Octave 3.8, jcobi/3

Percentage Accurate: 94.5% → 99.5%

Time: 18.5s

Alternatives: 19

Speedup: 3.2×

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: 94.5% 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: 99.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 3\right)\\ t_1 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 72000000000000:\\ \;\;\;\;\frac{1 + \alpha}{t_1} \cdot \frac{1 + \beta}{t_0 \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-1 - \alpha\right) \cdot \frac{-1 - \frac{-1 - \alpha}{\beta}}{t_0}}{2 + \left(\beta + \alpha\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 3.0))) (t_1 (+ alpha (+ beta 2.0))))
   (if (<= beta 72000000000000.0)
     (* (/ (+ 1.0 alpha) t_1) (/ (+ 1.0 beta) (* t_0 t_1)))
     (/
      (* (- -1.0 alpha) (/ (- -1.0 (/ (- -1.0 alpha) beta)) t_0))
      (+ 2.0 (+ beta alpha))))))

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 <= 72000000000000.0) {
		tmp = ((1.0 + alpha) / t_1) * ((1.0 + beta) / (t_0 * t_1));
	} else {
		tmp = ((-1.0 - alpha) * ((-1.0 - ((-1.0 - alpha) / beta)) / t_0)) / (2.0 + (beta + 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) :: t_1
    real(8) :: tmp
    t_0 = alpha + (beta + 3.0d0)
    t_1 = alpha + (beta + 2.0d0)
    if (beta <= 72000000000000.0d0) then
        tmp = ((1.0d0 + alpha) / t_1) * ((1.0d0 + beta) / (t_0 * t_1))
    else
        tmp = (((-1.0d0) - alpha) * (((-1.0d0) - (((-1.0d0) - alpha) / beta)) / t_0)) / (2.0d0 + (beta + alpha))
    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 <= 72000000000000.0) {
		tmp = ((1.0 + alpha) / t_1) * ((1.0 + beta) / (t_0 * t_1));
	} else {
		tmp = ((-1.0 - alpha) * ((-1.0 - ((-1.0 - alpha) / beta)) / t_0)) / (2.0 + (beta + alpha));
	}
	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 <= 72000000000000.0:
		tmp = ((1.0 + alpha) / t_1) * ((1.0 + beta) / (t_0 * t_1))
	else:
		tmp = ((-1.0 - alpha) * ((-1.0 - ((-1.0 - alpha) / beta)) / t_0)) / (2.0 + (beta + alpha))
	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 <= 72000000000000.0)
		tmp = Float64(Float64(Float64(1.0 + alpha) / t_1) * Float64(Float64(1.0 + beta) / Float64(t_0 * t_1)));
	else
		tmp = Float64(Float64(Float64(-1.0 - alpha) * Float64(Float64(-1.0 - Float64(Float64(-1.0 - alpha) / beta)) / t_0)) / Float64(2.0 + Float64(beta + alpha)));
	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 <= 72000000000000.0)
		tmp = ((1.0 + alpha) / t_1) * ((1.0 + beta) / (t_0 * t_1));
	else
		tmp = ((-1.0 - alpha) * ((-1.0 - ((-1.0 - alpha) / beta)) / t_0)) / (2.0 + (beta + 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 + 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 72000000000000.0], N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$1), $MachinePrecision] * N[(N[(1.0 + beta), $MachinePrecision] / N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-1.0 - alpha), $MachinePrecision] * N[(N[(-1.0 - N[(N[(-1.0 - alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 7.2e13

   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 99.7%

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

   if 7.2e13 < beta

   1. Initial program 77.9%

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

   3. Simplified 92.7%

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

      1. clear-num 92.7%

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

      2. associate-+r+ 92.7%

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

      3. *-commutative 92.7%

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

      4. frac-times 85.4%

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

      5. *-un-lft-identity 85.4%

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

      6. +-commutative 85.4%

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

      7. *-commutative 85.4%

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

      8. associate-+r+ 85.4%

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

   5. Applied egg-rr 85.4%

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

      1. associate-/r* 92.7%

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

      2. associate-/l* 77.3%

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

      3. associate-*l/ 92.7%

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

      4. *-commutative 92.7%

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

      5. times-frac 99.7%

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

      6. associate-/r* 92.7%

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

      7. *-commutative 92.7%

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

      8. associate-/r* 99.7%

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

      9. +-commutative 99.7%

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

      10. +-commutative 99.7%

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

      11. +-commutative 99.7%

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

      12. +-commutative 99.7%

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

   7. Simplified 99.7%

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

      1. associate-*r/ 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+r+ 99.9%

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

      4. +-commutative 99.9%

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

      5. +-commutative 99.9%

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

      6. +-commutative 99.9%

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

      7. +-commutative 99.9%

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

      8. associate-+r+ 99.9%

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

      9. +-commutative 99.9%

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

      10. +-commutative 99.9%

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

   9. Applied egg-rr 99.9%

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

   10. Taylor expanded in beta around inf 74.7%

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

       1. mul-1-neg 74.7%

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

       2. unsub-neg 74.7%

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

   12. Simplified 74.7%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 72000000000000:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-1 - \alpha\right) \cdot \frac{-1 - \frac{-1 - \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}}{2 + \left(\beta + \alpha\right)}\\ \end{array} \]

Alternative 2: 98.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 3.6:\\ \;\;\;\;\frac{1 + \alpha}{t_0 \cdot \left(\left(\alpha + 3\right) \cdot \left(2 + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{t_0} \cdot \frac{1 + \frac{-1 - \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 3.6)
     (/ (+ 1.0 alpha) (* t_0 (* (+ alpha 3.0) (+ 2.0 alpha))))
     (*
      (/ (+ 1.0 alpha) t_0)
      (/ (+ 1.0 (/ (- -1.0 alpha) beta)) (+ alpha (+ beta 3.0)))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 3.6) {
		tmp = (1.0 + alpha) / (t_0 * ((alpha + 3.0) * (2.0 + alpha)));
	} else {
		tmp = ((1.0 + alpha) / t_0) * ((1.0 + ((-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) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    if (beta <= 3.6d0) then
        tmp = (1.0d0 + alpha) / (t_0 * ((alpha + 3.0d0) * (2.0d0 + alpha)))
    else
        tmp = ((1.0d0 + alpha) / t_0) * ((1.0d0 + (((-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 t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 3.6) {
		tmp = (1.0 + alpha) / (t_0 * ((alpha + 3.0) * (2.0 + alpha)));
	} else {
		tmp = ((1.0 + alpha) / t_0) * ((1.0 + ((-1.0 - alpha) / beta)) / (alpha + (beta + 3.0)));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 3.6:
		tmp = (1.0 + alpha) / (t_0 * ((alpha + 3.0) * (2.0 + alpha)))
	else:
		tmp = ((1.0 + alpha) / t_0) * ((1.0 + ((-1.0 - alpha) / beta)) / (alpha + (beta + 3.0)))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 3.6)
		tmp = Float64(Float64(1.0 + alpha) / Float64(t_0 * Float64(Float64(alpha + 3.0) * Float64(2.0 + alpha))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / t_0) * Float64(Float64(1.0 + 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)
	t_0 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 3.6)
		tmp = (1.0 + alpha) / (t_0 * ((alpha + 3.0) * (2.0 + alpha)));
	else
		tmp = ((1.0 + alpha) / t_0) * ((1.0 + ((-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_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 3.6], N[(N[(1.0 + alpha), $MachinePrecision] / N[(t$95$0 * N[(N[(alpha + 3.0), $MachinePrecision] * N[(2.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(N[(1.0 + N[(N[(-1.0 - alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.60000000000000009

   1. Initial program 99.9%

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

   3. Simplified 99.7%

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

      1. *-commutative 99.7%

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

      2. clear-num 99.7%

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

      3. frac-times 92.5%

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

      4. *-un-lft-identity 92.5%

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

      5. +-commutative 92.5%

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

   5. Applied egg-rr 92.5%

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

   6. Taylor expanded in beta around 0 90.6%

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

      1. +-commutative 90.6%

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

      2. +-commutative 90.6%

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

   8. Simplified 90.6%

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

   if 3.60000000000000009 < beta

   1. Initial program 78.1%

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

   3. Simplified 92.8%

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

      1. clear-num 92.7%

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

      2. associate-+r+ 92.7%

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

      3. *-commutative 92.7%

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

      4. frac-times 85.6%

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

      5. *-un-lft-identity 85.6%

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

      6. +-commutative 85.6%

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

      7. *-commutative 85.6%

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

      8. associate-+r+ 85.6%

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

   5. Applied egg-rr 85.6%

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

      1. associate-/r* 92.8%

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

      2. associate-/l* 77.6%

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

      3. associate-*l/ 92.8%

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

      4. *-commutative 92.8%

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

      5. times-frac 99.7%

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

      6. associate-/r* 92.8%

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

      7. *-commutative 92.8%

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

      8. associate-/r* 99.7%

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

      9. +-commutative 99.7%

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

      10. +-commutative 99.7%

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

      11. +-commutative 99.7%

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

      12. +-commutative 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in beta around inf 73.6%

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

      1. associate-*r/ 73.6%

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

      2. distribute-lft-in 73.6%

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

      3. metadata-eval 73.6%

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

      4. neg-mul-1 73.6%

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

      5. unsub-neg 73.6%

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

   10. Simplified 73.6%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.6:\\ \;\;\;\;\frac{1 + \alpha}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\alpha + 3\right) \cdot \left(2 + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \frac{-1 - \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 3: 98.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.35:\\ \;\;\;\;\frac{1 + \alpha}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\alpha + 3\right) \cdot \left(2 + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-1 - \alpha\right) \cdot \frac{-1 - \frac{-1 - \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}}{2 + \left(\beta + \alpha\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.35)
   (/ (+ 1.0 alpha) (* (+ alpha (+ beta 2.0)) (* (+ alpha 3.0) (+ 2.0 alpha))))
   (/
    (*
     (- -1.0 alpha)
     (/ (- -1.0 (/ (- -1.0 alpha) beta)) (+ alpha (+ beta 3.0))))
    (+ 2.0 (+ beta alpha)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.35) {
		tmp = (1.0 + alpha) / ((alpha + (beta + 2.0)) * ((alpha + 3.0) * (2.0 + alpha)));
	} else {
		tmp = ((-1.0 - alpha) * ((-1.0 - ((-1.0 - alpha) / beta)) / (alpha + (beta + 3.0)))) / (2.0 + (beta + 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) :: tmp
    if (beta <= 3.35d0) then
        tmp = (1.0d0 + alpha) / ((alpha + (beta + 2.0d0)) * ((alpha + 3.0d0) * (2.0d0 + alpha)))
    else
        tmp = (((-1.0d0) - alpha) * (((-1.0d0) - (((-1.0d0) - alpha) / beta)) / (alpha + (beta + 3.0d0)))) / (2.0d0 + (beta + alpha))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.35) {
		tmp = (1.0 + alpha) / ((alpha + (beta + 2.0)) * ((alpha + 3.0) * (2.0 + alpha)));
	} else {
		tmp = ((-1.0 - alpha) * ((-1.0 - ((-1.0 - alpha) / beta)) / (alpha + (beta + 3.0)))) / (2.0 + (beta + alpha));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 3.35:
		tmp = (1.0 + alpha) / ((alpha + (beta + 2.0)) * ((alpha + 3.0) * (2.0 + alpha)))
	else:
		tmp = ((-1.0 - alpha) * ((-1.0 - ((-1.0 - alpha) / beta)) / (alpha + (beta + 3.0)))) / (2.0 + (beta + alpha))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.35)
		tmp = Float64(Float64(1.0 + alpha) / Float64(Float64(alpha + Float64(beta + 2.0)) * Float64(Float64(alpha + 3.0) * Float64(2.0 + alpha))));
	else
		tmp = Float64(Float64(Float64(-1.0 - alpha) * Float64(Float64(-1.0 - Float64(Float64(-1.0 - alpha) / beta)) / Float64(alpha + Float64(beta + 3.0)))) / Float64(2.0 + Float64(beta + alpha)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.35)
		tmp = (1.0 + alpha) / ((alpha + (beta + 2.0)) * ((alpha + 3.0) * (2.0 + alpha)));
	else
		tmp = ((-1.0 - alpha) * ((-1.0 - ((-1.0 - alpha) / beta)) / (alpha + (beta + 3.0)))) / (2.0 + (beta + alpha));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 3.35], N[(N[(1.0 + alpha), $MachinePrecision] / N[(N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(alpha + 3.0), $MachinePrecision] * N[(2.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-1.0 - alpha), $MachinePrecision] * N[(N[(-1.0 - N[(N[(-1.0 - alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.35000000000000009

   1. Initial program 99.9%

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

   3. Simplified 99.7%

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

      1. *-commutative 99.7%

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

      2. clear-num 99.7%

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

      3. frac-times 92.5%

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

      4. *-un-lft-identity 92.5%

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

      5. +-commutative 92.5%

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

   5. Applied egg-rr 92.5%

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

   6. Taylor expanded in beta around 0 90.6%

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

      1. +-commutative 90.6%

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

      2. +-commutative 90.6%

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

   8. Simplified 90.6%

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

   if 3.35000000000000009 < beta

   1. Initial program 78.1%

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

   3. Simplified 92.8%

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

      1. clear-num 92.7%

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

      2. associate-+r+ 92.7%

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

      3. *-commutative 92.7%

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

      4. frac-times 85.6%

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

      5. *-un-lft-identity 85.6%

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

      6. +-commutative 85.6%

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

      7. *-commutative 85.6%

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

      8. associate-+r+ 85.6%

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

   5. Applied egg-rr 85.6%

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

      1. associate-/r* 92.8%

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

      2. associate-/l* 77.6%

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

      3. associate-*l/ 92.8%

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

      4. *-commutative 92.8%

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

      5. times-frac 99.7%

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

      6. associate-/r* 92.8%

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

      7. *-commutative 92.8%

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

      8. associate-/r* 99.7%

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

      9. +-commutative 99.7%

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

      10. +-commutative 99.7%

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

      11. +-commutative 99.7%

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

      12. +-commutative 99.7%

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

   7. Simplified 99.7%

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

      1. associate-*r/ 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+r+ 99.9%

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

      4. +-commutative 99.9%

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

      5. +-commutative 99.9%

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

      6. +-commutative 99.9%

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

      7. +-commutative 99.9%

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

      8. associate-+r+ 99.9%

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

      9. +-commutative 99.9%

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

      10. +-commutative 99.9%

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

   9. Applied egg-rr 99.9%

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

   10. Taylor expanded in beta around inf 73.7%

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

       1. mul-1-neg 73.7%

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

       2. unsub-neg 73.7%

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

   12. Simplified 73.7%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.35:\\ \;\;\;\;\frac{1 + \alpha}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\alpha + 3\right) \cdot \left(2 + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-1 - \alpha\right) \cdot \frac{-1 - \frac{-1 - \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}}{2 + \left(\beta + \alpha\right)}\\ \end{array} \]

Alternative 4: 99.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.3%

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

3. Simplified 97.6%

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

   1. clear-num 97.6%

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

   2. associate-+r+ 97.6%

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

   3. *-commutative 97.6%

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

   4. frac-times 95.4%

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

   5. *-un-lft-identity 95.4%

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

   6. +-commutative 95.4%

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

   7. *-commutative 95.4%

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

   8. associate-+r+ 95.4%

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

5. Applied egg-rr 95.4%

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

   1. associate-/r* 97.6%

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

   2. associate-/l* 93.1%

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

   3. associate-*l/ 97.6%

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

   4. *-commutative 97.6%

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

   5. times-frac 99.8%

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

   6. associate-/r* 97.6%

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

   7. *-commutative 97.6%

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

   8. associate-/r* 99.8%

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

   9. +-commutative 99.8%

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

   10. +-commutative 99.8%

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

   11. +-commutative 99.8%

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

   12. +-commutative 99.8%

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

7. Simplified 99.8%

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

   1. associate-*r/ 99.8%

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

   2. +-commutative 99.8%

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

   3. associate-+r+ 99.8%

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

   4. +-commutative 99.8%

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

   5. +-commutative 99.8%

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

   6. +-commutative 99.8%

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

   7. +-commutative 99.8%

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

   8. associate-+r+ 99.8%

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

   9. +-commutative 99.8%

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

   10. +-commutative 99.8%

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

9. Applied egg-rr 99.8%

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

10. Final simplification 99.8%

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

Alternative 5: 99.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.3%

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

3. Simplified 97.6%

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

   1. clear-num 97.6%

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

   2. associate-+r+ 97.6%

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

   3. *-commutative 97.6%

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

   4. frac-times 95.4%

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

   5. *-un-lft-identity 95.4%

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

   6. +-commutative 95.4%

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

   7. *-commutative 95.4%

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

   8. associate-+r+ 95.4%

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

5. Applied egg-rr 95.4%

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

   1. associate-/r* 97.6%

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

   2. associate-/l* 93.1%

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

   3. associate-*l/ 97.6%

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

   4. *-commutative 97.6%

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

   5. times-frac 99.8%

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

   6. associate-/r* 97.6%

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

   7. *-commutative 97.6%

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

   8. associate-/r* 99.8%

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

   9. +-commutative 99.8%

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

   10. +-commutative 99.8%

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

   11. +-commutative 99.8%

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

   12. +-commutative 99.8%

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

7. Simplified 99.8%

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

8. Final simplification 99.8%

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

Alternative 6: 97.8% accurate, 1.5× speedup

Math

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

FPCore

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

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 7.2) {
		tmp = (1.0 + alpha) / ((alpha + (beta + 2.0)) * ((alpha + 3.0) * (2.0 + alpha)));
	} else {
		tmp = ((1.0 + alpha) * ((1.0 - (alpha / beta)) / (alpha + (beta + 3.0)))) / (2.0 + (beta + 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) :: tmp
    if (beta <= 7.2d0) then
        tmp = (1.0d0 + alpha) / ((alpha + (beta + 2.0d0)) * ((alpha + 3.0d0) * (2.0d0 + alpha)))
    else
        tmp = ((1.0d0 + alpha) * ((1.0d0 - (alpha / beta)) / (alpha + (beta + 3.0d0)))) / (2.0d0 + (beta + alpha))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if beta < 7.20000000000000018

   1. Initial program 99.9%

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

   3. Simplified 99.7%

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

      1. *-commutative 99.7%

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

      2. clear-num 99.7%

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

      3. frac-times 92.5%

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

      4. *-un-lft-identity 92.5%

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

      5. +-commutative 92.5%

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

   5. Applied egg-rr 92.5%

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

   6. Taylor expanded in beta around 0 90.6%

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

      1. +-commutative 90.6%

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

      2. +-commutative 90.6%

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

   8. Simplified 90.6%

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

   if 7.20000000000000018 < beta

   1. Initial program 78.1%

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

   3. Simplified 92.8%

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

      1. clear-num 92.7%

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

      2. associate-+r+ 92.7%

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

      3. *-commutative 92.7%

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

      4. frac-times 85.6%

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

      5. *-un-lft-identity 85.6%

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

      6. +-commutative 85.6%

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

      7. *-commutative 85.6%

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

      8. associate-+r+ 85.6%

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

   5. Applied egg-rr 85.6%

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

      1. associate-/r* 92.8%

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

      2. associate-/l* 77.6%

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

      3. associate-*l/ 92.8%

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

      4. *-commutative 92.8%

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

      5. times-frac 99.7%

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

      6. associate-/r* 92.8%

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

      7. *-commutative 92.8%

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

      8. associate-/r* 99.7%

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

      9. +-commutative 99.7%

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

      10. +-commutative 99.7%

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

      11. +-commutative 99.7%

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

      12. +-commutative 99.7%

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

   7. Simplified 99.7%

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

      1. associate-*r/ 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+r+ 99.9%

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

      4. +-commutative 99.9%

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

      5. +-commutative 99.9%

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

      6. +-commutative 99.9%

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

      7. +-commutative 99.9%

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

      8. associate-+r+ 99.9%

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

      9. +-commutative 99.9%

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

      10. +-commutative 99.9%

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

   9. Applied egg-rr 99.9%

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

   10. Taylor expanded in beta around inf 73.7%

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

       1. mul-1-neg 73.7%

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

       2. unsub-neg 73.7%

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

   12. Simplified 73.7%

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

   13. Taylor expanded in alpha around inf 73.7%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 7.2:\\ \;\;\;\;\frac{1 + \alpha}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\alpha + 3\right) \cdot \left(2 + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \alpha\right) \cdot \frac{1 - \frac{\alpha}{\beta}}{\alpha + \left(\beta + 3\right)}}{2 + \left(\beta + \alpha\right)}\\ \end{array} \]

Alternative 7: 97.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 3.35:\\ \;\;\;\;\frac{1 + \alpha}{t_0 \cdot \left(\left(\alpha + 3\right) \cdot \left(2 + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \alpha\right) \cdot \frac{1}{t_0}}{1 + \left(2 + \left(\beta + \alpha\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 3.35)
     (/ (+ 1.0 alpha) (* t_0 (* (+ alpha 3.0) (+ 2.0 alpha))))
     (/ (* (+ 1.0 alpha) (/ 1.0 t_0)) (+ 1.0 (+ 2.0 (+ beta alpha)))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 3.35) {
		tmp = (1.0 + alpha) / (t_0 * ((alpha + 3.0) * (2.0 + alpha)));
	} else {
		tmp = ((1.0 + alpha) * (1.0 / t_0)) / (1.0 + (2.0 + (beta + 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 <= 3.35d0) then
        tmp = (1.0d0 + alpha) / (t_0 * ((alpha + 3.0d0) * (2.0d0 + alpha)))
    else
        tmp = ((1.0d0 + alpha) * (1.0d0 / t_0)) / (1.0d0 + (2.0d0 + (beta + 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 <= 3.35) {
		tmp = (1.0 + alpha) / (t_0 * ((alpha + 3.0) * (2.0 + alpha)));
	} else {
		tmp = ((1.0 + alpha) * (1.0 / t_0)) / (1.0 + (2.0 + (beta + alpha)));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 3.35:
		tmp = (1.0 + alpha) / (t_0 * ((alpha + 3.0) * (2.0 + alpha)))
	else:
		tmp = ((1.0 + alpha) * (1.0 / t_0)) / (1.0 + (2.0 + (beta + 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 <= 3.35)
		tmp = Float64(Float64(1.0 + alpha) / Float64(t_0 * Float64(Float64(alpha + 3.0) * Float64(2.0 + alpha))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) * Float64(1.0 / t_0)) / Float64(1.0 + Float64(2.0 + Float64(beta + 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 <= 3.35)
		tmp = (1.0 + alpha) / (t_0 * ((alpha + 3.0) * (2.0 + alpha)));
	else
		tmp = ((1.0 + alpha) * (1.0 / t_0)) / (1.0 + (2.0 + (beta + 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, 3.35], N[(N[(1.0 + alpha), $MachinePrecision] / N[(t$95$0 * N[(N[(alpha + 3.0), $MachinePrecision] * N[(2.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.35000000000000009

   1. Initial program 99.9%

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

   3. Simplified 99.7%

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

      1. *-commutative 99.7%

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

      2. clear-num 99.7%

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

      3. frac-times 92.5%

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

      4. *-un-lft-identity 92.5%

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

      5. +-commutative 92.5%

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

   5. Applied egg-rr 92.5%

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

   6. Taylor expanded in beta around 0 90.6%

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

      1. +-commutative 90.6%

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

      2. +-commutative 90.6%

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

   8. Simplified 90.6%

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

   if 3.35000000000000009 < beta

   1. Initial program 78.1%

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

      1. div-inv 78.1%

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

      2. +-commutative 78.1%

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

      3. *-commutative 78.1%

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

      4. associate-+r+ 78.1%

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

      5. +-commutative 78.1%

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

      6. associate-+r+ 78.1%

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

      7. +-commutative 78.1%

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

      8. +-commutative 78.1%

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

      9. *-commutative 78.1%

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

      10. fma-def 78.1%

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

      11. metadata-eval 78.1%

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

      12. associate-+r+ 78.1%

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

      13. metadata-eval 78.1%

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

      14. associate-+r+ 78.1%

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

   3. Applied egg-rr 78.1%

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

   4. Taylor expanded in beta around inf 74.9%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.35:\\ \;\;\;\;\frac{1 + \alpha}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\alpha + 3\right) \cdot \left(2 + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \alpha\right) \cdot \frac{1}{\alpha + \left(\beta + 2\right)}}{1 + \left(2 + \left(\beta + \alpha\right)\right)}\\ \end{array} \]

Alternative 8: 97.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 1.7:\\ \;\;\;\;\frac{0.16666666666666666 + \beta \cdot 0.027777777777777776}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \alpha\right) \cdot \frac{1}{\alpha + \left(\beta + 3\right)}}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (+ beta alpha))))
   (if (<= beta 1.7)
     (/ (+ 0.16666666666666666 (* beta 0.027777777777777776)) t_0)
     (/ (* (+ 1.0 alpha) (/ 1.0 (+ alpha (+ beta 3.0)))) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta + alpha);
	double tmp;
	if (beta <= 1.7) {
		tmp = (0.16666666666666666 + (beta * 0.027777777777777776)) / t_0;
	} else {
		tmp = ((1.0 + alpha) * (1.0 / (alpha + (beta + 3.0)))) / t_0;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 + (beta + alpha)
    if (beta <= 1.7d0) then
        tmp = (0.16666666666666666d0 + (beta * 0.027777777777777776d0)) / t_0
    else
        tmp = ((1.0d0 + alpha) * (1.0d0 / (alpha + (beta + 3.0d0)))) / t_0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta + alpha);
	double tmp;
	if (beta <= 1.7) {
		tmp = (0.16666666666666666 + (beta * 0.027777777777777776)) / t_0;
	} else {
		tmp = ((1.0 + alpha) * (1.0 / (alpha + (beta + 3.0)))) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = 2.0 + (beta + alpha)
	tmp = 0
	if beta <= 1.7:
		tmp = (0.16666666666666666 + (beta * 0.027777777777777776)) / t_0
	else:
		tmp = ((1.0 + alpha) * (1.0 / (alpha + (beta + 3.0)))) / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(beta + alpha))
	tmp = 0.0
	if (beta <= 1.7)
		tmp = Float64(Float64(0.16666666666666666 + Float64(beta * 0.027777777777777776)) / t_0);
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) * Float64(1.0 / Float64(alpha + Float64(beta + 3.0)))) / t_0);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = 2.0 + (beta + alpha);
	tmp = 0.0;
	if (beta <= 1.7)
		tmp = (0.16666666666666666 + (beta * 0.027777777777777776)) / t_0;
	else
		tmp = ((1.0 + alpha) * (1.0 / (alpha + (beta + 3.0)))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1.7], N[(N[(0.16666666666666666 + N[(beta * 0.027777777777777776), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] * N[(1.0 / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.69999999999999996

   1. Initial program 99.9%

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

   3. Simplified 99.7%

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

      1. clear-num 99.6%

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

      2. associate-+r+ 99.6%

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

      3. *-commutative 99.6%

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

      4. frac-times 99.7%

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

      5. *-un-lft-identity 99.7%

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

      6. +-commutative 99.7%

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

      7. *-commutative 99.7%

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

      8. associate-+r+ 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. associate-/r* 99.7%

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

      2. associate-/l* 99.7%

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

      3. associate-*l/ 99.7%

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

      4. *-commutative 99.7%

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

      5. times-frac 99.8%

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

      6. associate-/r* 99.7%

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

      7. *-commutative 99.7%

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

      8. associate-/r* 99.8%

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

      9. +-commutative 99.8%

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

      10. +-commutative 99.8%

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

      11. +-commutative 99.8%

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

      12. +-commutative 99.8%

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

   7. Simplified 99.8%

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

      1. associate-*r/ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+r+ 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. associate-+r+ 99.8%

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

      9. +-commutative 99.8%

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

      10. +-commutative 99.8%

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

   9. Applied egg-rr 99.8%

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

   10. Taylor expanded in alpha around 0 64.8%

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

   11. Taylor expanded in beta around 0 64.5%

       \[\leadsto \frac{\color{blue}{0.16666666666666666 + 0.027777777777777776 \cdot \beta}}{2 + \left(\beta + \alpha\right)} \]
   12. Step-by-step derivation

       1. *-commutative 64.5%

          \[\leadsto \frac{0.16666666666666666 + \color{blue}{\beta \cdot 0.027777777777777776}}{2 + \left(\beta + \alpha\right)} \]

   13. Simplified 64.5%

       \[\leadsto \frac{\color{blue}{0.16666666666666666 + \beta \cdot 0.027777777777777776}}{2 + \left(\beta + \alpha\right)} \]

   if 1.69999999999999996 < beta

   1. Initial program 78.1%

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

   3. Simplified 92.8%

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

      1. clear-num 92.7%

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

      2. associate-+r+ 92.7%

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

      3. *-commutative 92.7%

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

      4. frac-times 85.6%

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

      5. *-un-lft-identity 85.6%

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

      6. +-commutative 85.6%

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

      7. *-commutative 85.6%

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

      8. associate-+r+ 85.6%

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

   5. Applied egg-rr 85.6%

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

      1. associate-/r* 92.8%

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

      2. associate-/l* 77.6%

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

      3. associate-*l/ 92.8%

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

      4. *-commutative 92.8%

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

      5. times-frac 99.7%

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

      6. associate-/r* 92.8%

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

      7. *-commutative 92.8%

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

      8. associate-/r* 99.7%

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

      9. +-commutative 99.7%

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

      10. +-commutative 99.7%

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

      11. +-commutative 99.7%

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

      12. +-commutative 99.7%

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

   7. Simplified 99.7%

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

      1. associate-*r/ 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+r+ 99.9%

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

      4. +-commutative 99.9%

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

      5. +-commutative 99.9%

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

      6. +-commutative 99.9%

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

      7. +-commutative 99.9%

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

      8. associate-+r+ 99.9%

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

      9. +-commutative 99.9%

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

      10. +-commutative 99.9%

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

   9. Applied egg-rr 99.9%

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

   10. Taylor expanded in beta around inf 74.9%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.7:\\ \;\;\;\;\frac{0.16666666666666666 + \beta \cdot 0.027777777777777776}{2 + \left(\beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \alpha\right) \cdot \frac{1}{\alpha + \left(\beta + 3\right)}}{2 + \left(\beta + \alpha\right)}\\ \end{array} \]

Alternative 9: 97.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 6.2], N[(N[(1.0 + alpha), $MachinePrecision] / N[(N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(alpha + 3.0), $MachinePrecision] * N[(2.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] * N[(1.0 / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.20000000000000018

   1. Initial program 99.9%

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

   3. Simplified 99.7%

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

      1. *-commutative 99.7%

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

      2. clear-num 99.7%

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

      3. frac-times 92.5%

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

      4. *-un-lft-identity 92.5%

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

      5. +-commutative 92.5%

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

   5. Applied egg-rr 92.5%

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

   6. Taylor expanded in beta around 0 90.6%

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

      1. +-commutative 90.6%

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

      2. +-commutative 90.6%

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

   8. Simplified 90.6%

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

   if 6.20000000000000018 < beta

   1. Initial program 78.1%

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

   3. Simplified 92.8%

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

      1. clear-num 92.7%

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

      2. associate-+r+ 92.7%

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

      3. *-commutative 92.7%

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

      4. frac-times 85.6%

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

      5. *-un-lft-identity 85.6%

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

      6. +-commutative 85.6%

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

      7. *-commutative 85.6%

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

      8. associate-+r+ 85.6%

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

   5. Applied egg-rr 85.6%

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

      1. associate-/r* 92.8%

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

      2. associate-/l* 77.6%

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

      3. associate-*l/ 92.8%

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

      4. *-commutative 92.8%

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

      5. times-frac 99.7%

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

      6. associate-/r* 92.8%

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

      7. *-commutative 92.8%

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

      8. associate-/r* 99.7%

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

      9. +-commutative 99.7%

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

      10. +-commutative 99.7%

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

      11. +-commutative 99.7%

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

      12. +-commutative 99.7%

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

   7. Simplified 99.7%

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

      1. associate-*r/ 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+r+ 99.9%

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

      4. +-commutative 99.9%

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

      5. +-commutative 99.9%

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

      6. +-commutative 99.9%

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

      7. +-commutative 99.9%

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

      8. associate-+r+ 99.9%

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

      9. +-commutative 99.9%

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

      10. +-commutative 99.9%

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

   9. Applied egg-rr 99.9%

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

   10. Taylor expanded in beta around inf 74.9%

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

4. Final simplification 85.8%

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

Alternative 10: 97.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.2:\\ \;\;\;\;\frac{0.16666666666666666 + \beta \cdot 0.027777777777777776}{2 + \left(\beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 - \alpha}{\beta} \cdot \frac{-1}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.2)
   (/
    (+ 0.16666666666666666 (* beta 0.027777777777777776))
    (+ 2.0 (+ beta alpha)))
   (* (/ (- -1.0 alpha) beta) (/ -1.0 beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.2) {
		tmp = (0.16666666666666666 + (beta * 0.027777777777777776)) / (2.0 + (beta + alpha));
	} else {
		tmp = ((-1.0 - alpha) / beta) * (-1.0 / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 6.2d0) then
        tmp = (0.16666666666666666d0 + (beta * 0.027777777777777776d0)) / (2.0d0 + (beta + alpha))
    else
        tmp = (((-1.0d0) - alpha) / beta) * ((-1.0d0) / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.2) {
		tmp = (0.16666666666666666 + (beta * 0.027777777777777776)) / (2.0 + (beta + alpha));
	} else {
		tmp = ((-1.0 - alpha) / beta) * (-1.0 / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 6.2:
		tmp = (0.16666666666666666 + (beta * 0.027777777777777776)) / (2.0 + (beta + alpha))
	else:
		tmp = ((-1.0 - alpha) / beta) * (-1.0 / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.2)
		tmp = Float64(Float64(0.16666666666666666 + Float64(beta * 0.027777777777777776)) / Float64(2.0 + Float64(beta + alpha)));
	else
		tmp = Float64(Float64(Float64(-1.0 - alpha) / beta) * Float64(-1.0 / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 6.2)
		tmp = (0.16666666666666666 + (beta * 0.027777777777777776)) / (2.0 + (beta + alpha));
	else
		tmp = ((-1.0 - alpha) / beta) * (-1.0 / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 6.2], N[(N[(0.16666666666666666 + N[(beta * 0.027777777777777776), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-1.0 - alpha), $MachinePrecision] / beta), $MachinePrecision] * N[(-1.0 / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.20000000000000018

   1. Initial program 99.9%

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

   3. Simplified 99.7%

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

      1. clear-num 99.6%

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

      2. associate-+r+ 99.6%

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

      3. *-commutative 99.6%

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

      4. frac-times 99.7%

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

      5. *-un-lft-identity 99.7%

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

      6. +-commutative 99.7%

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

      7. *-commutative 99.7%

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

      8. associate-+r+ 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. associate-/r* 99.7%

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

      2. associate-/l* 99.7%

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

      3. associate-*l/ 99.7%

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

      4. *-commutative 99.7%

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

      5. times-frac 99.8%

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

      6. associate-/r* 99.7%

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

      7. *-commutative 99.7%

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

      8. associate-/r* 99.8%

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

      9. +-commutative 99.8%

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

      10. +-commutative 99.8%

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

      11. +-commutative 99.8%

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

      12. +-commutative 99.8%

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

   7. Simplified 99.8%

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

      1. associate-*r/ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+r+ 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. associate-+r+ 99.8%

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

      9. +-commutative 99.8%

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

      10. +-commutative 99.8%

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

   9. Applied egg-rr 99.8%

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

   10. Taylor expanded in alpha around 0 64.8%

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

   11. Taylor expanded in beta around 0 64.5%

       \[\leadsto \frac{\color{blue}{0.16666666666666666 + 0.027777777777777776 \cdot \beta}}{2 + \left(\beta + \alpha\right)} \]
   12. Step-by-step derivation

       1. *-commutative 64.5%

          \[\leadsto \frac{0.16666666666666666 + \color{blue}{\beta \cdot 0.027777777777777776}}{2 + \left(\beta + \alpha\right)} \]

   13. Simplified 64.5%

       \[\leadsto \frac{\color{blue}{0.16666666666666666 + \beta \cdot 0.027777777777777776}}{2 + \left(\beta + \alpha\right)} \]

   if 6.20000000000000018 < beta

   1. Initial program 78.1%

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

   3. Simplified 92.8%

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

   4. Taylor expanded in beta around inf 74.1%

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

   5. Taylor expanded in beta around inf 73.8%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.2:\\ \;\;\;\;\frac{0.16666666666666666 + \beta \cdot 0.027777777777777776}{2 + \left(\beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 - \alpha}{\beta} \cdot \frac{-1}{\beta}\\ \end{array} \]

Alternative 11: 97.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 4.2:\\ \;\;\;\;\frac{0.16666666666666666 + \beta \cdot 0.027777777777777776}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (+ beta alpha))))
   (if (<= beta 4.2)
     (/ (+ 0.16666666666666666 (* beta 0.027777777777777776)) t_0)
     (/ (/ (+ 1.0 alpha) beta) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta + alpha);
	double tmp;
	if (beta <= 4.2) {
		tmp = (0.16666666666666666 + (beta * 0.027777777777777776)) / t_0;
	} 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) :: tmp
    t_0 = 2.0d0 + (beta + alpha)
    if (beta <= 4.2d0) then
        tmp = (0.16666666666666666d0 + (beta * 0.027777777777777776d0)) / t_0
    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 = 2.0 + (beta + alpha);
	double tmp;
	if (beta <= 4.2) {
		tmp = (0.16666666666666666 + (beta * 0.027777777777777776)) / t_0;
	} else {
		tmp = ((1.0 + alpha) / beta) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = 2.0 + (beta + alpha)
	tmp = 0
	if beta <= 4.2:
		tmp = (0.16666666666666666 + (beta * 0.027777777777777776)) / t_0
	else:
		tmp = ((1.0 + alpha) / beta) / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(beta + alpha))
	tmp = 0.0
	if (beta <= 4.2)
		tmp = Float64(Float64(0.16666666666666666 + Float64(beta * 0.027777777777777776)) / t_0);
	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 = 2.0 + (beta + alpha);
	tmp = 0.0;
	if (beta <= 4.2)
		tmp = (0.16666666666666666 + (beta * 0.027777777777777776)) / t_0;
	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[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 4.2], N[(N[(0.16666666666666666 + N[(beta * 0.027777777777777776), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.20000000000000018

   1. Initial program 99.9%

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

   3. Simplified 99.7%

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

      1. clear-num 99.6%

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

      2. associate-+r+ 99.6%

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

      3. *-commutative 99.6%

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

      4. frac-times 99.7%

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

      5. *-un-lft-identity 99.7%

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

      6. +-commutative 99.7%

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

      7. *-commutative 99.7%

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

      8. associate-+r+ 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. associate-/r* 99.7%

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

      2. associate-/l* 99.7%

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

      3. associate-*l/ 99.7%

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

      4. *-commutative 99.7%

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

      5. times-frac 99.8%

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

      6. associate-/r* 99.7%

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

      7. *-commutative 99.7%

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

      8. associate-/r* 99.8%

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

      9. +-commutative 99.8%

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

      10. +-commutative 99.8%

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

      11. +-commutative 99.8%

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

      12. +-commutative 99.8%

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

   7. Simplified 99.8%

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

      1. associate-*r/ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+r+ 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. associate-+r+ 99.8%

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

      9. +-commutative 99.8%

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

      10. +-commutative 99.8%

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

   9. Applied egg-rr 99.8%

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

   10. Taylor expanded in alpha around 0 64.8%

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

   11. Taylor expanded in beta around 0 64.5%

       \[\leadsto \frac{\color{blue}{0.16666666666666666 + 0.027777777777777776 \cdot \beta}}{2 + \left(\beta + \alpha\right)} \]
   12. Step-by-step derivation

       1. *-commutative 64.5%

          \[\leadsto \frac{0.16666666666666666 + \color{blue}{\beta \cdot 0.027777777777777776}}{2 + \left(\beta + \alpha\right)} \]

   13. Simplified 64.5%

       \[\leadsto \frac{\color{blue}{0.16666666666666666 + \beta \cdot 0.027777777777777776}}{2 + \left(\beta + \alpha\right)} \]

   if 4.20000000000000018 < beta

   1. Initial program 78.1%

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

   3. Simplified 92.8%

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

   4. Taylor expanded in beta around inf 74.1%

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

      1. associate-*l/ 74.2%

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

      2. +-commutative 74.2%

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

      3. associate-+r+ 74.2%

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

      4. +-commutative 74.2%

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

      5. +-commutative 74.2%

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

   6. Applied egg-rr 74.2%

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

      1. associate-*r/ 74.2%

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

      2. *-rgt-identity 74.2%

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

      3. +-commutative 74.2%

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

   8. Simplified 74.2%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.2:\\ \;\;\;\;\frac{0.16666666666666666 + \beta \cdot 0.027777777777777776}{2 + \left(\beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{2 + \left(\beta + \alpha\right)}\\ \end{array} \]

Alternative 12: 97.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.5:\\ \;\;\;\;\frac{0.16666666666666666 + \beta \cdot 0.027777777777777776}{2 + \left(\beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.5)
   (/
    (+ 0.16666666666666666 (* beta 0.027777777777777776))
    (+ 2.0 (+ beta alpha)))
   (/ (/ (+ 1.0 alpha) beta) (+ beta 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.5) {
		tmp = (0.16666666666666666 + (beta * 0.027777777777777776)) / (2.0 + (beta + 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 <= 3.5d0) then
        tmp = (0.16666666666666666d0 + (beta * 0.027777777777777776d0)) / (2.0d0 + (beta + 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 <= 3.5) {
		tmp = (0.16666666666666666 + (beta * 0.027777777777777776)) / (2.0 + (beta + 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 <= 3.5:
		tmp = (0.16666666666666666 + (beta * 0.027777777777777776)) / (2.0 + (beta + 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 <= 3.5)
		tmp = Float64(Float64(0.16666666666666666 + Float64(beta * 0.027777777777777776)) / Float64(2.0 + Float64(beta + 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 <= 3.5)
		tmp = (0.16666666666666666 + (beta * 0.027777777777777776)) / (2.0 + (beta + 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, 3.5], N[(N[(0.16666666666666666 + N[(beta * 0.027777777777777776), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(beta + 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 < 3.5

   1. Initial program 99.9%

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

   3. Simplified 99.7%

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

      1. clear-num 99.6%

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

      2. associate-+r+ 99.6%

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

      3. *-commutative 99.6%

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

      4. frac-times 99.7%

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

      5. *-un-lft-identity 99.7%

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

      6. +-commutative 99.7%

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

      7. *-commutative 99.7%

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

      8. associate-+r+ 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. associate-/r* 99.7%

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

      2. associate-/l* 99.7%

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

      3. associate-*l/ 99.7%

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

      4. *-commutative 99.7%

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

      5. times-frac 99.8%

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

      6. associate-/r* 99.7%

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

      7. *-commutative 99.7%

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

      8. associate-/r* 99.8%

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

      9. +-commutative 99.8%

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

      10. +-commutative 99.8%

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

      11. +-commutative 99.8%

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

      12. +-commutative 99.8%

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

   7. Simplified 99.8%

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

      1. associate-*r/ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+r+ 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. associate-+r+ 99.8%

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

      9. +-commutative 99.8%

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

      10. +-commutative 99.8%

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

   9. Applied egg-rr 99.8%

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

   10. Taylor expanded in alpha around 0 64.8%

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

   11. Taylor expanded in beta around 0 64.5%

       \[\leadsto \frac{\color{blue}{0.16666666666666666 + 0.027777777777777776 \cdot \beta}}{2 + \left(\beta + \alpha\right)} \]
   12. Step-by-step derivation

       1. *-commutative 64.5%

          \[\leadsto \frac{0.16666666666666666 + \color{blue}{\beta \cdot 0.027777777777777776}}{2 + \left(\beta + \alpha\right)} \]

   13. Simplified 64.5%

       \[\leadsto \frac{\color{blue}{0.16666666666666666 + \beta \cdot 0.027777777777777776}}{2 + \left(\beta + \alpha\right)} \]

   if 3.5 < beta

   1. Initial program 78.1%

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

   2. Taylor expanded in beta around -inf 74.2%

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

   3. Taylor expanded in alpha around 0 73.9%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.5:\\ \;\;\;\;\frac{0.16666666666666666 + \beta \cdot 0.027777777777777776}{2 + \left(\beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + 3}\\ \end{array} \]

Alternative 13: 97.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.3:\\ \;\;\;\;\frac{0.16666666666666666 + \beta \cdot 0.027777777777777776}{2 + \left(\beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.3)
   (/
    (+ 0.16666666666666666 (* beta 0.027777777777777776))
    (+ 2.0 (+ beta alpha)))
   (/ (/ (+ 1.0 alpha) beta) (+ alpha (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.3) {
		tmp = (0.16666666666666666 + (beta * 0.027777777777777776)) / (2.0 + (beta + 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 <= 3.3d0) then
        tmp = (0.16666666666666666d0 + (beta * 0.027777777777777776d0)) / (2.0d0 + (beta + 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 <= 3.3) {
		tmp = (0.16666666666666666 + (beta * 0.027777777777777776)) / (2.0 + (beta + 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 <= 3.3:
		tmp = (0.16666666666666666 + (beta * 0.027777777777777776)) / (2.0 + (beta + 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 <= 3.3)
		tmp = Float64(Float64(0.16666666666666666 + Float64(beta * 0.027777777777777776)) / Float64(2.0 + Float64(beta + 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 <= 3.3)
		tmp = (0.16666666666666666 + (beta * 0.027777777777777776)) / (2.0 + (beta + 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, 3.3], N[(N[(0.16666666666666666 + N[(beta * 0.027777777777777776), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(beta + 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 < 3.2999999999999998

   1. Initial program 99.9%

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

   3. Simplified 99.7%

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

      1. clear-num 99.6%

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

      2. associate-+r+ 99.6%

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

      3. *-commutative 99.6%

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

      4. frac-times 99.7%

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

      5. *-un-lft-identity 99.7%

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

      6. +-commutative 99.7%

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

      7. *-commutative 99.7%

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

      8. associate-+r+ 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. associate-/r* 99.7%

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

      2. associate-/l* 99.7%

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

      3. associate-*l/ 99.7%

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

      4. *-commutative 99.7%

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

      5. times-frac 99.8%

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

      6. associate-/r* 99.7%

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

      7. *-commutative 99.7%

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

      8. associate-/r* 99.8%

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

      9. +-commutative 99.8%

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

      10. +-commutative 99.8%

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

      11. +-commutative 99.8%

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

      12. +-commutative 99.8%

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

   7. Simplified 99.8%

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

      1. associate-*r/ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+r+ 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. associate-+r+ 99.8%

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

      9. +-commutative 99.8%

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

      10. +-commutative 99.8%

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

   9. Applied egg-rr 99.8%

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

   10. Taylor expanded in alpha around 0 64.8%

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

   11. Taylor expanded in beta around 0 64.5%

       \[\leadsto \frac{\color{blue}{0.16666666666666666 + 0.027777777777777776 \cdot \beta}}{2 + \left(\beta + \alpha\right)} \]
   12. Step-by-step derivation

       1. *-commutative 64.5%

          \[\leadsto \frac{0.16666666666666666 + \color{blue}{\beta \cdot 0.027777777777777776}}{2 + \left(\beta + \alpha\right)} \]

   13. Simplified 64.5%

       \[\leadsto \frac{\color{blue}{0.16666666666666666 + \beta \cdot 0.027777777777777776}}{2 + \left(\beta + \alpha\right)} \]

   if 3.2999999999999998 < beta

   1. Initial program 78.1%

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

   2. Taylor expanded in beta around -inf 74.2%

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

   3. Taylor expanded in alpha around 0 74.2%

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

      1. +-commutative 74.2%

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

      2. associate-+r+ 74.2%

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

      3. +-commutative 74.2%

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

   5. Simplified 74.2%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.3:\\ \;\;\;\;\frac{0.16666666666666666 + \beta \cdot 0.027777777777777776}{2 + \left(\beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 14: 96.6% accurate, 3.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 8.2:\\ \;\;\;\;\frac{0.16666666666666666}{2 + \left(\beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 - \alpha}{\beta} \cdot \frac{-1}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 8.2)
   (/ 0.16666666666666666 (+ 2.0 (+ beta alpha)))
   (* (/ (- -1.0 alpha) beta) (/ -1.0 beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 8.2) {
		tmp = 0.16666666666666666 / (2.0 + (beta + alpha));
	} else {
		tmp = ((-1.0 - alpha) / beta) * (-1.0 / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 8.2d0) then
        tmp = 0.16666666666666666d0 / (2.0d0 + (beta + alpha))
    else
        tmp = (((-1.0d0) - alpha) / beta) * ((-1.0d0) / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 8.2) {
		tmp = 0.16666666666666666 / (2.0 + (beta + alpha));
	} else {
		tmp = ((-1.0 - alpha) / beta) * (-1.0 / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 8.2:
		tmp = 0.16666666666666666 / (2.0 + (beta + alpha))
	else:
		tmp = ((-1.0 - alpha) / beta) * (-1.0 / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 8.2)
		tmp = Float64(0.16666666666666666 / Float64(2.0 + Float64(beta + alpha)));
	else
		tmp = Float64(Float64(Float64(-1.0 - alpha) / beta) * Float64(-1.0 / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 8.2)
		tmp = 0.16666666666666666 / (2.0 + (beta + alpha));
	else
		tmp = ((-1.0 - alpha) / beta) * (-1.0 / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 8.2], N[(0.16666666666666666 / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-1.0 - alpha), $MachinePrecision] / beta), $MachinePrecision] * N[(-1.0 / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 8.1999999999999993

   1. Initial program 99.9%

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

   3. Simplified 99.7%

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

      1. clear-num 99.6%

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

      2. associate-+r+ 99.6%

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

      3. *-commutative 99.6%

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

      4. frac-times 99.7%

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

      5. *-un-lft-identity 99.7%

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

      6. +-commutative 99.7%

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

      7. *-commutative 99.7%

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

      8. associate-+r+ 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. associate-/r* 99.7%

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

      2. associate-/l* 99.7%

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

      3. associate-*l/ 99.7%

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

      4. *-commutative 99.7%

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

      5. times-frac 99.8%

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

      6. associate-/r* 99.7%

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

      7. *-commutative 99.7%

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

      8. associate-/r* 99.8%

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

      9. +-commutative 99.8%

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

      10. +-commutative 99.8%

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

      11. +-commutative 99.8%

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

      12. +-commutative 99.8%

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

   7. Simplified 99.8%

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

      1. associate-*r/ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+r+ 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. associate-+r+ 99.8%

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

      9. +-commutative 99.8%

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

      10. +-commutative 99.8%

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

   9. Applied egg-rr 99.8%

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

   10. Taylor expanded in alpha around 0 64.8%

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

   11. Taylor expanded in beta around 0 63.7%

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

   if 8.1999999999999993 < beta

   1. Initial program 78.1%

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

   3. Simplified 92.8%

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

   4. Taylor expanded in beta around inf 74.1%

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

   5. Taylor expanded in beta around inf 73.8%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 8.2:\\ \;\;\;\;\frac{0.16666666666666666}{2 + \left(\beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 - \alpha}{\beta} \cdot \frac{-1}{\beta}\\ \end{array} \]

Alternative 15: 91.0% accurate, 3.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6:\\ \;\;\;\;\frac{0.16666666666666666}{2 + \left(\beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.0)
   (/ 0.16666666666666666 (+ 2.0 (+ beta alpha)))
   (/ 1.0 (* beta (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.0) {
		tmp = 0.16666666666666666 / (2.0 + (beta + 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 <= 6.0d0) then
        tmp = 0.16666666666666666d0 / (2.0d0 + (beta + 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 <= 6.0) {
		tmp = 0.16666666666666666 / (2.0 + (beta + 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 <= 6.0:
		tmp = 0.16666666666666666 / (2.0 + (beta + 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 <= 6.0)
		tmp = Float64(0.16666666666666666 / Float64(2.0 + Float64(beta + alpha)));
	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 <= 6.0)
		tmp = 0.16666666666666666 / (2.0 + (beta + 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, 6.0], N[(0.16666666666666666 / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(beta * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6

   1. Initial program 99.9%

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

   3. Simplified 99.7%

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

      1. clear-num 99.6%

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

      2. associate-+r+ 99.6%

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

      3. *-commutative 99.6%

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

      4. frac-times 99.7%

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

      5. *-un-lft-identity 99.7%

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

      6. +-commutative 99.7%

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

      7. *-commutative 99.7%

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

      8. associate-+r+ 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. associate-/r* 99.7%

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

      2. associate-/l* 99.7%

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

      3. associate-*l/ 99.7%

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

      4. *-commutative 99.7%

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

      5. times-frac 99.8%

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

      6. associate-/r* 99.7%

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

      7. *-commutative 99.7%

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

      8. associate-/r* 99.8%

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

      9. +-commutative 99.8%

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

      10. +-commutative 99.8%

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

      11. +-commutative 99.8%

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

      12. +-commutative 99.8%

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

   7. Simplified 99.8%

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

      1. associate-*r/ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+r+ 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. associate-+r+ 99.8%

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

      9. +-commutative 99.8%

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

      10. +-commutative 99.8%

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

   9. Applied egg-rr 99.8%

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

   10. Taylor expanded in alpha around 0 64.8%

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

   11. Taylor expanded in beta around 0 63.7%

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

   if 6 < beta

   1. Initial program 78.1%

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

   2. Taylor expanded in beta around -inf 74.2%

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

   3. Taylor expanded in alpha around 0 74.5%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6:\\ \;\;\;\;\frac{0.16666666666666666}{2 + \left(\beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \end{array} \]

Alternative 16: 91.4% accurate, 3.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 7.2:\\ \;\;\;\;\frac{0.16666666666666666}{2 + \left(\beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta + 2}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 7.2)
   (/ 0.16666666666666666 (+ 2.0 (+ beta alpha)))
   (/ (/ 1.0 beta) (+ beta 2.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 7.2) {
		tmp = 0.16666666666666666 / (2.0 + (beta + alpha));
	} else {
		tmp = (1.0 / beta) / (beta + 2.0);
	}
	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 <= 7.2d0) then
        tmp = 0.16666666666666666d0 / (2.0d0 + (beta + alpha))
    else
        tmp = (1.0d0 / beta) / (beta + 2.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 7.2) {
		tmp = 0.16666666666666666 / (2.0 + (beta + alpha));
	} else {
		tmp = (1.0 / beta) / (beta + 2.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 7.2:
		tmp = 0.16666666666666666 / (2.0 + (beta + alpha))
	else:
		tmp = (1.0 / beta) / (beta + 2.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 7.2)
		tmp = Float64(0.16666666666666666 / Float64(2.0 + Float64(beta + alpha)));
	else
		tmp = Float64(Float64(1.0 / beta) / Float64(beta + 2.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 7.2)
		tmp = 0.16666666666666666 / (2.0 + (beta + alpha));
	else
		tmp = (1.0 / beta) / (beta + 2.0);
	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, 7.2], N[(0.16666666666666666 / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / beta), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 7.20000000000000018

   1. Initial program 99.9%

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

   3. Simplified 99.7%

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

      1. clear-num 99.6%

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

      2. associate-+r+ 99.6%

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

      3. *-commutative 99.6%

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

      4. frac-times 99.7%

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

      5. *-un-lft-identity 99.7%

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

      6. +-commutative 99.7%

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

      7. *-commutative 99.7%

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

      8. associate-+r+ 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. associate-/r* 99.7%

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

      2. associate-/l* 99.7%

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

      3. associate-*l/ 99.7%

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

      4. *-commutative 99.7%

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

      5. times-frac 99.8%

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

      6. associate-/r* 99.7%

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

      7. *-commutative 99.7%

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

      8. associate-/r* 99.8%

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

      9. +-commutative 99.8%

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

      10. +-commutative 99.8%

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

      11. +-commutative 99.8%

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

      12. +-commutative 99.8%

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

   7. Simplified 99.8%

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

      1. associate-*r/ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+r+ 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. associate-+r+ 99.8%

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

      9. +-commutative 99.8%

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

      10. +-commutative 99.8%

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

   9. Applied egg-rr 99.8%

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

   10. Taylor expanded in alpha around 0 64.8%

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

   11. Taylor expanded in beta around 0 63.7%

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

   if 7.20000000000000018 < beta

   1. Initial program 78.1%

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

   3. Simplified 92.8%

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

   4. Taylor expanded in beta around inf 74.1%

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

      1. associate-*l/ 74.2%

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

      2. +-commutative 74.2%

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

      3. associate-+r+ 74.2%

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

      4. +-commutative 74.2%

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

      5. +-commutative 74.2%

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

   6. Applied egg-rr 74.2%

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

   7. Taylor expanded in alpha around 0 74.5%

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

      1. associate-/r* 74.7%

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

   9. Simplified 74.7%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 7.2:\\ \;\;\;\;\frac{0.16666666666666666}{2 + \left(\beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta + 2}\\ \end{array} \]

Alternative 17: 47.1% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.3%

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

3. Simplified 97.6%

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

   1. clear-num 97.6%

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

   2. associate-+r+ 97.6%

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

   3. *-commutative 97.6%

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

   4. frac-times 95.4%

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

   5. *-un-lft-identity 95.4%

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

   6. +-commutative 95.4%

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

   7. *-commutative 95.4%

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

   8. associate-+r+ 95.4%

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

5. Applied egg-rr 95.4%

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

   1. associate-/r* 97.6%

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

   2. associate-/l* 93.1%

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

   3. associate-*l/ 97.6%

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

   4. *-commutative 97.6%

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

   5. times-frac 99.8%

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

   6. associate-/r* 97.6%

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

   7. *-commutative 97.6%

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

   8. associate-/r* 99.8%

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

   9. +-commutative 99.8%

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

   10. +-commutative 99.8%

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

   11. +-commutative 99.8%

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

   12. +-commutative 99.8%

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

7. Simplified 99.8%

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

   1. associate-*r/ 99.8%

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

   2. +-commutative 99.8%

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

   3. associate-+r+ 99.8%

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

   4. +-commutative 99.8%

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

   5. +-commutative 99.8%

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

   6. +-commutative 99.8%

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

   7. +-commutative 99.8%

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

   8. associate-+r+ 99.8%

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

   9. +-commutative 99.8%

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

   10. +-commutative 99.8%

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

9. Applied egg-rr 99.8%

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

10. Taylor expanded in alpha around 0 70.0%

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

11. Taylor expanded in beta around 0 46.8%

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

12. Final simplification 46.8%

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

Alternative 18: 45.6% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.3%

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

3. Simplified 97.6%

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

   1. clear-num 97.6%

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

   2. associate-+r+ 97.6%

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

   3. *-commutative 97.6%

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

   4. frac-times 95.4%

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

   5. *-un-lft-identity 95.4%

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

   6. +-commutative 95.4%

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

   7. *-commutative 95.4%

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

   8. associate-+r+ 95.4%

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

5. Applied egg-rr 95.4%

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

   1. associate-/r* 97.6%

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

   2. associate-/l* 93.1%

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

   3. associate-*l/ 97.6%

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

   4. *-commutative 97.6%

      \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

   5. times-frac 99.8%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{\frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 3\right)}} \]

   6. associate-/r* 97.6%

      \[\leadsto \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \color{blue}{\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]

   7. *-commutative 97.6%

      \[\leadsto \color{blue}{\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}} \]

   8. associate-/r* 99.8%

      \[\leadsto \color{blue}{\frac{\frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 3\right)}} \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \]

   9. +-commutative 99.8%

      \[\leadsto \frac{\frac{\color{blue}{1 + \beta}}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 3\right)} \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \]

   10. +-commutative 99.8%

       \[\leadsto \frac{\frac{1 + \beta}{\alpha + \color{blue}{\left(2 + \beta\right)}}}{\alpha + \left(\beta + 3\right)} \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \]

   11. +-commutative 99.8%

       \[\leadsto \frac{\frac{1 + \beta}{\alpha + \left(2 + \beta\right)}}{\alpha + \color{blue}{\left(3 + \beta\right)}} \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \]

   12. +-commutative 99.8%

       \[\leadsto \frac{\frac{1 + \beta}{\alpha + \left(2 + \beta\right)}}{\alpha + \left(3 + \beta\right)} \cdot \frac{1 + \alpha}{\alpha + \color{blue}{\left(2 + \beta\right)}} \]

7. Simplified 99.8%

   \[\leadsto \color{blue}{\frac{\frac{1 + \beta}{\alpha + \left(2 + \beta\right)}}{\alpha + \left(3 + \beta\right)} \cdot \frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}} \]
8. Step-by-step derivation

   1. associate-*r/ 99.8%

      \[\leadsto \color{blue}{\frac{\frac{\frac{1 + \beta}{\alpha + \left(2 + \beta\right)}}{\alpha + \left(3 + \beta\right)} \cdot \left(1 + \alpha\right)}{\alpha + \left(2 + \beta\right)}} \]

   2. +-commutative 99.8%

      \[\leadsto \frac{\frac{\frac{1 + \beta}{\alpha + \color{blue}{\left(\beta + 2\right)}}}{\alpha + \left(3 + \beta\right)} \cdot \left(1 + \alpha\right)}{\alpha + \left(2 + \beta\right)} \]

   3. associate-+r+ 99.8%

      \[\leadsto \frac{\frac{\frac{1 + \beta}{\color{blue}{\left(\alpha + \beta\right) + 2}}}{\alpha + \left(3 + \beta\right)} \cdot \left(1 + \alpha\right)}{\alpha + \left(2 + \beta\right)} \]

   4. +-commutative 99.8%

      \[\leadsto \frac{\frac{\frac{1 + \beta}{\color{blue}{2 + \left(\alpha + \beta\right)}}}{\alpha + \left(3 + \beta\right)} \cdot \left(1 + \alpha\right)}{\alpha + \left(2 + \beta\right)} \]

   5. +-commutative 99.8%

      \[\leadsto \frac{\frac{\frac{1 + \beta}{2 + \color{blue}{\left(\beta + \alpha\right)}}}{\alpha + \left(3 + \beta\right)} \cdot \left(1 + \alpha\right)}{\alpha + \left(2 + \beta\right)} \]

   6. +-commutative 99.8%

      \[\leadsto \frac{\frac{\frac{1 + \beta}{2 + \left(\beta + \alpha\right)}}{\alpha + \color{blue}{\left(\beta + 3\right)}} \cdot \left(1 + \alpha\right)}{\alpha + \left(2 + \beta\right)} \]

   7. +-commutative 99.8%

      \[\leadsto \frac{\frac{\frac{1 + \beta}{2 + \left(\beta + \alpha\right)}}{\alpha + \left(\beta + 3\right)} \cdot \left(1 + \alpha\right)}{\alpha + \color{blue}{\left(\beta + 2\right)}} \]

   8. associate-+r+ 99.8%

      \[\leadsto \frac{\frac{\frac{1 + \beta}{2 + \left(\beta + \alpha\right)}}{\alpha + \left(\beta + 3\right)} \cdot \left(1 + \alpha\right)}{\color{blue}{\left(\alpha + \beta\right) + 2}} \]

   9. +-commutative 99.8%

      \[\leadsto \frac{\frac{\frac{1 + \beta}{2 + \left(\beta + \alpha\right)}}{\alpha + \left(\beta + 3\right)} \cdot \left(1 + \alpha\right)}{\color{blue}{2 + \left(\alpha + \beta\right)}} \]

   10. +-commutative 99.8%

       \[\leadsto \frac{\frac{\frac{1 + \beta}{2 + \left(\beta + \alpha\right)}}{\alpha + \left(\beta + 3\right)} \cdot \left(1 + \alpha\right)}{2 + \color{blue}{\left(\beta + \alpha\right)}} \]

9. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\frac{\frac{\frac{1 + \beta}{2 + \left(\beta + \alpha\right)}}{\alpha + \left(\beta + 3\right)} \cdot \left(1 + \alpha\right)}{2 + \left(\beta + \alpha\right)}} \]

10. Taylor expanded in alpha around 0 70.0%

    \[\leadsto \frac{\color{blue}{\frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(3 + \beta\right)}}}{2 + \left(\beta + \alpha\right)} \]

11. Taylor expanded in beta around 0 46.1%

    \[\leadsto \color{blue}{\frac{0.16666666666666666}{2 + \alpha}} \]

12. Final simplification 46.1%

    \[\leadsto \frac{0.16666666666666666}{2 + \alpha} \]

Alternative 19: 6.1% accurate, 11.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \frac{0.3333333333333333}{\beta} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta) :precision binary64 (/ 0.3333333333333333 beta))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return 0.3333333333333333 / beta;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = 0.3333333333333333d0 / beta
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return 0.3333333333333333 / beta;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return 0.3333333333333333 / beta

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return Float64(0.3333333333333333 / beta)
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = 0.3333333333333333 / beta;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := N[(0.3333333333333333 / beta), $MachinePrecision]

Derivation

1. Initial program 93.3%

   \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

2. Taylor expanded in beta around -inf 24.5%

   \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \alpha - 1}{\beta}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

3. Taylor expanded in alpha around 0 24.6%

   \[\leadsto \color{blue}{\frac{1}{\beta \cdot \left(3 + \beta\right)}} \]

4. Taylor expanded in beta around 0 4.2%

   \[\leadsto \color{blue}{\frac{0.3333333333333333}{\beta}} \]

5. Final simplification 4.2%

   \[\leadsto \frac{0.3333333333333333}{\beta} \]

Reproduce

herbie shell --seed 2023318 
(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)))