Octave 3.8, jcobi/3

Percentage Accurate: 94.2% → 99.6%

Time: 15.0s

Alternatives: 17

Speedup: 2.5×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.2% 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: 99.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 10^{+142}:\\ \;\;\;\;\frac{\frac{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}{t\_0}}{\left(3 + \left(\beta + \alpha\right)\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \left(\beta + \alpha\right)} \cdot \frac{1}{\frac{\beta + \left(\alpha + 3\right)}{1 + \alpha}}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 1e+142)
     (/ (/ (* (+ 1.0 beta) (+ 1.0 alpha)) t_0) (* (+ 3.0 (+ beta alpha)) t_0))
     (*
      (/ 1.0 (+ 2.0 (+ beta alpha)))
      (/ 1.0 (/ (+ beta (+ alpha 3.0)) (+ 1.0 alpha)))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 1e+142) {
		tmp = (((1.0 + beta) * (1.0 + alpha)) / t_0) / ((3.0 + (beta + alpha)) * t_0);
	} else {
		tmp = (1.0 / (2.0 + (beta + alpha))) * (1.0 / ((beta + (alpha + 3.0)) / (1.0 + alpha)));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    if (beta <= 1d+142) then
        tmp = (((1.0d0 + beta) * (1.0d0 + alpha)) / t_0) / ((3.0d0 + (beta + alpha)) * t_0)
    else
        tmp = (1.0d0 / (2.0d0 + (beta + alpha))) * (1.0d0 / ((beta + (alpha + 3.0d0)) / (1.0d0 + alpha)))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 1e+142) {
		tmp = (((1.0 + beta) * (1.0 + alpha)) / t_0) / ((3.0 + (beta + alpha)) * t_0);
	} else {
		tmp = (1.0 / (2.0 + (beta + alpha))) * (1.0 / ((beta + (alpha + 3.0)) / (1.0 + alpha)));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 1e+142:
		tmp = (((1.0 + beta) * (1.0 + alpha)) / t_0) / ((3.0 + (beta + alpha)) * t_0)
	else:
		tmp = (1.0 / (2.0 + (beta + alpha))) * (1.0 / ((beta + (alpha + 3.0)) / (1.0 + alpha)))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 1e+142)
		tmp = Float64(Float64(Float64(Float64(1.0 + beta) * Float64(1.0 + alpha)) / t_0) / Float64(Float64(3.0 + Float64(beta + alpha)) * t_0));
	else
		tmp = Float64(Float64(1.0 / Float64(2.0 + Float64(beta + alpha))) * Float64(1.0 / Float64(Float64(beta + Float64(alpha + 3.0)) / Float64(1.0 + alpha))));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 1e+142)
		tmp = (((1.0 + beta) * (1.0 + alpha)) / t_0) / ((3.0 + (beta + alpha)) * t_0);
	else
		tmp = (1.0 / (2.0 + (beta + alpha))) * (1.0 / ((beta + (alpha + 3.0)) / (1.0 + alpha)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1e+142], N[(N[(N[(N[(1.0 + beta), $MachinePrecision] * N[(1.0 + alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.00000000000000005e142

   1. Initial program 99.4%

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

      1. associate-/l/ 98.0%

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

      2. +-commutative 98.0%

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

      3. associate-+l+ 98.0%

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

      4. *-commutative 98.0%

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

      5. metadata-eval 98.0%

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

      6. associate-+l+ 98.0%

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

      7. metadata-eval 98.0%

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

      8. +-commutative 98.0%

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

      9. +-commutative 98.0%

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

      10. +-commutative 98.0%

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

      11. metadata-eval 98.0%

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

      12. metadata-eval 98.0%

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

      13. associate-+l+ 98.0%

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

   3. Simplified 98.0%

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

      1. +-commutative 98.0%

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

      2. associate-+r+ 98.0%

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

      3. *-commutative 98.0%

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

      4. associate-+r+ 98.0%

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

      5. metadata-eval 98.0%

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

      6. *-un-lft-identity 98.0%

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

      7. +-commutative 98.0%

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

      8. *-commutative 98.0%

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

      9. associate-+r+ 98.0%

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

      10. +-commutative 98.0%

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

      11. distribute-rgt1-in 98.0%

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

      12. fma-define 98.0%

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

      13. metadata-eval 98.0%

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

      14. associate-+r+ 98.0%

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

   6. Applied egg-rr 98.0%

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

      1. *-lft-identity 98.0%

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

      2. +-commutative 98.0%

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

      3. fma-undefine 98.0%

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

      4. +-commutative 98.0%

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

      5. *-commutative 98.0%

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

      6. +-commutative 98.0%

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

      7. associate-+r+ 98.0%

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

      8. distribute-lft1-in 98.0%

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

      9. +-commutative 98.0%

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

      10. +-commutative 98.0%

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

      11. +-commutative 98.0%

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

      12. +-commutative 98.0%

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

      13. +-commutative 98.0%

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

   8. Simplified 98.0%

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

   if 1.00000000000000005e142 < beta

   1. Initial program 76.6%

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

      1. associate-/l/ 72.9%

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

      2. +-commutative 72.9%

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

      3. associate-+l+ 72.9%

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

      4. *-commutative 72.9%

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

      5. metadata-eval 72.9%

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

      6. associate-+l+ 72.9%

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

      7. metadata-eval 72.9%

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

      8. +-commutative 72.9%

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

      9. +-commutative 72.9%

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

      10. +-commutative 72.9%

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

      11. metadata-eval 72.9%

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

      12. metadata-eval 72.9%

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

      13. associate-+l+ 72.9%

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

   3. Simplified 72.9%

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

      1. clear-num 72.9%

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

      2. inv-pow 72.9%

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

   6. Applied egg-rr 72.9%

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

      1. unpow-1 72.9%

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

      2. associate-/l* 75.8%

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

      3. +-commutative 75.8%

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

      4. +-commutative 75.8%

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

      5. +-commutative 75.8%

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

      6. +-commutative 75.8%

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

      7. +-commutative 75.8%

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

      8. +-commutative 75.8%

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

      9. +-commutative 75.8%

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

      10. fma-undefine 75.8%

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

      11. +-commutative 75.8%

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

      12. *-commutative 75.8%

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

      13. +-commutative 75.8%

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

      14. associate-+r+ 75.8%

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

      15. distribute-lft1-in 75.8%

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

      16. +-commutative 75.8%

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

      17. +-commutative 75.8%

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

      18. +-commutative 75.8%

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

      19. +-commutative 75.8%

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

   8. Simplified 75.8%

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

   9. Taylor expanded in beta around inf 95.2%

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

       1. inv-pow 95.2%

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

       2. unpow-prod-down 96.0%

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

       3. inv-pow 96.0%

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

       4. +-commutative 96.0%

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

       5. associate-+r+ 96.0%

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

       6. +-commutative 96.0%

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

       7. associate-+l+ 96.0%

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

   11. Applied egg-rr 96.0%

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

       1. +-commutative 96.0%

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

       2. +-commutative 96.0%

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

       3. unpow-1 96.0%

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

       4. +-commutative 96.0%

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

       5. +-commutative 96.0%

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

   13. Simplified 96.0%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 10^{+142}:\\ \;\;\;\;\frac{\frac{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}{\alpha + \left(\beta + 2\right)}}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \left(\beta + \alpha\right)} \cdot \frac{1}{\frac{\beta + \left(\alpha + 3\right)}{1 + \alpha}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 3\right)\\ t_1 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 3.2 \cdot 10^{+58}:\\ \;\;\;\;\frac{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}{t\_1 \cdot \left(t\_0 \cdot t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 3.0))) (t_1 (+ alpha (+ beta 2.0))))
   (if (<= beta 3.2e+58)
     (/ (* (+ 1.0 beta) (+ 1.0 alpha)) (* t_1 (* t_0 t_1)))
     (/ (/ (+ 1.0 alpha) beta) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 3.0);
	double t_1 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 3.2e+58) {
		tmp = ((1.0 + beta) * (1.0 + alpha)) / (t_1 * (t_0 * t_1));
	} else {
		tmp = ((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 = alpha + (beta + 3.0d0)
    t_1 = alpha + (beta + 2.0d0)
    if (beta <= 3.2d+58) then
        tmp = ((1.0d0 + beta) * (1.0d0 + alpha)) / (t_1 * (t_0 * t_1))
    else
        tmp = ((1.0d0 + alpha) / beta) / t_0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 3.0);
	double t_1 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 3.2e+58) {
		tmp = ((1.0 + beta) * (1.0 + alpha)) / (t_1 * (t_0 * t_1));
	} else {
		tmp = ((1.0 + alpha) / beta) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 3.0)
	t_1 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 3.2e+58:
		tmp = ((1.0 + beta) * (1.0 + alpha)) / (t_1 * (t_0 * t_1))
	else:
		tmp = ((1.0 + alpha) / beta) / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 3.0))
	t_1 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 3.2e+58)
		tmp = Float64(Float64(Float64(1.0 + beta) * Float64(1.0 + alpha)) / Float64(t_1 * Float64(t_0 * t_1)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / t_0);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 3.0);
	t_1 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 3.2e+58)
		tmp = ((1.0 + beta) * (1.0 + alpha)) / (t_1 * (t_0 * t_1));
	else
		tmp = ((1.0 + alpha) / beta) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 3.2e+58], N[(N[(N[(1.0 + beta), $MachinePrecision] * N[(1.0 + alpha), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.20000000000000015e58

   1. Initial program 99.8%

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

   3. Simplified 94.3%

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

   if 3.20000000000000015e58 < beta

   1. Initial program 85.6%

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

   3. Taylor expanded in beta around inf 90.7%

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

   4. Taylor expanded in alpha around 0 90.7%

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

      1. +-commutative 90.7%

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

      2. associate-+r+ 90.7%

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

      3. +-commutative 90.7%

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

      4. +-commutative 90.7%

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

      5. +-commutative 90.7%

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

   6. Simplified 90.7%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.2 \cdot 10^{+58}:\\ \;\;\;\;\frac{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\alpha + \left(\beta + 3\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.55 \cdot 10^{+30}:\\ \;\;\;\;\frac{\left(1 + \beta\right) \cdot \frac{1 + \alpha}{\beta + \left(2 + \alpha\right)}}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.55e+30)
   (/
    (* (+ 1.0 beta) (/ (+ 1.0 alpha) (+ beta (+ 2.0 alpha))))
    (* (+ 3.0 (+ beta alpha)) (+ alpha (+ beta 2.0))))
   (/ (/ (+ 1.0 alpha) beta) (+ alpha (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.55e+30) {
		tmp = ((1.0 + beta) * ((1.0 + alpha) / (beta + (2.0 + alpha)))) / ((3.0 + (beta + alpha)) * (alpha + (beta + 2.0)));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.55d+30) then
        tmp = ((1.0d0 + beta) * ((1.0d0 + alpha) / (beta + (2.0d0 + alpha)))) / ((3.0d0 + (beta + alpha)) * (alpha + (beta + 2.0d0)))
    else
        tmp = ((1.0d0 + alpha) / beta) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.55e+30) {
		tmp = ((1.0 + beta) * ((1.0 + alpha) / (beta + (2.0 + alpha)))) / ((3.0 + (beta + alpha)) * (alpha + (beta + 2.0)));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.55e+30:
		tmp = ((1.0 + beta) * ((1.0 + alpha) / (beta + (2.0 + alpha)))) / ((3.0 + (beta + alpha)) * (alpha + (beta + 2.0)))
	else:
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.55e+30)
		tmp = Float64(Float64(Float64(1.0 + beta) * Float64(Float64(1.0 + alpha) / Float64(beta + Float64(2.0 + alpha)))) / Float64(Float64(3.0 + Float64(beta + alpha)) * Float64(alpha + Float64(beta + 2.0))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.55e+30)
		tmp = ((1.0 + beta) * ((1.0 + alpha) / (beta + (2.0 + alpha)))) / ((3.0 + (beta + alpha)) * (alpha + (beta + 2.0)));
	else
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.55e+30], N[(N[(N[(1.0 + beta), $MachinePrecision] * N[(N[(1.0 + alpha), $MachinePrecision] / N[(beta + N[(2.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] * N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.5499999999999999e30

   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/ 98.7%

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

      2. +-commutative 98.7%

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

      3. associate-+l+ 98.7%

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

      4. *-commutative 98.7%

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

      5. metadata-eval 98.7%

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

      6. associate-+l+ 98.7%

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

      7. metadata-eval 98.7%

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

      8. +-commutative 98.7%

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

      9. +-commutative 98.7%

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

      10. +-commutative 98.7%

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

      11. metadata-eval 98.7%

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

      12. metadata-eval 98.7%

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

      13. associate-+l+ 98.7%

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

   3. Simplified 98.7%

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

      1. +-commutative 98.7%

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

      2. associate-+r+ 98.7%

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

      3. *-commutative 98.7%

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

      4. associate-+r+ 98.7%

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

      5. metadata-eval 98.7%

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

      6. *-un-lft-identity 98.7%

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

      7. +-commutative 98.7%

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

      8. *-commutative 98.7%

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

      9. associate-+r+ 98.7%

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

      10. +-commutative 98.7%

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

      11. distribute-rgt1-in 98.7%

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

      12. fma-define 98.7%

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

      13. metadata-eval 98.7%

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

      14. associate-+r+ 98.7%

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

   6. Applied egg-rr 98.7%

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

      1. *-lft-identity 98.7%

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

      2. +-commutative 98.7%

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

      3. fma-undefine 98.7%

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

      4. +-commutative 98.7%

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

      5. *-commutative 98.7%

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

      6. +-commutative 98.7%

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

      7. associate-+r+ 98.7%

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

      8. distribute-lft1-in 98.7%

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

      9. +-commutative 98.7%

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

      10. +-commutative 98.7%

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

      11. +-commutative 98.7%

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

      12. +-commutative 98.7%

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

      13. +-commutative 98.7%

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

   8. Simplified 98.7%

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

      1. *-un-lft-identity 99.2%

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

      2. associate-+l+ 99.2%

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

   10. Applied egg-rr 98.7%

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

       1. *-lft-identity 99.2%

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

       2. associate-*r/ 99.2%

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

       3. +-commutative 99.2%

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

       4. +-commutative 99.2%

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

   12. Simplified 98.7%

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

   if 1.5499999999999999e30 < beta

   1. Initial program 86.4%

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

   3. Taylor expanded in beta around inf 91.2%

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

   4. Taylor expanded in alpha around 0 91.2%

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

      1. +-commutative 91.2%

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

      2. associate-+r+ 91.2%

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

      3. +-commutative 91.2%

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

      4. +-commutative 91.2%

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

      5. +-commutative 91.2%

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

   6. Simplified 91.2%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.55 \cdot 10^{+30}:\\ \;\;\;\;\frac{\left(1 + \beta\right) \cdot \frac{1 + \alpha}{\beta + \left(2 + \alpha\right)}}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.1% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.0%

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

   1. associate-/l/ 94.3%

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

   2. +-commutative 94.3%

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

   3. associate-+l+ 94.3%

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

   4. *-commutative 94.3%

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

   5. metadata-eval 94.3%

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

   6. associate-+l+ 94.3%

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

   7. metadata-eval 94.3%

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

   8. +-commutative 94.3%

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

   9. +-commutative 94.3%

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

   10. +-commutative 94.3%

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

   11. metadata-eval 94.3%

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

   12. metadata-eval 94.3%

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

   13. associate-+l+ 94.3%

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

3. Simplified 94.3%

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

   1. clear-num 94.3%

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

   2. inv-pow 94.3%

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

6. Applied egg-rr 94.3%

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

   1. unpow-1 94.3%

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

   2. associate-/l* 95.4%

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

   3. +-commutative 95.4%

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

   4. +-commutative 95.4%

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

   5. +-commutative 95.4%

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

   6. +-commutative 95.4%

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

   7. +-commutative 95.4%

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

   8. +-commutative 95.4%

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

   9. +-commutative 95.4%

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

   10. fma-undefine 95.4%

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

   11. +-commutative 95.4%

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

   12. *-commutative 95.4%

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

   13. +-commutative 95.4%

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

   14. associate-+r+ 95.4%

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

   15. distribute-lft1-in 95.4%

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

   16. +-commutative 95.4%

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

   17. +-commutative 95.4%

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

   18. +-commutative 95.4%

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

   19. +-commutative 95.4%

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

8. Simplified 95.4%

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

   1. div-inv 95.3%

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

   2. associate-+l+ 95.3%

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

10. Applied egg-rr 95.3%

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

    1. associate-*l* 99.2%

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

    2. +-commutative 99.2%

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

    3. +-commutative 99.2%

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

12. Simplified 99.2%

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

13. Final simplification 99.2%

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

Alternative 5: 99.2% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.0%

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

   1. associate-/l/ 94.3%

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

   2. +-commutative 94.3%

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

   3. associate-+l+ 94.3%

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

   4. *-commutative 94.3%

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

   5. metadata-eval 94.3%

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

   6. associate-+l+ 94.3%

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

   7. metadata-eval 94.3%

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

   8. +-commutative 94.3%

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

   9. +-commutative 94.3%

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

   10. +-commutative 94.3%

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

   11. metadata-eval 94.3%

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

   12. metadata-eval 94.3%

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

   13. associate-+l+ 94.3%

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

3. Simplified 94.3%

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

   1. clear-num 94.3%

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

   2. inv-pow 94.3%

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

6. Applied egg-rr 94.3%

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

   1. unpow-1 94.3%

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

   2. associate-/l* 95.4%

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

   3. +-commutative 95.4%

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

   4. +-commutative 95.4%

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

   5. +-commutative 95.4%

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

   6. +-commutative 95.4%

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

   7. +-commutative 95.4%

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

   8. +-commutative 95.4%

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

   9. +-commutative 95.4%

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

   10. fma-undefine 95.4%

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

   11. +-commutative 95.4%

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

   12. *-commutative 95.4%

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

   13. +-commutative 95.4%

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

   14. associate-+r+ 95.4%

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

   15. distribute-lft1-in 95.4%

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

   16. +-commutative 95.4%

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

   17. +-commutative 95.4%

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

   18. +-commutative 95.4%

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

   19. +-commutative 95.4%

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

8. Simplified 95.4%

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

   1. *-un-lft-identity 95.4%

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

   2. associate-+l+ 95.4%

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

10. Applied egg-rr 95.4%

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

    1. *-lft-identity 95.4%

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

    2. associate-*r/ 99.2%

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

    3. +-commutative 99.2%

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

    4. +-commutative 99.2%

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

12. Simplified 99.2%

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

13. Final simplification 99.2%

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

Alternative 6: 98.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 8.2 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 2}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 8.2e+15)
   (/ (/ (+ 1.0 beta) (+ beta 2.0)) (* (+ alpha (+ beta 2.0)) (+ beta 3.0)))
   (/ (/ (+ 1.0 alpha) beta) (+ alpha (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 8.2e+15) {
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((alpha + (beta + 2.0)) * (beta + 3.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 8.2d+15) then
        tmp = ((1.0d0 + beta) / (beta + 2.0d0)) / ((alpha + (beta + 2.0d0)) * (beta + 3.0d0))
    else
        tmp = ((1.0d0 + alpha) / beta) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 8.2e+15) {
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((alpha + (beta + 2.0)) * (beta + 3.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 8.2e+15:
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((alpha + (beta + 2.0)) * (beta + 3.0))
	else:
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 8.2e+15)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(beta + 2.0)) / Float64(Float64(alpha + Float64(beta + 2.0)) * Float64(beta + 3.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 8.2e+15)
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((alpha + (beta + 2.0)) * (beta + 3.0));
	else
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 8.2e+15], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 8.2e15

   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/ 99.2%

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

      2. +-commutative 99.2%

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

      3. associate-+l+ 99.2%

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

      4. *-commutative 99.2%

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

      5. metadata-eval 99.2%

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

      6. associate-+l+ 99.2%

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

      7. metadata-eval 99.2%

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

      8. +-commutative 99.2%

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

      9. +-commutative 99.2%

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

      10. +-commutative 99.2%

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

      11. metadata-eval 99.2%

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

      12. metadata-eval 99.2%

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

      13. associate-+l+ 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in alpha around 0 83.1%

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

      1. +-commutative 83.1%

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

   7. Simplified 83.1%

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

   8. Taylor expanded in alpha around 0 65.2%

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

   if 8.2e15 < beta

   1. Initial program 87.3%

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

   3. Taylor expanded in beta around inf 89.7%

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

   4. Taylor expanded in alpha around 0 89.7%

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

      1. +-commutative 89.7%

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

      2. associate-+r+ 89.7%

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

      3. +-commutative 89.7%

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

      4. +-commutative 89.7%

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

      5. +-commutative 89.7%

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

   6. Simplified 89.7%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 8.2 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 2}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.8 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 2}}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.8e+15)
   (/ (/ (+ 1.0 beta) (+ beta 2.0)) (* (+ beta 3.0) (+ beta 2.0)))
   (/ (/ (+ 1.0 alpha) beta) (+ alpha (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.8e+15) {
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 3.0) * (beta + 2.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 6.8d+15) then
        tmp = ((1.0d0 + beta) / (beta + 2.0d0)) / ((beta + 3.0d0) * (beta + 2.0d0))
    else
        tmp = ((1.0d0 + alpha) / beta) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.8e+15) {
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 3.0) * (beta + 2.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 6.8e+15:
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 3.0) * (beta + 2.0))
	else:
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.8e+15)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(beta + 2.0)) / Float64(Float64(beta + 3.0) * Float64(beta + 2.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 6.8e+15)
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 3.0) * (beta + 2.0));
	else
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 6.8e+15], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + 3.0), $MachinePrecision] * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.8e15

   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/ 99.2%

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

      2. +-commutative 99.2%

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

      3. associate-+l+ 99.2%

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

      4. *-commutative 99.2%

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

      5. metadata-eval 99.2%

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

      6. associate-+l+ 99.2%

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

      7. metadata-eval 99.2%

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

      8. +-commutative 99.2%

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

      9. +-commutative 99.2%

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

      10. +-commutative 99.2%

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

      11. metadata-eval 99.2%

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

      12. metadata-eval 99.2%

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

      13. associate-+l+ 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in alpha around 0 83.1%

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

      1. +-commutative 83.1%

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

   7. Simplified 83.1%

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

   8. Taylor expanded in alpha around 0 64.2%

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

   if 6.8e15 < beta

   1. Initial program 87.3%

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

   3. Taylor expanded in beta around inf 89.7%

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

   4. Taylor expanded in alpha around 0 89.7%

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

      1. +-commutative 89.7%

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

      2. associate-+r+ 89.7%

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

      3. +-commutative 89.7%

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

      4. +-commutative 89.7%

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

      5. +-commutative 89.7%

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

   6. Simplified 89.7%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.8 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 2}}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if beta < 5.5

   1. Initial program 99.8%

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

      1. associate-/l/ 99.2%

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

      2. +-commutative 99.2%

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

      3. associate-+l+ 99.2%

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

      4. *-commutative 99.2%

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

      5. metadata-eval 99.2%

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

      6. associate-+l+ 99.2%

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

      7. metadata-eval 99.2%

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

      8. +-commutative 99.2%

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

      9. +-commutative 99.2%

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

      10. +-commutative 99.2%

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

      11. metadata-eval 99.2%

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

      12. metadata-eval 99.2%

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

      13. associate-+l+ 99.2%

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

   3. Simplified 99.2%

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

      1. clear-num 99.2%

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

      2. inv-pow 99.2%

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

   6. Applied egg-rr 99.2%

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

      1. unpow-1 99.2%

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

      2. associate-/l* 99.2%

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

      3. +-commutative 99.2%

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

      4. +-commutative 99.2%

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

      5. +-commutative 99.2%

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

      6. +-commutative 99.2%

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

      7. +-commutative 99.2%

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

      8. +-commutative 99.2%

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

      9. +-commutative 99.2%

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

      10. fma-undefine 99.2%

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

      11. +-commutative 99.2%

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

      12. *-commutative 99.2%

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

      13. +-commutative 99.2%

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

      14. associate-+r+ 99.2%

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

      15. distribute-lft1-in 99.2%

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

      16. +-commutative 99.2%

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

      17. +-commutative 99.2%

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

      18. +-commutative 99.2%

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

      19. +-commutative 99.2%

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

   8. Simplified 99.2%

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

   9. Taylor expanded in beta around 0 98.3%

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

       1. associate-/l* 98.3%

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

       2. +-commutative 98.3%

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

       3. +-commutative 98.3%

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

   11. Simplified 98.3%

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

   12. Taylor expanded in alpha around 0 81.4%

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

       1. mul-1-neg 81.4%

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

   14. Simplified 81.4%

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

       1. inv-pow 81.4%

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

       2. unpow-prod-down 81.4%

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

       3. inv-pow 81.4%

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

       4. +-commutative 81.4%

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

       5. associate-+r+ 81.4%

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

       6. +-commutative 81.4%

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

       7. unsub-neg 81.4%

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

   16. Applied egg-rr 81.4%

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

       1. unpow-1 81.4%

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

       2. *-commutative 81.4%

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

       3. +-commutative 81.4%

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

       4. +-commutative 81.4%

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

   18. Simplified 81.4%

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

   if 5.5 < beta

   1. Initial program 87.5%

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

   3. Taylor expanded in beta around inf 88.7%

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

   4. Taylor expanded in alpha around 0 88.7%

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

      1. +-commutative 88.7%

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

      2. associate-+r+ 88.7%

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

      3. +-commutative 88.7%

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

      4. +-commutative 88.7%

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

      5. +-commutative 88.7%

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

   6. Simplified 88.7%

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

4. Final simplification 83.6%

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

Alternative 9: 97.2% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if beta < 6

   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/ 99.2%

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

      2. +-commutative 99.2%

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

      3. associate-+l+ 99.2%

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

      4. *-commutative 99.2%

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

      5. metadata-eval 99.2%

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

      6. associate-+l+ 99.2%

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

      7. metadata-eval 99.2%

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

      8. +-commutative 99.2%

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

      9. +-commutative 99.2%

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

      10. +-commutative 99.2%

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

      11. metadata-eval 99.2%

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

      12. metadata-eval 99.2%

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

      13. associate-+l+ 99.2%

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

   3. Simplified 99.2%

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

      1. clear-num 99.2%

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

      2. inv-pow 99.2%

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

   6. Applied egg-rr 99.2%

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

      1. unpow-1 99.2%

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

      2. associate-/l* 99.2%

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

      3. +-commutative 99.2%

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

      4. +-commutative 99.2%

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

      5. +-commutative 99.2%

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

      6. +-commutative 99.2%

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

      7. +-commutative 99.2%

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

      8. +-commutative 99.2%

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

      9. +-commutative 99.2%

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

      10. fma-undefine 99.2%

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

      11. +-commutative 99.2%

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

      12. *-commutative 99.2%

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

      13. +-commutative 99.2%

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

      14. associate-+r+ 99.2%

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

      15. distribute-lft1-in 99.2%

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

      16. +-commutative 99.2%

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

      17. +-commutative 99.2%

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

      18. +-commutative 99.2%

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

      19. +-commutative 99.2%

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

   8. Simplified 99.2%

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

   9. Taylor expanded in beta around 0 98.3%

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

       1. associate-/l* 98.3%

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

       2. +-commutative 98.3%

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

       3. +-commutative 98.3%

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

   11. Simplified 98.3%

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

   12. Taylor expanded in alpha around 0 81.4%

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

       1. mul-1-neg 81.4%

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

   14. Simplified 81.4%

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

   if 6 < beta

   1. Initial program 87.5%

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

   3. Taylor expanded in beta around inf 88.7%

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

   4. Taylor expanded in alpha around 0 88.7%

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

      1. +-commutative 88.7%

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

      2. associate-+r+ 88.7%

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

      3. +-commutative 88.7%

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

      4. +-commutative 88.7%

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

      5. +-commutative 88.7%

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

   6. Simplified 88.7%

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

4. Final simplification 83.7%

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

Alternative 10: 97.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.3:\\ \;\;\;\;\frac{1}{\left(2 + \alpha\right) \cdot \left(6 - \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.3)
   (/ 1.0 (* (+ 2.0 alpha) (- 6.0 alpha)))
   (/ (/ (+ 1.0 alpha) beta) (+ alpha (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.3) {
		tmp = 1.0 / ((2.0 + alpha) * (6.0 - alpha));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.3d0) then
        tmp = 1.0d0 / ((2.0d0 + alpha) * (6.0d0 - alpha))
    else
        tmp = ((1.0d0 + alpha) / beta) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.3) {
		tmp = 1.0 / ((2.0 + alpha) * (6.0 - alpha));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.3:
		tmp = 1.0 / ((2.0 + alpha) * (6.0 - alpha))
	else:
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.3)
		tmp = Float64(1.0 / Float64(Float64(2.0 + alpha) * Float64(6.0 - alpha)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.3)
		tmp = 1.0 / ((2.0 + alpha) * (6.0 - alpha));
	else
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.3], N[(1.0 / N[(N[(2.0 + alpha), $MachinePrecision] * N[(6.0 - alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.2999999999999998

   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/ 99.2%

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

      2. +-commutative 99.2%

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

      3. associate-+l+ 99.2%

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

      4. *-commutative 99.2%

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

      5. metadata-eval 99.2%

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

      6. associate-+l+ 99.2%

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

      7. metadata-eval 99.2%

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

      8. +-commutative 99.2%

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

      9. +-commutative 99.2%

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

      10. +-commutative 99.2%

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

      11. metadata-eval 99.2%

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

      12. metadata-eval 99.2%

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

      13. associate-+l+ 99.2%

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

   3. Simplified 99.2%

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

      1. clear-num 99.2%

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

      2. inv-pow 99.2%

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

   6. Applied egg-rr 99.2%

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

      1. unpow-1 99.2%

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

      2. associate-/l* 99.2%

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

      3. +-commutative 99.2%

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

      4. +-commutative 99.2%

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

      5. +-commutative 99.2%

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

      6. +-commutative 99.2%

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

      7. +-commutative 99.2%

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

      8. +-commutative 99.2%

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

      9. +-commutative 99.2%

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

      10. fma-undefine 99.2%

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

      11. +-commutative 99.2%

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

      12. *-commutative 99.2%

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

      13. +-commutative 99.2%

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

      14. associate-+r+ 99.2%

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

      15. distribute-lft1-in 99.2%

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

      16. +-commutative 99.2%

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

      17. +-commutative 99.2%

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

      18. +-commutative 99.2%

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

      19. +-commutative 99.2%

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

   8. Simplified 99.2%

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

   9. Taylor expanded in beta around 0 98.7%

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

       1. associate-/l* 98.7%

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

       2. +-commutative 98.7%

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

       3. +-commutative 98.7%

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

   11. Simplified 98.7%

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

   12. Taylor expanded in alpha around 0 81.7%

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

       1. mul-1-neg 81.7%

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

   14. Simplified 81.7%

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

   15. Taylor expanded in beta around 0 81.7%

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

       1. +-commutative 81.7%

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

   17. Simplified 81.7%

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

   if 2.2999999999999998 < beta

   1. Initial program 87.6%

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

   3. Taylor expanded in beta around inf 87.8%

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

   4. Taylor expanded in alpha around 0 87.8%

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

      1. +-commutative 87.8%

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

      2. associate-+r+ 87.8%

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

      3. +-commutative 87.8%

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

      4. +-commutative 87.8%

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

      5. +-commutative 87.8%

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

   6. Simplified 87.8%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.3:\\ \;\;\;\;\frac{1}{\left(2 + \alpha\right) \cdot \left(6 - \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 97.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.2:\\ \;\;\;\;\frac{1}{\left(2 + \alpha\right) \cdot \left(6 - \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.2)
   (/ 1.0 (* (+ 2.0 alpha) (- 6.0 alpha)))
   (/ (/ (+ 1.0 alpha) beta) (+ beta 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.2) {
		tmp = 1.0 / ((2.0 + alpha) * (6.0 - alpha));
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.2d0) then
        tmp = 1.0d0 / ((2.0d0 + alpha) * (6.0d0 - alpha))
    else
        tmp = ((1.0d0 + alpha) / beta) / (beta + 3.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.2) {
		tmp = 1.0 / ((2.0 + alpha) * (6.0 - alpha));
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.2:
		tmp = 1.0 / ((2.0 + alpha) * (6.0 - alpha))
	else:
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.2)
		tmp = Float64(1.0 / Float64(Float64(2.0 + alpha) * Float64(6.0 - alpha)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(beta + 3.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.2)
		tmp = 1.0 / ((2.0 + alpha) * (6.0 - alpha));
	else
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.2], N[(1.0 / N[(N[(2.0 + alpha), $MachinePrecision] * N[(6.0 - alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.2000000000000002

   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/ 99.2%

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

      2. +-commutative 99.2%

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

      3. associate-+l+ 99.2%

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

      4. *-commutative 99.2%

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

      5. metadata-eval 99.2%

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

      6. associate-+l+ 99.2%

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

      7. metadata-eval 99.2%

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

      8. +-commutative 99.2%

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

      9. +-commutative 99.2%

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

      10. +-commutative 99.2%

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

      11. metadata-eval 99.2%

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

      12. metadata-eval 99.2%

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

      13. associate-+l+ 99.2%

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

   3. Simplified 99.2%

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

      1. clear-num 99.2%

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

      2. inv-pow 99.2%

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

   6. Applied egg-rr 99.2%

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

      1. unpow-1 99.2%

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

      2. associate-/l* 99.2%

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

      3. +-commutative 99.2%

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

      4. +-commutative 99.2%

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

      5. +-commutative 99.2%

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

      6. +-commutative 99.2%

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

      7. +-commutative 99.2%

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

      8. +-commutative 99.2%

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

      9. +-commutative 99.2%

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

      10. fma-undefine 99.2%

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

      11. +-commutative 99.2%

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

      12. *-commutative 99.2%

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

      13. +-commutative 99.2%

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

      14. associate-+r+ 99.2%

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

      15. distribute-lft1-in 99.2%

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

      16. +-commutative 99.2%

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

      17. +-commutative 99.2%

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

      18. +-commutative 99.2%

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

      19. +-commutative 99.2%

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

   8. Simplified 99.2%

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

   9. Taylor expanded in beta around 0 98.7%

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

       1. associate-/l* 98.7%

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

       2. +-commutative 98.7%

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

       3. +-commutative 98.7%

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

   11. Simplified 98.7%

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

   12. Taylor expanded in alpha around 0 81.7%

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

       1. mul-1-neg 81.7%

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

   14. Simplified 81.7%

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

   15. Taylor expanded in beta around 0 81.7%

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

       1. +-commutative 81.7%

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

   17. Simplified 81.7%

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

   if 2.2000000000000002 < beta

   1. Initial program 87.6%

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

   3. Taylor expanded in beta around inf 87.8%

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

   4. Taylor expanded in alpha around 0 87.7%

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

      1. +-commutative 87.7%

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

   6. Simplified 87.7%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.2:\\ \;\;\;\;\frac{1}{\left(2 + \alpha\right) \cdot \left(6 - \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 92.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.3:\\ \;\;\;\;\frac{1}{\left(2 + \alpha\right) \cdot \left(6 - \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.3)
   (/ 1.0 (* (+ 2.0 alpha) (- 6.0 alpha)))
   (/ (/ 1.0 beta) (+ beta 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.3) {
		tmp = 1.0 / ((2.0 + alpha) * (6.0 - alpha));
	} else {
		tmp = (1.0 / beta) / (beta + 3.0);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.3d0) then
        tmp = 1.0d0 / ((2.0d0 + alpha) * (6.0d0 - alpha))
    else
        tmp = (1.0d0 / beta) / (beta + 3.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.3) {
		tmp = 1.0 / ((2.0 + alpha) * (6.0 - alpha));
	} else {
		tmp = (1.0 / beta) / (beta + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.3:
		tmp = 1.0 / ((2.0 + alpha) * (6.0 - alpha))
	else:
		tmp = (1.0 / beta) / (beta + 3.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.3)
		tmp = Float64(1.0 / Float64(Float64(2.0 + alpha) * Float64(6.0 - alpha)));
	else
		tmp = Float64(Float64(1.0 / beta) / Float64(beta + 3.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.3)
		tmp = 1.0 / ((2.0 + alpha) * (6.0 - alpha));
	else
		tmp = (1.0 / beta) / (beta + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.3], N[(1.0 / N[(N[(2.0 + alpha), $MachinePrecision] * N[(6.0 - alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.2999999999999998

   1. Initial program 99.8%

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

      1. associate-/l/ 99.2%

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

      2. +-commutative 99.2%

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

      3. associate-+l+ 99.2%

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

      4. *-commutative 99.2%

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

      5. metadata-eval 99.2%

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

      6. associate-+l+ 99.2%

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

      7. metadata-eval 99.2%

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

      8. +-commutative 99.2%

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

      9. +-commutative 99.2%

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

      10. +-commutative 99.2%

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

      11. metadata-eval 99.2%

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

      12. metadata-eval 99.2%

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

      13. associate-+l+ 99.2%

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

   3. Simplified 99.2%

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

      1. clear-num 99.2%

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

      2. inv-pow 99.2%

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

   6. Applied egg-rr 99.2%

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

      1. unpow-1 99.2%

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

      2. associate-/l* 99.2%

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

      3. +-commutative 99.2%

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

      4. +-commutative 99.2%

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

      5. +-commutative 99.2%

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

      6. +-commutative 99.2%

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

      7. +-commutative 99.2%

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

      8. +-commutative 99.2%

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

      9. +-commutative 99.2%

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

      10. fma-undefine 99.2%

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

      11. +-commutative 99.2%

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

      12. *-commutative 99.2%

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

      13. +-commutative 99.2%

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

      14. associate-+r+ 99.2%

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

      15. distribute-lft1-in 99.2%

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

      16. +-commutative 99.2%

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

      17. +-commutative 99.2%

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

      18. +-commutative 99.2%

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

      19. +-commutative 99.2%

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

   8. Simplified 99.2%

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

   9. Taylor expanded in beta around 0 98.7%

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

       1. associate-/l* 98.7%

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

       2. +-commutative 98.7%

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

       3. +-commutative 98.7%

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

   11. Simplified 98.7%

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

   12. Taylor expanded in alpha around 0 81.7%

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

       1. mul-1-neg 81.7%

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

   14. Simplified 81.7%

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

   15. Taylor expanded in beta around 0 81.7%

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

       1. +-commutative 81.7%

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

   17. Simplified 81.7%

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

   if 2.2999999999999998 < beta

   1. Initial program 87.6%

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

   3. Taylor expanded in beta around inf 87.8%

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

      1. clear-num 87.3%

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

      2. inv-pow 87.3%

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

      3. metadata-eval 87.3%

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

      4. associate-+l+ 87.3%

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

      5. metadata-eval 87.3%

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

      6. associate-+l+ 87.3%

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

   5. Applied egg-rr 87.3%

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

      1. unpow-1 87.3%

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

      2. associate-/r/ 87.4%

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

      3. +-commutative 87.4%

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

   7. Simplified 87.4%

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

   8. Taylor expanded in alpha around 0 75.0%

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

      1. associate-/r* 75.4%

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

   10. Simplified 75.4%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.3:\\ \;\;\;\;\frac{1}{\left(2 + \alpha\right) \cdot \left(6 - \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 91.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.4:\\ \;\;\;\;\frac{1}{\left(\beta + 2\right) \cdot 6}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5.4) (/ 1.0 (* (+ beta 2.0) 6.0)) (/ (/ 1.0 beta) (+ beta 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.4) {
		tmp = 1.0 / ((beta + 2.0) * 6.0);
	} else {
		tmp = (1.0 / beta) / (beta + 3.0);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 5.4d0) then
        tmp = 1.0d0 / ((beta + 2.0d0) * 6.0d0)
    else
        tmp = (1.0d0 / beta) / (beta + 3.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.4) {
		tmp = 1.0 / ((beta + 2.0) * 6.0);
	} else {
		tmp = (1.0 / beta) / (beta + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 5.4:
		tmp = 1.0 / ((beta + 2.0) * 6.0)
	else:
		tmp = (1.0 / beta) / (beta + 3.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 5.4)
		tmp = Float64(1.0 / Float64(Float64(beta + 2.0) * 6.0));
	else
		tmp = Float64(Float64(1.0 / beta) / Float64(beta + 3.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 5.4)
		tmp = 1.0 / ((beta + 2.0) * 6.0);
	else
		tmp = (1.0 / beta) / (beta + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 5.4], N[(1.0 / N[(N[(beta + 2.0), $MachinePrecision] * 6.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.4000000000000004

   1. Initial program 99.8%

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

      1. associate-/l/ 99.2%

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

      2. +-commutative 99.2%

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

      3. associate-+l+ 99.2%

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

      4. *-commutative 99.2%

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

      5. metadata-eval 99.2%

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

      6. associate-+l+ 99.2%

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

      7. metadata-eval 99.2%

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

      8. +-commutative 99.2%

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

      9. +-commutative 99.2%

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

      10. +-commutative 99.2%

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

      11. metadata-eval 99.2%

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

      12. metadata-eval 99.2%

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

      13. associate-+l+ 99.2%

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

   3. Simplified 99.2%

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

      1. clear-num 99.2%

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

      2. inv-pow 99.2%

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

   6. Applied egg-rr 99.2%

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

      1. unpow-1 99.2%

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

      2. associate-/l* 99.2%

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

      3. +-commutative 99.2%

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

      4. +-commutative 99.2%

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

      5. +-commutative 99.2%

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

      6. +-commutative 99.2%

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

      7. +-commutative 99.2%

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

      8. +-commutative 99.2%

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

      9. +-commutative 99.2%

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

      10. fma-undefine 99.2%

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

      11. +-commutative 99.2%

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

      12. *-commutative 99.2%

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

      13. +-commutative 99.2%

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

      14. associate-+r+ 99.2%

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

      15. distribute-lft1-in 99.2%

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

      16. +-commutative 99.2%

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

      17. +-commutative 99.2%

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

      18. +-commutative 99.2%

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

      19. +-commutative 99.2%

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

   8. Simplified 99.2%

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

   9. Taylor expanded in beta around 0 98.3%

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

       1. associate-/l* 98.3%

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

       2. +-commutative 98.3%

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

       3. +-commutative 98.3%

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

   11. Simplified 98.3%

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

   12. Taylor expanded in alpha around 0 63.8%

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

   if 5.4000000000000004 < beta

   1. Initial program 87.5%

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

   3. Taylor expanded in beta around inf 88.7%

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

      1. clear-num 88.1%

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

      2. inv-pow 88.1%

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

      3. metadata-eval 88.1%

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

      4. associate-+l+ 88.1%

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

      5. metadata-eval 88.1%

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

      6. associate-+l+ 88.1%

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

   5. Applied egg-rr 88.1%

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

      1. unpow-1 88.1%

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

      2. associate-/r/ 88.2%

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

      3. +-commutative 88.2%

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

   7. Simplified 88.2%

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

   8. Taylor expanded in alpha around 0 75.7%

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

      1. associate-/r* 76.1%

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

   10. Simplified 76.1%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.4:\\ \;\;\;\;\frac{1}{\left(\beta + 2\right) \cdot 6}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 91.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.4:\\ \;\;\;\;\frac{1}{\left(\beta + 2\right) \cdot 6}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5.4) (/ 1.0 (* (+ beta 2.0) 6.0)) (/ 1.0 (* beta (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.4) {
		tmp = 1.0 / ((beta + 2.0) * 6.0);
	} else {
		tmp = 1.0 / (beta * (beta + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 5.4d0) then
        tmp = 1.0d0 / ((beta + 2.0d0) * 6.0d0)
    else
        tmp = 1.0d0 / (beta * (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.4) {
		tmp = 1.0 / ((beta + 2.0) * 6.0);
	} else {
		tmp = 1.0 / (beta * (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 5.4:
		tmp = 1.0 / ((beta + 2.0) * 6.0)
	else:
		tmp = 1.0 / (beta * (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 5.4)
		tmp = Float64(1.0 / Float64(Float64(beta + 2.0) * 6.0));
	else
		tmp = Float64(1.0 / Float64(beta * Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 5.4)
		tmp = 1.0 / ((beta + 2.0) * 6.0);
	else
		tmp = 1.0 / (beta * (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 5.4], N[(1.0 / N[(N[(beta + 2.0), $MachinePrecision] * 6.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(beta * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.4000000000000004

   1. Initial program 99.8%

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

      1. associate-/l/ 99.2%

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

      2. +-commutative 99.2%

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

      3. associate-+l+ 99.2%

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

      4. *-commutative 99.2%

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

      5. metadata-eval 99.2%

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

      6. associate-+l+ 99.2%

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

      7. metadata-eval 99.2%

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

      8. +-commutative 99.2%

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

      9. +-commutative 99.2%

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

      10. +-commutative 99.2%

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

      11. metadata-eval 99.2%

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

      12. metadata-eval 99.2%

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

      13. associate-+l+ 99.2%

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

   3. Simplified 99.2%

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

      1. clear-num 99.2%

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

      2. inv-pow 99.2%

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

   6. Applied egg-rr 99.2%

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

      1. unpow-1 99.2%

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

      2. associate-/l* 99.2%

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

      3. +-commutative 99.2%

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

      4. +-commutative 99.2%

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

      5. +-commutative 99.2%

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

      6. +-commutative 99.2%

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

      7. +-commutative 99.2%

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

      8. +-commutative 99.2%

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

      9. +-commutative 99.2%

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

      10. fma-undefine 99.2%

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

      11. +-commutative 99.2%

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

      12. *-commutative 99.2%

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

      13. +-commutative 99.2%

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

      14. associate-+r+ 99.2%

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

      15. distribute-lft1-in 99.2%

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

      16. +-commutative 99.2%

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

      17. +-commutative 99.2%

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

      18. +-commutative 99.2%

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

      19. +-commutative 99.2%

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

   8. Simplified 99.2%

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

   9. Taylor expanded in beta around 0 98.3%

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

       1. associate-/l* 98.3%

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

       2. +-commutative 98.3%

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

       3. +-commutative 98.3%

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

   11. Simplified 98.3%

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

   12. Taylor expanded in alpha around 0 63.8%

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

   if 5.4000000000000004 < beta

   1. Initial program 87.5%

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

   3. Taylor expanded in beta around inf 88.7%

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

   4. Taylor expanded in alpha around 0 75.7%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.4:\\ \;\;\;\;\frac{1}{\left(\beta + 2\right) \cdot 6}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 47.1% accurate, 5.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \frac{1}{\left(\beta + 2\right) \cdot 6} \end{array} \]

FPCore

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

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return 1.0 / ((beta + 2.0) * 6.0);
}

Fortran

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

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return 1.0 / ((beta + 2.0) * 6.0);
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return 1.0 / ((beta + 2.0) * 6.0)

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return Float64(1.0 / Float64(Float64(beta + 2.0) * 6.0))
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = 1.0 / ((beta + 2.0) * 6.0);
end

Wolfram

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

Derivation

1. Initial program 96.0%

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

   1. associate-/l/ 94.3%

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

   2. +-commutative 94.3%

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

   3. associate-+l+ 94.3%

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

   4. *-commutative 94.3%

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

   5. metadata-eval 94.3%

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

   6. associate-+l+ 94.3%

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

   7. metadata-eval 94.3%

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

   8. +-commutative 94.3%

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

   9. +-commutative 94.3%

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

   10. +-commutative 94.3%

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

   11. metadata-eval 94.3%

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

   12. metadata-eval 94.3%

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

   13. associate-+l+ 94.3%

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

3. Simplified 94.3%

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

   1. clear-num 94.3%

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

   2. inv-pow 94.3%

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

6. Applied egg-rr 94.3%

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

   1. unpow-1 94.3%

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

   2. associate-/l* 95.4%

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

   3. +-commutative 95.4%

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

   4. +-commutative 95.4%

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

   5. +-commutative 95.4%

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

   6. +-commutative 95.4%

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

   7. +-commutative 95.4%

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

   8. +-commutative 95.4%

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

   9. +-commutative 95.4%

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

   10. fma-undefine 95.4%

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

   11. +-commutative 95.4%

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

   12. *-commutative 95.4%

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

   13. +-commutative 95.4%

       \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{\left(\beta + 3\right) + \alpha}{\frac{\left(\beta \cdot \color{blue}{\left(\alpha + 1\right)} + \alpha\right) + 1}{\alpha + \left(\beta + 2\right)}}} \]

   14. associate-+r+ 95.4%

       \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{\left(\beta + 3\right) + \alpha}{\frac{\color{blue}{\beta \cdot \left(\alpha + 1\right) + \left(\alpha + 1\right)}}{\alpha + \left(\beta + 2\right)}}} \]

   15. distribute-lft1-in 95.4%

       \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{\left(\beta + 3\right) + \alpha}{\frac{\color{blue}{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}}{\alpha + \left(\beta + 2\right)}}} \]

   16. +-commutative 95.4%

       \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{\left(\beta + 3\right) + \alpha}{\frac{\color{blue}{\left(1 + \beta\right)} \cdot \left(\alpha + 1\right)}{\alpha + \left(\beta + 2\right)}}} \]

   17. +-commutative 95.4%

       \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{\left(\beta + 3\right) + \alpha}{\frac{\left(1 + \beta\right) \cdot \color{blue}{\left(1 + \alpha\right)}}{\alpha + \left(\beta + 2\right)}}} \]

   18. +-commutative 95.4%

       \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{\left(\beta + 3\right) + \alpha}{\frac{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}{\alpha + \color{blue}{\left(2 + \beta\right)}}}} \]

   19. +-commutative 95.4%

       \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{\left(\beta + 3\right) + \alpha}{\frac{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}{\color{blue}{\left(2 + \beta\right) + \alpha}}}} \]

8. Simplified 95.4%

   \[\leadsto \color{blue}{\frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{\left(\beta + 3\right) + \alpha}{\frac{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}{\left(\beta + 2\right) + \alpha}}}} \]

9. Taylor expanded in beta around 0 71.6%

   \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \color{blue}{\frac{\left(2 + \alpha\right) \cdot \left(3 + \alpha\right)}{1 + \alpha}}} \]
10. Step-by-step derivation

    1. associate-/l* 71.6%

       \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \color{blue}{\left(\left(2 + \alpha\right) \cdot \frac{3 + \alpha}{1 + \alpha}\right)}} \]

    2. +-commutative 71.6%

       \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \left(\color{blue}{\left(\alpha + 2\right)} \cdot \frac{3 + \alpha}{1 + \alpha}\right)} \]

    3. +-commutative 71.6%

       \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \left(\left(\alpha + 2\right) \cdot \frac{\color{blue}{\alpha + 3}}{1 + \alpha}\right)} \]

11. Simplified 71.6%

    \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \color{blue}{\left(\left(\alpha + 2\right) \cdot \frac{\alpha + 3}{1 + \alpha}\right)}} \]

12. Taylor expanded in alpha around 0 46.0%

    \[\leadsto \frac{1}{\color{blue}{6 \cdot \left(2 + \beta\right)}} \]

13. Final simplification 46.0%

    \[\leadsto \frac{1}{\left(\beta + 2\right) \cdot 6} \]
14. Add Preprocessing

Alternative 16: 46.9% accurate, 7.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \frac{0.16666666666666666}{\beta + 2} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta) :precision binary64 (/ 0.16666666666666666 (+ beta 2.0)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return 0.16666666666666666 / (beta + 2.0);
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = 0.16666666666666666d0 / (beta + 2.0d0)
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return 0.16666666666666666 / (beta + 2.0);
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return 0.16666666666666666 / (beta + 2.0)

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return Float64(0.16666666666666666 / Float64(beta + 2.0))
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = 0.16666666666666666 / (beta + 2.0);
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := N[(0.16666666666666666 / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.0%

   \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
2. Step-by-step derivation

   1. associate-/l/ 94.3%

      \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   2. +-commutative 94.3%

      \[\leadsto \frac{\frac{\color{blue}{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   3. associate-+l+ 94.3%

      \[\leadsto \frac{\frac{1 + \color{blue}{\left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   4. *-commutative 94.3%

      \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \color{blue}{\alpha \cdot \beta}\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   5. metadata-eval 94.3%

      \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\left(\alpha + \beta\right) + \color{blue}{2}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   6. associate-+l+ 94.3%

      \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\color{blue}{\alpha + \left(\beta + 2\right)}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   7. metadata-eval 94.3%

      \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   8. +-commutative 94.3%

      \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}{\left(\color{blue}{\left(2 + \left(\alpha + \beta\right)\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   9. +-commutative 94.3%

      \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}{\left(\left(2 + \color{blue}{\left(\beta + \alpha\right)}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   10. +-commutative 94.3%

       \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}{\left(\color{blue}{\left(\left(\beta + \alpha\right) + 2\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   11. metadata-eval 94.3%

       \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}{\left(\left(\left(\beta + \alpha\right) + \color{blue}{2 \cdot 1}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   12. metadata-eval 94.3%

       \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}{\left(\left(\left(\beta + \alpha\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + \color{blue}{2}\right)} \]

   13. associate-+l+ 94.3%

       \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}{\left(\left(\left(\beta + \alpha\right) + 2 \cdot 1\right) + 1\right) \cdot \color{blue}{\left(\alpha + \left(\beta + 2\right)\right)}} \]

3. Simplified 94.3%

   \[\leadsto \color{blue}{\frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. clear-num 94.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}}} \]

   2. inv-pow 94.3%

      \[\leadsto \color{blue}{{\left(\frac{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}\right)}^{-1}} \]

6. Applied egg-rr 94.3%

   \[\leadsto \color{blue}{{\left(\frac{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}\right)}^{-1}} \]
7. Step-by-step derivation

   1. unpow-1 94.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}}} \]

   2. associate-/l* 95.4%

      \[\leadsto \frac{1}{\color{blue}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \frac{\alpha + \left(\beta + 3\right)}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}}} \]

   3. +-commutative 95.4%

      \[\leadsto \frac{1}{\left(\alpha + \color{blue}{\left(2 + \beta\right)}\right) \cdot \frac{\alpha + \left(\beta + 3\right)}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}} \]

   4. +-commutative 95.4%

      \[\leadsto \frac{1}{\color{blue}{\left(\left(2 + \beta\right) + \alpha\right)} \cdot \frac{\alpha + \left(\beta + 3\right)}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}} \]

   5. +-commutative 95.4%

      \[\leadsto \frac{1}{\left(\color{blue}{\left(\beta + 2\right)} + \alpha\right) \cdot \frac{\alpha + \left(\beta + 3\right)}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}} \]

   6. +-commutative 95.4%

      \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{\alpha + \color{blue}{\left(3 + \beta\right)}}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}} \]

   7. +-commutative 95.4%

      \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{\color{blue}{\left(3 + \beta\right) + \alpha}}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}} \]

   8. +-commutative 95.4%

      \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{\color{blue}{\left(\beta + 3\right)} + \alpha}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}} \]

   9. +-commutative 95.4%

      \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{\left(\beta + 3\right) + \alpha}{\frac{\color{blue}{\mathsf{fma}\left(\alpha + 1, \beta, \alpha\right) + 1}}{\alpha + \left(\beta + 2\right)}}} \]

   10. fma-undefine 95.4%

       \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{\left(\beta + 3\right) + \alpha}{\frac{\color{blue}{\left(\left(\alpha + 1\right) \cdot \beta + \alpha\right)} + 1}{\alpha + \left(\beta + 2\right)}}} \]

   11. +-commutative 95.4%

       \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{\left(\beta + 3\right) + \alpha}{\frac{\left(\color{blue}{\left(1 + \alpha\right)} \cdot \beta + \alpha\right) + 1}{\alpha + \left(\beta + 2\right)}}} \]

   12. *-commutative 95.4%

       \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{\left(\beta + 3\right) + \alpha}{\frac{\left(\color{blue}{\beta \cdot \left(1 + \alpha\right)} + \alpha\right) + 1}{\alpha + \left(\beta + 2\right)}}} \]

   13. +-commutative 95.4%

       \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{\left(\beta + 3\right) + \alpha}{\frac{\left(\beta \cdot \color{blue}{\left(\alpha + 1\right)} + \alpha\right) + 1}{\alpha + \left(\beta + 2\right)}}} \]

   14. associate-+r+ 95.4%

       \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{\left(\beta + 3\right) + \alpha}{\frac{\color{blue}{\beta \cdot \left(\alpha + 1\right) + \left(\alpha + 1\right)}}{\alpha + \left(\beta + 2\right)}}} \]

   15. distribute-lft1-in 95.4%

       \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{\left(\beta + 3\right) + \alpha}{\frac{\color{blue}{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}}{\alpha + \left(\beta + 2\right)}}} \]

   16. +-commutative 95.4%

       \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{\left(\beta + 3\right) + \alpha}{\frac{\color{blue}{\left(1 + \beta\right)} \cdot \left(\alpha + 1\right)}{\alpha + \left(\beta + 2\right)}}} \]

   17. +-commutative 95.4%

       \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{\left(\beta + 3\right) + \alpha}{\frac{\left(1 + \beta\right) \cdot \color{blue}{\left(1 + \alpha\right)}}{\alpha + \left(\beta + 2\right)}}} \]

   18. +-commutative 95.4%

       \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{\left(\beta + 3\right) + \alpha}{\frac{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}{\alpha + \color{blue}{\left(2 + \beta\right)}}}} \]

   19. +-commutative 95.4%

       \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{\left(\beta + 3\right) + \alpha}{\frac{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}{\color{blue}{\left(2 + \beta\right) + \alpha}}}} \]

8. Simplified 95.4%

   \[\leadsto \color{blue}{\frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{\left(\beta + 3\right) + \alpha}{\frac{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}{\left(\beta + 2\right) + \alpha}}}} \]

9. Taylor expanded in beta around 0 71.6%

   \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \color{blue}{\frac{\left(2 + \alpha\right) \cdot \left(3 + \alpha\right)}{1 + \alpha}}} \]
10. Step-by-step derivation

    1. associate-/l* 71.6%

       \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \color{blue}{\left(\left(2 + \alpha\right) \cdot \frac{3 + \alpha}{1 + \alpha}\right)}} \]

    2. +-commutative 71.6%

       \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \left(\color{blue}{\left(\alpha + 2\right)} \cdot \frac{3 + \alpha}{1 + \alpha}\right)} \]

    3. +-commutative 71.6%

       \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \left(\left(\alpha + 2\right) \cdot \frac{\color{blue}{\alpha + 3}}{1 + \alpha}\right)} \]

11. Simplified 71.6%

    \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \color{blue}{\left(\left(\alpha + 2\right) \cdot \frac{\alpha + 3}{1 + \alpha}\right)}} \]

12. Taylor expanded in alpha around 0 46.0%

    \[\leadsto \color{blue}{\frac{0.16666666666666666}{2 + \beta}} \]

13. Final simplification 46.0%

    \[\leadsto \frac{0.16666666666666666}{\beta + 2} \]
14. Add Preprocessing

Alternative 17: 45.1% accurate, 35.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ 0.08333333333333333 \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta) :precision binary64 0.08333333333333333)

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return 0.08333333333333333;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = 0.08333333333333333d0
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return 0.08333333333333333;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return 0.08333333333333333

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return 0.08333333333333333
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = 0.08333333333333333;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := 0.08333333333333333

Derivation

1. Initial program 96.0%

   \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
2. Step-by-step derivation

   1. associate-/l/ 94.3%

      \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   2. +-commutative 94.3%

      \[\leadsto \frac{\frac{\color{blue}{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   3. associate-+l+ 94.3%

      \[\leadsto \frac{\frac{1 + \color{blue}{\left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   4. *-commutative 94.3%

      \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \color{blue}{\alpha \cdot \beta}\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   5. metadata-eval 94.3%

      \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\left(\alpha + \beta\right) + \color{blue}{2}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   6. associate-+l+ 94.3%

      \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\color{blue}{\alpha + \left(\beta + 2\right)}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   7. metadata-eval 94.3%

      \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   8. +-commutative 94.3%

      \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}{\left(\color{blue}{\left(2 + \left(\alpha + \beta\right)\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   9. +-commutative 94.3%

      \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}{\left(\left(2 + \color{blue}{\left(\beta + \alpha\right)}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   10. +-commutative 94.3%

       \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}{\left(\color{blue}{\left(\left(\beta + \alpha\right) + 2\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   11. metadata-eval 94.3%

       \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}{\left(\left(\left(\beta + \alpha\right) + \color{blue}{2 \cdot 1}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   12. metadata-eval 94.3%

       \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}{\left(\left(\left(\beta + \alpha\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + \color{blue}{2}\right)} \]

   13. associate-+l+ 94.3%

       \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}{\left(\left(\left(\beta + \alpha\right) + 2 \cdot 1\right) + 1\right) \cdot \color{blue}{\left(\alpha + \left(\beta + 2\right)\right)}} \]

3. Simplified 94.3%

   \[\leadsto \color{blue}{\frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. clear-num 94.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}}} \]

   2. inv-pow 94.3%

      \[\leadsto \color{blue}{{\left(\frac{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}\right)}^{-1}} \]

6. Applied egg-rr 94.3%

   \[\leadsto \color{blue}{{\left(\frac{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}\right)}^{-1}} \]
7. Step-by-step derivation

   1. unpow-1 94.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}}} \]

   2. associate-/l* 95.4%

      \[\leadsto \frac{1}{\color{blue}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \frac{\alpha + \left(\beta + 3\right)}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}}} \]

   3. +-commutative 95.4%

      \[\leadsto \frac{1}{\left(\alpha + \color{blue}{\left(2 + \beta\right)}\right) \cdot \frac{\alpha + \left(\beta + 3\right)}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}} \]

   4. +-commutative 95.4%

      \[\leadsto \frac{1}{\color{blue}{\left(\left(2 + \beta\right) + \alpha\right)} \cdot \frac{\alpha + \left(\beta + 3\right)}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}} \]

   5. +-commutative 95.4%

      \[\leadsto \frac{1}{\left(\color{blue}{\left(\beta + 2\right)} + \alpha\right) \cdot \frac{\alpha + \left(\beta + 3\right)}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}} \]

   6. +-commutative 95.4%

      \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{\alpha + \color{blue}{\left(3 + \beta\right)}}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}} \]

   7. +-commutative 95.4%

      \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{\color{blue}{\left(3 + \beta\right) + \alpha}}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}} \]

   8. +-commutative 95.4%

      \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{\color{blue}{\left(\beta + 3\right)} + \alpha}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}} \]

   9. +-commutative 95.4%

      \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{\left(\beta + 3\right) + \alpha}{\frac{\color{blue}{\mathsf{fma}\left(\alpha + 1, \beta, \alpha\right) + 1}}{\alpha + \left(\beta + 2\right)}}} \]

   10. fma-undefine 95.4%

       \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{\left(\beta + 3\right) + \alpha}{\frac{\color{blue}{\left(\left(\alpha + 1\right) \cdot \beta + \alpha\right)} + 1}{\alpha + \left(\beta + 2\right)}}} \]

   11. +-commutative 95.4%

       \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{\left(\beta + 3\right) + \alpha}{\frac{\left(\color{blue}{\left(1 + \alpha\right)} \cdot \beta + \alpha\right) + 1}{\alpha + \left(\beta + 2\right)}}} \]

   12. *-commutative 95.4%

       \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{\left(\beta + 3\right) + \alpha}{\frac{\left(\color{blue}{\beta \cdot \left(1 + \alpha\right)} + \alpha\right) + 1}{\alpha + \left(\beta + 2\right)}}} \]

   13. +-commutative 95.4%

       \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{\left(\beta + 3\right) + \alpha}{\frac{\left(\beta \cdot \color{blue}{\left(\alpha + 1\right)} + \alpha\right) + 1}{\alpha + \left(\beta + 2\right)}}} \]

   14. associate-+r+ 95.4%

       \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{\left(\beta + 3\right) + \alpha}{\frac{\color{blue}{\beta \cdot \left(\alpha + 1\right) + \left(\alpha + 1\right)}}{\alpha + \left(\beta + 2\right)}}} \]

   15. distribute-lft1-in 95.4%

       \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{\left(\beta + 3\right) + \alpha}{\frac{\color{blue}{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}}{\alpha + \left(\beta + 2\right)}}} \]

   16. +-commutative 95.4%

       \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{\left(\beta + 3\right) + \alpha}{\frac{\color{blue}{\left(1 + \beta\right)} \cdot \left(\alpha + 1\right)}{\alpha + \left(\beta + 2\right)}}} \]

   17. +-commutative 95.4%

       \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{\left(\beta + 3\right) + \alpha}{\frac{\left(1 + \beta\right) \cdot \color{blue}{\left(1 + \alpha\right)}}{\alpha + \left(\beta + 2\right)}}} \]

   18. +-commutative 95.4%

       \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{\left(\beta + 3\right) + \alpha}{\frac{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}{\alpha + \color{blue}{\left(2 + \beta\right)}}}} \]

   19. +-commutative 95.4%

       \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{\left(\beta + 3\right) + \alpha}{\frac{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}{\color{blue}{\left(2 + \beta\right) + \alpha}}}} \]

8. Simplified 95.4%

   \[\leadsto \color{blue}{\frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{\left(\beta + 3\right) + \alpha}{\frac{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}{\left(\beta + 2\right) + \alpha}}}} \]

9. Taylor expanded in beta around 0 71.6%

   \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \color{blue}{\frac{\left(2 + \alpha\right) \cdot \left(3 + \alpha\right)}{1 + \alpha}}} \]
10. Step-by-step derivation

    1. associate-/l* 71.6%

       \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \color{blue}{\left(\left(2 + \alpha\right) \cdot \frac{3 + \alpha}{1 + \alpha}\right)}} \]

    2. +-commutative 71.6%

       \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \left(\color{blue}{\left(\alpha + 2\right)} \cdot \frac{3 + \alpha}{1 + \alpha}\right)} \]

    3. +-commutative 71.6%

       \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \left(\left(\alpha + 2\right) \cdot \frac{\color{blue}{\alpha + 3}}{1 + \alpha}\right)} \]

11. Simplified 71.6%

    \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \color{blue}{\left(\left(\alpha + 2\right) \cdot \frac{\alpha + 3}{1 + \alpha}\right)}} \]

12. Taylor expanded in alpha around 0 46.0%

    \[\leadsto \color{blue}{\frac{0.16666666666666666}{2 + \beta}} \]

13. Taylor expanded in beta around 0 45.1%

    \[\leadsto \color{blue}{0.08333333333333333} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024148 
(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)))