 + alpha) * 0.16666666666666666) / (alpha + 2.0);
	else
		tmp = (1.0 + alpha) / (beta * beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 3.6], N[(N[(N[(1.0 + alpha), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + alpha), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.60000000000000009

   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/ 99.6%

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. associate-/r* 96.5%

         \[\leadsto \color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}} \]

      3. associate-+l+ 96.5%

         \[\leadsto \frac{\color{blue}{\left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)} + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      4. +-commutative 96.5%

         \[\leadsto \frac{\left(\alpha + \color{blue}{\left(\beta \cdot \alpha + \beta\right)}\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      5. associate-+r+ 96.5%

         \[\leadsto \frac{\color{blue}{\left(\left(\alpha + \beta \cdot \alpha\right) + \beta\right)} + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      6. associate-+l+ 96.5%

         \[\leadsto \frac{\color{blue}{\left(\alpha + \beta \cdot \alpha\right) + \left(\beta + 1\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      7. distribute-rgt1-in 96.5%

         \[\leadsto \frac{\color{blue}{\left(\beta + 1\right) \cdot \alpha} + \left(\beta + 1\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      8. *-rgt-identity 96.5%

         \[\leadsto \frac{\left(\beta + 1\right) \cdot \alpha + \color{blue}{\left(\beta + 1\right) \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      9. distribute-lft-out 96.5%

         \[\leadsto \frac{\color{blue}{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      10. *-commutative 96.5%

          \[\leadsto \frac{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)\right)}} \]

      11. metadata-eval 96.5%

          \[\leadsto \frac{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}{\left(\left(\alpha + \beta\right) + \color{blue}{2}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)\right)} \]

      12. associate-+l+ 96.5%

          \[\leadsto \frac{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}{\color{blue}{\left(\alpha + \left(\beta + 2\right)\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)\right)} \]

      13. +-commutative 96.5%

          \[\leadsto \frac{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \color{blue}{\left(1 + \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}\right)} \]

   3. Simplified 96.5%

      \[\leadsto \color{blue}{\frac{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)\right)}} \]

   4. Taylor expanded in alpha around 0 66.1%

      \[\leadsto \frac{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \color{blue}{\left(\left(\beta + 3\right) \cdot \left(\beta + 2\right)\right)}} \]

   5. Taylor expanded in beta around 0 66.1%

      \[\leadsto \frac{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \color{blue}{\left(6 + 5 \cdot \beta\right)}} \]
   6. Step-by-step derivation

      1. *-commutative 66.1%

         \[\leadsto \frac{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(6 + \color{blue}{\beta \cdot 5}\right)} \]

   7. Simplified 66.1%

      \[\leadsto \frac{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \color{blue}{\left(6 + \beta \cdot 5\right)}} \]

   8. Taylor expanded in beta around 0 66.0%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot \frac{1 + \alpha}{2 + \alpha}} \]
   9. Step-by-step derivation

      1. associate-*r/ 66.0%

         \[\leadsto \color{blue}{\frac{0.16666666666666666 \cdot \left(1 + \alpha\right)}{2 + \alpha}} \]

      2. +-commutative 66.0%

         \[\leadsto \frac{0.16666666666666666 \cdot \left(1 + \alpha\right)}{\color{blue}{\alpha + 2}} \]

   10. Simplified 66.0%

       \[\leadsto \color{blue}{\frac{0.16666666666666666 \cdot \left(1 + \alpha\right)}{\alpha + 2}} \]

   if 3.60000000000000009 < beta

   1. Initial program 88.0%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Step-by-step derivation

      1. associate-/l/ 84.7%

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. associate-+l+ 84.7%

         \[\leadsto \frac{\frac{\color{blue}{\left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)} + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      3. +-commutative 84.7%

         \[\leadsto \frac{\frac{\left(\alpha + \color{blue}{\left(\beta \cdot \alpha + \beta\right)}\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      4. associate-+r+ 84.7%

         \[\leadsto \frac{\frac{\color{blue}{\left(\left(\alpha + \beta \cdot \alpha\right) + \beta\right)} + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      5. associate-+l+ 84.7%

         \[\leadsto \frac{\frac{\color{blue}{\left(\alpha + \beta \cdot \alpha\right) + \left(\beta + 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      6. distribute-rgt1-in 84.7%

         \[\leadsto \frac{\frac{\color{blue}{\left(\beta + 1\right) \cdot \alpha} + \left(\beta + 1\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      7. *-rgt-identity 84.7%

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \alpha + \color{blue}{\left(\beta + 1\right) \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      8. distribute-lft-out 84.7%

         \[\leadsto \frac{\frac{\color{blue}{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      9. +-commutative 84.7%

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \color{blue}{\left(1 + \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      10. associate-*l/ 90.6%

          \[\leadsto \frac{\color{blue}{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      11. *-commutative 90.6%

          \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right) \cdot \frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      12. associate-*r/ 88.8%

          \[\leadsto \color{blue}{\left(1 + \alpha\right) \cdot \frac{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   3. Simplified 88.8%

      \[\leadsto \color{blue}{\left(\alpha + 1\right) \cdot \frac{\frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]

   4. Taylor expanded in beta around inf 80.2%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 80.2%

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

   6. Simplified 80.2%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.6:\\ \;\;\;\;\frac{\left(1 + \alpha\right) \cdot 0.16666666666666666}{\alpha + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \end{array} \]

Alternative 18: 52.5% accurate, 5.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 4.1 \cdot 10^{+15}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 4.1e+15) (/ 1.0 (* beta beta)) (/ alpha (* beta beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 4.1e+15) {
		tmp = 1.0 / (beta * beta);
	} else {
		tmp = 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 (alpha <= 4.1d+15) then
        tmp = 1.0d0 / (beta * beta)
    else
        tmp = alpha / (beta * beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 4.1e+15) {
		tmp = 1.0 / (beta * beta);
	} else {
		tmp = alpha / (beta * beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if alpha <= 4.1e+15:
		tmp = 1.0 / (beta * beta)
	else:
		tmp = alpha / (beta * beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 4.1e+15)
		tmp = Float64(1.0 / Float64(beta * beta));
	else
		tmp = Float64(alpha / Float64(beta * beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 4.1e+15)
		tmp = 1.0 / (beta * beta);
	else
		tmp = 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[alpha, 4.1e+15], N[(1.0 / N[(beta * beta), $MachinePrecision]), $MachinePrecision], N[(alpha / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 4.1e15

   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/ 99.6%

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. associate-+l+ 99.6%

         \[\leadsto \frac{\frac{\color{blue}{\left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)} + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      3. +-commutative 99.6%

         \[\leadsto \frac{\frac{\left(\alpha + \color{blue}{\left(\beta \cdot \alpha + \beta\right)}\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      4. associate-+r+ 99.6%

         \[\leadsto \frac{\frac{\color{blue}{\left(\left(\alpha + \beta \cdot \alpha\right) + \beta\right)} + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      5. associate-+l+ 99.6%

         \[\leadsto \frac{\frac{\color{blue}{\left(\alpha + \beta \cdot \alpha\right) + \left(\beta + 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      6. distribute-rgt1-in 99.6%

         \[\leadsto \frac{\frac{\color{blue}{\left(\beta + 1\right) \cdot \alpha} + \left(\beta + 1\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      7. *-rgt-identity 99.6%

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \alpha + \color{blue}{\left(\beta + 1\right) \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      8. distribute-lft-out 99.6%

         \[\leadsto \frac{\frac{\color{blue}{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      9. +-commutative 99.6%

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \color{blue}{\left(1 + \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      10. associate-*l/ 99.6%

          \[\leadsto \frac{\color{blue}{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      11. *-commutative 99.6%

          \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right) \cdot \frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      12. associate-*r/ 99.5%

          \[\leadsto \color{blue}{\left(1 + \alpha\right) \cdot \frac{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\left(\alpha + 1\right) \cdot \frac{\frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]

   4. Taylor expanded in beta around inf 32.3%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 32.3%

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

   6. Simplified 32.3%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]

   7. Taylor expanded in alpha around 0 32.3%

      \[\leadsto \color{blue}{\frac{1}{{\beta}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 32.3%

         \[\leadsto \frac{1}{\color{blue}{\beta \cdot \beta}} \]

   9. Simplified 32.3%

      \[\leadsto \color{blue}{\frac{1}{\beta \cdot \beta}} \]

   if 4.1e15 < alpha

   1. Initial program 88.7%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Step-by-step derivation

      1. associate-/l/ 85.6%

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. associate-+l+ 85.6%

         \[\leadsto \frac{\frac{\color{blue}{\left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)} + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      3. +-commutative 85.6%

         \[\leadsto \frac{\frac{\left(\alpha + \color{blue}{\left(\beta \cdot \alpha + \beta\right)}\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      4. associate-+r+ 85.6%

         \[\leadsto \frac{\frac{\color{blue}{\left(\left(\alpha + \beta \cdot \alpha\right) + \beta\right)} + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      5. associate-+l+ 85.6%

         \[\leadsto \frac{\frac{\color{blue}{\left(\alpha + \beta \cdot \alpha\right) + \left(\beta + 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      6. distribute-rgt1-in 85.6%

         \[\leadsto \frac{\frac{\color{blue}{\left(\beta + 1\right) \cdot \alpha} + \left(\beta + 1\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      7. *-rgt-identity 85.6%

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \alpha + \color{blue}{\left(\beta + 1\right) \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      8. distribute-lft-out 85.6%

         \[\leadsto \frac{\frac{\color{blue}{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      9. +-commutative 85.6%

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \color{blue}{\left(1 + \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      10. associate-*l/ 91.1%

          \[\leadsto \frac{\color{blue}{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      11. *-commutative 91.1%

          \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right) \cdot \frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      12. associate-*r/ 83.4%

          \[\leadsto \color{blue}{\left(1 + \alpha\right) \cdot \frac{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   3. Simplified 83.4%

      \[\leadsto \color{blue}{\left(\alpha + 1\right) \cdot \frac{\frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]

   4. Taylor expanded in beta around inf 11.8%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 11.8%

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

   6. Simplified 11.8%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]

   7. Taylor expanded in alpha around inf 11.8%

      \[\leadsto \color{blue}{\frac{\alpha}{{\beta}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 11.8%

         \[\leadsto \frac{\alpha}{\color{blue}{\beta \cdot \beta}} \]

   9. Simplified 11.8%

      \[\leadsto \color{blue}{\frac{\alpha}{\beta \cdot \beta}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 25.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 4.1 \cdot 10^{+15}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha}{\beta \cdot \beta}\\ \end{array} \]

Alternative 19: 53.5% accurate, 5.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \frac{1 + \alpha}{\beta \cdot \beta} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta) :precision binary64 (/ (+ 1.0 alpha) (* beta beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return (1.0 + alpha) / (beta * beta);
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = (1.0d0 + alpha) / (beta * beta)
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return (1.0 + alpha) / (beta * beta);
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return (1.0 + alpha) / (beta * beta)

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return Float64(Float64(1.0 + alpha) / Float64(beta * beta))
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = (1.0 + alpha) / (beta * beta);
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := N[(N[(1.0 + alpha), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.4%

   \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
2. Step-by-step derivation

   1. associate-/l/ 95.2%

      \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   2. associate-+l+ 95.2%

      \[\leadsto \frac{\frac{\color{blue}{\left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)} + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   3. +-commutative 95.2%

      \[\leadsto \frac{\frac{\left(\alpha + \color{blue}{\left(\beta \cdot \alpha + \beta\right)}\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   4. associate-+r+ 95.2%

      \[\leadsto \frac{\frac{\color{blue}{\left(\left(\alpha + \beta \cdot \alpha\right) + \beta\right)} + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   5. associate-+l+ 95.2%

      \[\leadsto \frac{\frac{\color{blue}{\left(\alpha + \beta \cdot \alpha\right) + \left(\beta + 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   6. distribute-rgt1-in 95.2%

      \[\leadsto \frac{\frac{\color{blue}{\left(\beta + 1\right) \cdot \alpha} + \left(\beta + 1\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   7. *-rgt-identity 95.2%

      \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \alpha + \color{blue}{\left(\beta + 1\right) \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   8. distribute-lft-out 95.2%

      \[\leadsto \frac{\frac{\color{blue}{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   9. +-commutative 95.2%

      \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \color{blue}{\left(1 + \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   10. associate-*l/ 96.9%

       \[\leadsto \frac{\color{blue}{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   11. *-commutative 96.9%

       \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right) \cdot \frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   12. associate-*r/ 94.5%

       \[\leadsto \color{blue}{\left(1 + \alpha\right) \cdot \frac{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

3. Simplified 94.5%

   \[\leadsto \color{blue}{\left(\alpha + 1\right) \cdot \frac{\frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]

4. Taylor expanded in beta around inf 25.9%

   \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
5. Step-by-step derivation

   1. unpow2 25.9%

      \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

6. Simplified 25.9%

   \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]

7. Final simplification 25.9%

   \[\leadsto \frac{1 + \alpha}{\beta \cdot \beta} \]

Alternative 20: 33.7% accurate, 7.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \frac{0.5}{\beta \cdot \beta} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta) :precision binary64 (/ 0.5 (* beta beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return 0.5 / (beta * beta);
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = 0.5d0 / (beta * beta)
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return 0.5 / (beta * beta);
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return 0.5 / (beta * beta)

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return Float64(0.5 / Float64(beta * beta))
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = 0.5 / (beta * beta);
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := N[(0.5 / N[(beta * beta), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.4%

   \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
2. Step-by-step derivation

   1. associate-/l/ 95.2%

      \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   2. associate-+l+ 95.2%

      \[\leadsto \frac{\frac{\color{blue}{\left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)} + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   3. +-commutative 95.2%

      \[\leadsto \frac{\frac{\left(\alpha + \color{blue}{\left(\beta \cdot \alpha + \beta\right)}\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   4. associate-+r+ 95.2%

      \[\leadsto \frac{\frac{\color{blue}{\left(\left(\alpha + \beta \cdot \alpha\right) + \beta\right)} + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   5. associate-+l+ 95.2%

      \[\leadsto \frac{\frac{\color{blue}{\left(\alpha + \beta \cdot \alpha\right) + \left(\beta + 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   6. distribute-rgt1-in 95.2%

      \[\leadsto \frac{\frac{\color{blue}{\left(\beta + 1\right) \cdot \alpha} + \left(\beta + 1\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   7. *-rgt-identity 95.2%

      \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \alpha + \color{blue}{\left(\beta + 1\right) \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   8. distribute-lft-out 95.2%

      \[\leadsto \frac{\frac{\color{blue}{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   9. +-commutative 95.2%

      \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \color{blue}{\left(1 + \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   10. associate-*r/ 96.9%

       \[\leadsto \frac{\color{blue}{\left(\beta + 1\right) \cdot \frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   11. associate-*r/ 94.6%

       \[\leadsto \color{blue}{\left(\beta + 1\right) \cdot \frac{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

3. Simplified 94.6%

   \[\leadsto \color{blue}{\left(\beta + 1\right) \cdot \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]

4. Taylor expanded in beta around inf 23.7%

   \[\leadsto \left(\beta + 1\right) \cdot \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\color{blue}{{\beta}^{2}}} \]
5. Step-by-step derivation

   1. unpow2 23.7%

      \[\leadsto \left(\beta + 1\right) \cdot \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\color{blue}{\beta \cdot \beta}} \]

6. Simplified 23.7%

   \[\leadsto \left(\beta + 1\right) \cdot \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\color{blue}{\beta \cdot \beta}} \]

7. Taylor expanded in alpha around 0 22.9%

   \[\leadsto \color{blue}{\frac{\beta + 1}{{\beta}^{2} \cdot \left(\beta + 2\right)}} \]
8. Step-by-step derivation

   1. +-commutative 22.9%

      \[\leadsto \frac{\color{blue}{1 + \beta}}{{\beta}^{2} \cdot \left(\beta + 2\right)} \]

   2. unpow2 22.9%

      \[\leadsto \frac{1 + \beta}{\color{blue}{\left(\beta \cdot \beta\right)} \cdot \left(\beta + 2\right)} \]

   3. +-commutative 22.9%

      \[\leadsto \frac{1 + \beta}{\left(\beta \cdot \beta\right) \cdot \color{blue}{\left(2 + \beta\right)}} \]

9. Simplified 22.9%

   \[\leadsto \color{blue}{\frac{1 + \beta}{\left(\beta \cdot \beta\right) \cdot \left(2 + \beta\right)}} \]

10. Taylor expanded in beta around 0 18.9%

    \[\leadsto \color{blue}{\frac{0.5}{{\beta}^{2}}} \]
11. Step-by-step derivation

    1. unpow2 18.9%

       \[\leadsto \frac{0.5}{\color{blue}{\beta \cdot \beta}} \]

12. Simplified 18.9%

    \[\leadsto \color{blue}{\frac{0.5}{\beta \cdot \beta}} \]

13. Final simplification 18.9%

    \[\leadsto \frac{0.5}{\beta \cdot \beta} \]

Alternative 21: 50.8% accurate, 7.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \frac{1}{\beta \cdot \beta} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta) :precision binary64 (/ 1.0 (* beta beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return 1.0 / (beta * beta);
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = 1.0d0 / (beta * beta)
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return 1.0 / (beta * beta);
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return 1.0 / (beta * beta)

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return Float64(1.0 / Float64(beta * beta))
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = 1.0 / (beta * beta);
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := N[(1.0 / N[(beta * beta), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.4%

   \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
2. Step-by-step derivation

   1. associate-/l/ 95.2%

      \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   2. associate-+l+ 95.2%

      \[\leadsto \frac{\frac{\color{blue}{\left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)} + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   3. +-commutative 95.2%

      \[\leadsto \frac{\frac{\left(\alpha + \color{blue}{\left(\beta \cdot \alpha + \beta\right)}\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   4. associate-+r+ 95.2%

      \[\leadsto \frac{\frac{\color{blue}{\left(\left(\alpha + \beta \cdot \alpha\right) + \beta\right)} + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   5. associate-+l+ 95.2%

      \[\leadsto \frac{\frac{\color{blue}{\left(\alpha + \beta \cdot \alpha\right) + \left(\beta + 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   6. distribute-rgt1-in 95.2%

      \[\leadsto \frac{\frac{\color{blue}{\left(\beta + 1\right) \cdot \alpha} + \left(\beta + 1\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   7. *-rgt-identity 95.2%

      \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \alpha + \color{blue}{\left(\beta + 1\right) \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   8. distribute-lft-out 95.2%

      \[\leadsto \frac{\frac{\color{blue}{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   9. +-commutative 95.2%

      \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \color{blue}{\left(1 + \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   10. associate-*l/ 96.9%

       \[\leadsto \frac{\color{blue}{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   11. *-commutative 96.9%

       \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right) \cdot \frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   12. associate-*r/ 94.5%

       \[\leadsto \color{blue}{\left(1 + \alpha\right) \cdot \frac{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

3. Simplified 94.5%

   \[\leadsto \color{blue}{\left(\alpha + 1\right) \cdot \frac{\frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]

4. Taylor expanded in beta around inf 25.9%

   \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
5. Step-by-step derivation

   1. unpow2 25.9%

      \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

6. Simplified 25.9%

   \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]

7. Taylor expanded in alpha around 0 24.7%

   \[\leadsto \color{blue}{\frac{1}{{\beta}^{2}}} \]
8. Step-by-step derivation

   1. unpow2 24.7%

      \[\leadsto \frac{1}{\color{blue}{\beta \cdot \beta}} \]

9. Simplified 24.7%

   \[\leadsto \color{blue}{\frac{1}{\beta \cdot \beta}} \]

10. Final simplification 24.7%

    \[\leadsto \frac{1}{\beta \cdot \beta} \]

Alternative 22: 6.1% accurate, 11.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \frac{0.25}{\beta} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta) :precision binary64 (/ 0.25 beta))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return 0.25 / beta;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = 0.25d0 / beta
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return 0.25 / beta;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return 0.25 / beta

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return Float64(0.25 / beta)
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = 0.25 / beta;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := N[(0.25 / beta), $MachinePrecision]

Derivation

1. Initial program 96.4%

   \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
2. Step-by-step derivation

   1. associate-/l/ 95.2%

      \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   2. associate-+l+ 95.2%

      \[\leadsto \frac{\frac{\color{blue}{\left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)} + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   3. +-commutative 95.2%

      \[\leadsto \frac{\frac{\left(\alpha + \color{blue}{\left(\beta \cdot \alpha + \beta\right)}\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   4. associate-+r+ 95.2%

      \[\leadsto \frac{\frac{\color{blue}{\left(\left(\alpha + \beta \cdot \alpha\right) + \beta\right)} + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   5. associate-+l+ 95.2%

      \[\leadsto \frac{\frac{\color{blue}{\left(\alpha + \beta \cdot \alpha\right) + \left(\beta + 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   6. distribute-rgt1-in 95.2%

      \[\leadsto \frac{\frac{\color{blue}{\left(\beta + 1\right) \cdot \alpha} + \left(\beta + 1\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   7. *-rgt-identity 95.2%

      \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \alpha + \color{blue}{\left(\beta + 1\right) \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   8. distribute-lft-out 95.2%

      \[\leadsto \frac{\frac{\color{blue}{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   9. +-commutative 95.2%

      \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \color{blue}{\left(1 + \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   10. associate-*r/ 96.9%

       \[\leadsto \frac{\color{blue}{\left(\beta + 1\right) \cdot \frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   11. associate-*r/ 94.6%

       \[\leadsto \color{blue}{\left(\beta + 1\right) \cdot \frac{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

3. Simplified 94.6%

   \[\leadsto \color{blue}{\left(\beta + 1\right) \cdot \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]

4. Taylor expanded in beta around inf 23.7%

   \[\leadsto \left(\beta + 1\right) \cdot \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\color{blue}{{\beta}^{2}}} \]
5. Step-by-step derivation

   1. unpow2 23.7%

      \[\leadsto \left(\beta + 1\right) \cdot \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\color{blue}{\beta \cdot \beta}} \]

6. Simplified 23.7%

   \[\leadsto \left(\beta + 1\right) \cdot \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\color{blue}{\beta \cdot \beta}} \]

7. Taylor expanded in alpha around 0 22.9%

   \[\leadsto \color{blue}{\frac{\beta + 1}{{\beta}^{2} \cdot \left(\beta + 2\right)}} \]
8. Step-by-step derivation

   1. +-commutative 22.9%

      \[\leadsto \frac{\color{blue}{1 + \beta}}{{\beta}^{2} \cdot \left(\beta + 2\right)} \]

   2. unpow2 22.9%

      \[\leadsto \frac{1 + \beta}{\color{blue}{\left(\beta \cdot \beta\right)} \cdot \left(\beta + 2\right)} \]

   3. +-commutative 22.9%

      \[\leadsto \frac{1 + \beta}{\left(\beta \cdot \beta\right) \cdot \color{blue}{\left(2 + \beta\right)}} \]

9. Simplified 22.9%

   \[\leadsto \color{blue}{\frac{1 + \beta}{\left(\beta \cdot \beta\right) \cdot \left(2 + \beta\right)}} \]

10. Taylor expanded in beta around 0 4.5%

    \[\leadsto \color{blue}{0.25 \cdot \frac{1}{\beta} + 0.5 \cdot \frac{1}{{\beta}^{2}}} \]
11. Step-by-step derivation

    1. associate-*r/ 4.5%

       \[\leadsto \color{blue}{\frac{0.25 \cdot 1}{\beta}} + 0.5 \cdot \frac{1}{{\beta}^{2}} \]

    2. metadata-eval 4.5%

       \[\leadsto \frac{\color{blue}{0.25}}{\beta} + 0.5 \cdot \frac{1}{{\beta}^{2}} \]

    3. associate-*r/ 4.5%

       \[\leadsto \frac{0.25}{\beta} + \color{blue}{\frac{0.5 \cdot 1}{{\beta}^{2}}} \]

    4. metadata-eval 4.5%

       \[\leadsto \frac{0.25}{\beta} + \frac{\color{blue}{0.5}}{{\beta}^{2}} \]

    5. unpow2 4.5%

       \[\leadsto \frac{0.25}{\beta} + \frac{0.5}{\color{blue}{\beta \cdot \beta}} \]

12. Simplified 4.5%

    \[\leadsto \color{blue}{\frac{0.25}{\beta} + \frac{0.5}{\beta \cdot \beta}} \]

13. Taylor expanded in beta around inf 4.2%

    \[\leadsto \color{blue}{\frac{0.25}{\beta}} \]

14. Final simplification 4.2%

    \[\leadsto \frac{0.25}{\beta} \]

Reproduce

herbie shell --seed 2023182 
(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)))