Octave 3.8, jcobi/3

Percentage Accurate: 94.5% → 99.5%

Time: 17.5s

Alternatives: 22

Speedup: 2.7×

• Unsound rule application detected in e-graph. Results may not be sound.

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, 0.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\beta + \alpha\right)\\ t_1 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\beta \leq 2 \cdot 10^{+135}:\\ \;\;\;\;\frac{\alpha + \left(\beta + \mathsf{fma}\left(\alpha, \beta, 1\right)\right)}{t_1} \cdot \frac{1}{t_1 \cdot \left(\alpha + \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{t_0}}{1 + t_0}\\ \end{array} \end{array} \]

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))) (t_1 (+ beta (+ alpha 2.0))))
   (if (<= beta 2e+135)
     (*
      (/ (+ alpha (+ beta (fma alpha beta 1.0))) t_1)
      (/ 1.0 (* t_1 (+ alpha (+ beta 3.0)))))
     (/ (/ (+ alpha 1.0) t_0) (+ 1.0 t_0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta + alpha);
	double t_1 = beta + (alpha + 2.0);
	double tmp;
	if (beta <= 2e+135) {
		tmp = ((alpha + (beta + fma(alpha, beta, 1.0))) / t_1) * (1.0 / (t_1 * (alpha + (beta + 3.0))));
	} else {
		tmp = ((alpha + 1.0) / t_0) / (1.0 + t_0);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(beta + alpha))
	t_1 = Float64(beta + Float64(alpha + 2.0))
	tmp = 0.0
	if (beta <= 2e+135)
		tmp = Float64(Float64(Float64(alpha + Float64(beta + fma(alpha, beta, 1.0))) / t_1) * Float64(1.0 / Float64(t_1 * Float64(alpha + Float64(beta + 3.0)))));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / t_0) / Float64(1.0 + t_0));
	end
	return 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]}, Block[{t$95$1 = N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2e+135], N[(N[(N[(alpha + N[(beta + N[(alpha * beta + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] * N[(1.0 / N[(t$95$1 * N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.99999999999999992e135

   1. Initial program 98.8%

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

      1. associate-/l/ 98.1%

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

      2. div-inv 98.2%

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

      3. associate-+l+ 98.2%

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

      4. associate-+l+ 98.2%

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

      5. *-commutative 98.2%

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

      6. fma-def 98.2%

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

      7. +-commutative 98.2%

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

      8. metadata-eval 98.2%

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

      9. associate-+l+ 98.2%

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

      10. *-commutative 98.2%

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

      11. +-commutative 98.2%

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

      12. metadata-eval 98.2%

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

      13. associate-+l+ 98.2%

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

      14. metadata-eval 98.2%

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

   3. Applied egg-rr 98.2%

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

   if 1.99999999999999992e135 < beta

   1. Initial program 75.9%

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

   2. Taylor expanded in beta around inf 94.0%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2 \cdot 10^{+135}:\\ \;\;\;\;\frac{\alpha + \left(\beta + \mathsf{fma}\left(\alpha, \beta, 1\right)\right)}{\beta + \left(\alpha + 2\right)} \cdot \frac{1}{\left(\beta + \left(\alpha + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{2 + \left(\beta + \alpha\right)}}{1 + \left(2 + \left(\beta + \alpha\right)\right)}\\ \end{array} \]

Alternative 2: 99.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 5 \cdot 10^{+103}:\\ \;\;\;\;\frac{\frac{1 + \left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)}{t_0}}{\left(3 + \left(\beta + \alpha\right)\right) \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta} + \left(\left(\frac{1}{\beta \cdot \beta} + \left(\frac{\alpha}{\beta} + \frac{\alpha}{\beta \cdot \beta}\right)\right) + \frac{-1 - \alpha}{\frac{\beta \cdot \beta}{4 + \alpha \cdot 2}}\right)}{1 + t_0}\\ \end{array} \end{array} \]

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 5e+103)
     (/
      (/ (+ 1.0 (+ alpha (+ beta (* beta alpha)))) t_0)
      (* (+ 3.0 (+ beta alpha)) t_0))
     (/
      (+
       (/ 1.0 beta)
       (+
        (+ (/ 1.0 (* beta beta)) (+ (/ alpha beta) (/ alpha (* beta beta))))
        (/ (- -1.0 alpha) (/ (* beta beta) (+ 4.0 (* alpha 2.0))))))
      (+ 1.0 t_0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta + alpha);
	double tmp;
	if (beta <= 5e+103) {
		tmp = ((1.0 + (alpha + (beta + (beta * alpha)))) / t_0) / ((3.0 + (beta + alpha)) * t_0);
	} else {
		tmp = ((1.0 / beta) + (((1.0 / (beta * beta)) + ((alpha / beta) + (alpha / (beta * beta)))) + ((-1.0 - alpha) / ((beta * beta) / (4.0 + (alpha * 2.0)))))) / (1.0 + 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 <= 5d+103) then
        tmp = ((1.0d0 + (alpha + (beta + (beta * alpha)))) / t_0) / ((3.0d0 + (beta + alpha)) * t_0)
    else
        tmp = ((1.0d0 / beta) + (((1.0d0 / (beta * beta)) + ((alpha / beta) + (alpha / (beta * beta)))) + (((-1.0d0) - alpha) / ((beta * beta) / (4.0d0 + (alpha * 2.0d0)))))) / (1.0d0 + 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 <= 5e+103) {
		tmp = ((1.0 + (alpha + (beta + (beta * alpha)))) / t_0) / ((3.0 + (beta + alpha)) * t_0);
	} else {
		tmp = ((1.0 / beta) + (((1.0 / (beta * beta)) + ((alpha / beta) + (alpha / (beta * beta)))) + ((-1.0 - alpha) / ((beta * beta) / (4.0 + (alpha * 2.0)))))) / (1.0 + t_0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = 2.0 + (beta + alpha)
	tmp = 0
	if beta <= 5e+103:
		tmp = ((1.0 + (alpha + (beta + (beta * alpha)))) / t_0) / ((3.0 + (beta + alpha)) * t_0)
	else:
		tmp = ((1.0 / beta) + (((1.0 / (beta * beta)) + ((alpha / beta) + (alpha / (beta * beta)))) + ((-1.0 - alpha) / ((beta * beta) / (4.0 + (alpha * 2.0)))))) / (1.0 + 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 <= 5e+103)
		tmp = Float64(Float64(Float64(1.0 + Float64(alpha + Float64(beta + Float64(beta * alpha)))) / t_0) / Float64(Float64(3.0 + Float64(beta + alpha)) * t_0));
	else
		tmp = Float64(Float64(Float64(1.0 / beta) + Float64(Float64(Float64(1.0 / Float64(beta * beta)) + Float64(Float64(alpha / beta) + Float64(alpha / Float64(beta * beta)))) + Float64(Float64(-1.0 - alpha) / Float64(Float64(beta * beta) / Float64(4.0 + Float64(alpha * 2.0)))))) / Float64(1.0 + 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 <= 5e+103)
		tmp = ((1.0 + (alpha + (beta + (beta * alpha)))) / t_0) / ((3.0 + (beta + alpha)) * t_0);
	else
		tmp = ((1.0 / beta) + (((1.0 / (beta * beta)) + ((alpha / beta) + (alpha / (beta * beta)))) + ((-1.0 - alpha) / ((beta * beta) / (4.0 + (alpha * 2.0)))))) / (1.0 + 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, 5e+103], N[(N[(N[(1.0 + N[(alpha + N[(beta + N[(beta * alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / beta), $MachinePrecision] + N[(N[(N[(1.0 / N[(beta * beta), $MachinePrecision]), $MachinePrecision] + N[(N[(alpha / beta), $MachinePrecision] + N[(alpha / N[(beta * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 - alpha), $MachinePrecision] / N[(N[(beta * beta), $MachinePrecision] / N[(4.0 + N[(alpha * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 5e103

   1. Initial program 99.3%

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

      1. metadata-eval 99.3%

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

      2. flip3-+ 79.8%

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

      3. div-inv 79.7%

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

      4. metadata-eval 79.7%

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left({\left(\alpha + \beta\right)}^{3} + \color{blue}{8}\right) \cdot \frac{1}{\left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right) + \left(2 \cdot 2 - \left(\alpha + \beta\right) \cdot 2\right)} + 1} \]

      5. metadata-eval 79.7%

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

      6. metadata-eval 79.7%

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

      7. +-commutative 79.7%

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

      8. metadata-eval 79.7%

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

      9. metadata-eval 79.7%

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{8} + {\left(\alpha + \beta\right)}^{3}\right) \cdot \frac{1}{\left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right) + \left(2 \cdot 2 - \left(\alpha + \beta\right) \cdot 2\right)} + 1} \]

      10. associate-+r- 79.7%

          \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(8 + {\left(\alpha + \beta\right)}^{3}\right) \cdot \frac{1}{\color{blue}{\left(\left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right) + 2 \cdot 2\right) - \left(\alpha + \beta\right) \cdot 2}} + 1} \]

      11. pow2 79.7%

          \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(8 + {\left(\alpha + \beta\right)}^{3}\right) \cdot \frac{1}{\left(\color{blue}{{\left(\alpha + \beta\right)}^{2}} + 2 \cdot 2\right) - \left(\alpha + \beta\right) \cdot 2} + 1} \]

      12. metadata-eval 79.7%

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

   3. Applied egg-rr 79.7%

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

      1. *-un-lft-identity 79.7%

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

      2. associate-/l/ 79.7%

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

      3. +-commutative 79.7%

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

      4. associate-+l+ 79.7%

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

      5. metadata-eval 79.7%

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

      6. +-commutative 79.7%

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

      7. +-commutative 79.7%

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

   5. Applied egg-rr 98.6%

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

   if 5e103 < beta

   1. Initial program 77.9%

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

   2. Taylor expanded in beta around inf 84.1%

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

      1. associate--l+ 84.1%

         \[\leadsto \frac{\color{blue}{\frac{1}{\beta} + \left(\left(\frac{1}{{\beta}^{2}} + \left(\frac{\alpha}{\beta} + \frac{\alpha}{{\beta}^{2}}\right)\right) - \frac{\left(1 + \alpha\right) \cdot \left(4 + 2 \cdot \alpha\right)}{{\beta}^{2}}\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      2. unpow2 84.1%

         \[\leadsto \frac{\frac{1}{\beta} + \left(\left(\frac{1}{\color{blue}{\beta \cdot \beta}} + \left(\frac{\alpha}{\beta} + \frac{\alpha}{{\beta}^{2}}\right)\right) - \frac{\left(1 + \alpha\right) \cdot \left(4 + 2 \cdot \alpha\right)}{{\beta}^{2}}\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      3. unpow2 84.1%

         \[\leadsto \frac{\frac{1}{\beta} + \left(\left(\frac{1}{\beta \cdot \beta} + \left(\frac{\alpha}{\beta} + \frac{\alpha}{\color{blue}{\beta \cdot \beta}}\right)\right) - \frac{\left(1 + \alpha\right) \cdot \left(4 + 2 \cdot \alpha\right)}{{\beta}^{2}}\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      4. associate-/l* 93.0%

         \[\leadsto \frac{\frac{1}{\beta} + \left(\left(\frac{1}{\beta \cdot \beta} + \left(\frac{\alpha}{\beta} + \frac{\alpha}{\beta \cdot \beta}\right)\right) - \color{blue}{\frac{1 + \alpha}{\frac{{\beta}^{2}}{4 + 2 \cdot \alpha}}}\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      5. unpow2 93.0%

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

   4. Simplified 93.0%

      \[\leadsto \frac{\color{blue}{\frac{1}{\beta} + \left(\left(\frac{1}{\beta \cdot \beta} + \left(\frac{\alpha}{\beta} + \frac{\alpha}{\beta \cdot \beta}\right)\right) - \frac{1 + \alpha}{\frac{\beta \cdot \beta}{4 + 2 \cdot \alpha}}\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5 \cdot 10^{+103}:\\ \;\;\;\;\frac{\frac{1 + \left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)}{2 + \left(\beta + \alpha\right)}}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(2 + \left(\beta + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta} + \left(\left(\frac{1}{\beta \cdot \beta} + \left(\frac{\alpha}{\beta} + \frac{\alpha}{\beta \cdot \beta}\right)\right) + \frac{-1 - \alpha}{\frac{\beta \cdot \beta}{4 + \alpha \cdot 2}}\right)}{1 + \left(2 + \left(\beta + \alpha\right)\right)}\\ \end{array} \]

Alternative 3: 99.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 5.5 \cdot 10^{+87}:\\ \;\;\;\;\frac{1 + \left(\left(\beta + \alpha\right) + \beta \cdot \alpha\right)}{t_0 \cdot \left(\left(3 + \left(\beta + \alpha\right)\right) \cdot t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{t_0}}{1 + t_0}\\ \end{array} \end{array} \]

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 5.5e+87)
     (/
      (+ 1.0 (+ (+ beta alpha) (* beta alpha)))
      (* t_0 (* (+ 3.0 (+ beta alpha)) t_0)))
     (/ (/ (+ alpha 1.0) t_0) (+ 1.0 t_0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta + alpha);
	double tmp;
	if (beta <= 5.5e+87) {
		tmp = (1.0 + ((beta + alpha) + (beta * alpha))) / (t_0 * ((3.0 + (beta + alpha)) * t_0));
	} else {
		tmp = ((alpha + 1.0) / t_0) / (1.0 + 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 <= 5.5d+87) then
        tmp = (1.0d0 + ((beta + alpha) + (beta * alpha))) / (t_0 * ((3.0d0 + (beta + alpha)) * t_0))
    else
        tmp = ((alpha + 1.0d0) / t_0) / (1.0d0 + 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 <= 5.5e+87) {
		tmp = (1.0 + ((beta + alpha) + (beta * alpha))) / (t_0 * ((3.0 + (beta + alpha)) * t_0));
	} else {
		tmp = ((alpha + 1.0) / t_0) / (1.0 + t_0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = 2.0 + (beta + alpha)
	tmp = 0
	if beta <= 5.5e+87:
		tmp = (1.0 + ((beta + alpha) + (beta * alpha))) / (t_0 * ((3.0 + (beta + alpha)) * t_0))
	else:
		tmp = ((alpha + 1.0) / t_0) / (1.0 + 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 <= 5.5e+87)
		tmp = Float64(Float64(1.0 + Float64(Float64(beta + alpha) + Float64(beta * alpha))) / Float64(t_0 * Float64(Float64(3.0 + Float64(beta + alpha)) * t_0)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / t_0) / Float64(1.0 + 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 <= 5.5e+87)
		tmp = (1.0 + ((beta + alpha) + (beta * alpha))) / (t_0 * ((3.0 + (beta + alpha)) * t_0));
	else
		tmp = ((alpha + 1.0) / t_0) / (1.0 + 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, 5.5e+87], N[(N[(1.0 + N[(N[(beta + alpha), $MachinePrecision] + N[(beta * alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[(N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.50000000000000022e87

   1. Initial program 99.8%

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

      1. metadata-eval 99.8%

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

      2. flip3-+ 79.8%

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

      3. div-inv 79.7%

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

      4. metadata-eval 79.7%

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left({\left(\alpha + \beta\right)}^{3} + \color{blue}{8}\right) \cdot \frac{1}{\left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right) + \left(2 \cdot 2 - \left(\alpha + \beta\right) \cdot 2\right)} + 1} \]

      5. metadata-eval 79.7%

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

      6. metadata-eval 79.7%

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

      7. +-commutative 79.7%

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

      8. metadata-eval 79.7%

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

      9. metadata-eval 79.7%

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{8} + {\left(\alpha + \beta\right)}^{3}\right) \cdot \frac{1}{\left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right) + \left(2 \cdot 2 - \left(\alpha + \beta\right) \cdot 2\right)} + 1} \]

      10. associate-+r- 79.7%

          \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(8 + {\left(\alpha + \beta\right)}^{3}\right) \cdot \frac{1}{\color{blue}{\left(\left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right) + 2 \cdot 2\right) - \left(\alpha + \beta\right) \cdot 2}} + 1} \]

      11. pow2 79.7%

          \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(8 + {\left(\alpha + \beta\right)}^{3}\right) \cdot \frac{1}{\left(\color{blue}{{\left(\alpha + \beta\right)}^{2}} + 2 \cdot 2\right) - \left(\alpha + \beta\right) \cdot 2} + 1} \]

      12. metadata-eval 79.7%

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

   3. Applied egg-rr 79.7%

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

      1. *-un-lft-identity 79.7%

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

      2. associate-/l/ 79.7%

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

      3. +-commutative 79.7%

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

      4. associate-+l+ 79.7%

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

      5. metadata-eval 79.7%

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

      6. +-commutative 79.7%

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

      7. +-commutative 79.7%

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

   5. Applied egg-rr 99.1%

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

      1. *-lft-identity 99.1%

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

      2. associate-/l/ 90.5%

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

      3. associate-+r+ 90.5%

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

      4. *-commutative 90.5%

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

      5. +-commutative 90.5%

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

      6. +-commutative 90.5%

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

      7. +-commutative 90.5%

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

      8. +-commutative 90.5%

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

   7. Simplified 90.5%

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

   if 5.50000000000000022e87 < beta

   1. Initial program 78.0%

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

   2. Taylor expanded in beta around inf 92.3%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.5 \cdot 10^{+87}:\\ \;\;\;\;\frac{1 + \left(\left(\beta + \alpha\right) + \beta \cdot \alpha\right)}{\left(2 + \left(\beta + \alpha\right)\right) \cdot \left(\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(2 + \left(\beta + \alpha\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{2 + \left(\beta + \alpha\right)}}{1 + \left(2 + \left(\beta + \alpha\right)\right)}\\ \end{array} \]

Alternative 4: 99.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 4 \cdot 10^{+103}:\\ \;\;\;\;\frac{\frac{1 + \left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)}{t_0}}{\left(3 + \left(\beta + \alpha\right)\right) \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{t_0}}{1 + t_0}\\ \end{array} \end{array} \]

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 4e+103)
     (/
      (/ (+ 1.0 (+ alpha (+ beta (* beta alpha)))) t_0)
      (* (+ 3.0 (+ beta alpha)) t_0))
     (/ (/ (+ alpha 1.0) t_0) (+ 1.0 t_0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta + alpha);
	double tmp;
	if (beta <= 4e+103) {
		tmp = ((1.0 + (alpha + (beta + (beta * alpha)))) / t_0) / ((3.0 + (beta + alpha)) * t_0);
	} else {
		tmp = ((alpha + 1.0) / t_0) / (1.0 + 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 <= 4d+103) then
        tmp = ((1.0d0 + (alpha + (beta + (beta * alpha)))) / t_0) / ((3.0d0 + (beta + alpha)) * t_0)
    else
        tmp = ((alpha + 1.0d0) / t_0) / (1.0d0 + 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 <= 4e+103) {
		tmp = ((1.0 + (alpha + (beta + (beta * alpha)))) / t_0) / ((3.0 + (beta + alpha)) * t_0);
	} else {
		tmp = ((alpha + 1.0) / t_0) / (1.0 + t_0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = 2.0 + (beta + alpha)
	tmp = 0
	if beta <= 4e+103:
		tmp = ((1.0 + (alpha + (beta + (beta * alpha)))) / t_0) / ((3.0 + (beta + alpha)) * t_0)
	else:
		tmp = ((alpha + 1.0) / t_0) / (1.0 + 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 <= 4e+103)
		tmp = Float64(Float64(Float64(1.0 + Float64(alpha + Float64(beta + Float64(beta * alpha)))) / t_0) / Float64(Float64(3.0 + Float64(beta + alpha)) * t_0));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / t_0) / Float64(1.0 + 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 <= 4e+103)
		tmp = ((1.0 + (alpha + (beta + (beta * alpha)))) / t_0) / ((3.0 + (beta + alpha)) * t_0);
	else
		tmp = ((alpha + 1.0) / t_0) / (1.0 + 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, 4e+103], N[(N[(N[(1.0 + N[(alpha + N[(beta + N[(beta * alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 4e103

   1. Initial program 99.3%

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

      1. metadata-eval 99.3%

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

      2. flip3-+ 79.8%

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

      3. div-inv 79.7%

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

      4. metadata-eval 79.7%

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left({\left(\alpha + \beta\right)}^{3} + \color{blue}{8}\right) \cdot \frac{1}{\left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right) + \left(2 \cdot 2 - \left(\alpha + \beta\right) \cdot 2\right)} + 1} \]

      5. metadata-eval 79.7%

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

      6. metadata-eval 79.7%

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

      7. +-commutative 79.7%

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

      8. metadata-eval 79.7%

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

      9. metadata-eval 79.7%

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{8} + {\left(\alpha + \beta\right)}^{3}\right) \cdot \frac{1}{\left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right) + \left(2 \cdot 2 - \left(\alpha + \beta\right) \cdot 2\right)} + 1} \]

      10. associate-+r- 79.7%

          \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(8 + {\left(\alpha + \beta\right)}^{3}\right) \cdot \frac{1}{\color{blue}{\left(\left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right) + 2 \cdot 2\right) - \left(\alpha + \beta\right) \cdot 2}} + 1} \]

      11. pow2 79.7%

          \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(8 + {\left(\alpha + \beta\right)}^{3}\right) \cdot \frac{1}{\left(\color{blue}{{\left(\alpha + \beta\right)}^{2}} + 2 \cdot 2\right) - \left(\alpha + \beta\right) \cdot 2} + 1} \]

      12. metadata-eval 79.7%

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

   3. Applied egg-rr 79.7%

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

      1. *-un-lft-identity 79.7%

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

      2. associate-/l/ 79.7%

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

      3. +-commutative 79.7%

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

      4. associate-+l+ 79.7%

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

      5. metadata-eval 79.7%

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

      6. +-commutative 79.7%

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

      7. +-commutative 79.7%

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

   5. Applied egg-rr 98.6%

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

   if 4e103 < beta

   1. Initial program 77.9%

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

   2. Taylor expanded in beta around inf 93.2%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4 \cdot 10^{+103}:\\ \;\;\;\;\frac{\frac{1 + \left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)}{2 + \left(\beta + \alpha\right)}}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(2 + \left(\beta + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{2 + \left(\beta + \alpha\right)}}{1 + \left(2 + \left(\beta + \alpha\right)\right)}\\ \end{array} \]

Alternative 5: 97.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 7.5 \cdot 10^{-57}:\\ \;\;\;\;\frac{\frac{\frac{\beta + 1}{\beta + 2}}{\beta + 2}}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{2 + \left(\beta + \alpha\right)} \cdot \frac{1}{\beta + \left(\alpha + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 7.5e-57)
   (/ (/ (/ (+ beta 1.0) (+ beta 2.0)) (+ beta 2.0)) (+ beta 3.0))
   (*
    (/ (+ alpha 1.0) (+ 2.0 (+ beta alpha)))
    (/ 1.0 (+ beta (+ alpha 3.0))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 7.5e-57) {
		tmp = (((beta + 1.0) / (beta + 2.0)) / (beta + 2.0)) / (beta + 3.0);
	} else {
		tmp = ((alpha + 1.0) / (2.0 + (beta + alpha))) * (1.0 / (beta + (alpha + 3.0)));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 7.5d-57) then
        tmp = (((beta + 1.0d0) / (beta + 2.0d0)) / (beta + 2.0d0)) / (beta + 3.0d0)
    else
        tmp = ((alpha + 1.0d0) / (2.0d0 + (beta + alpha))) * (1.0d0 / (beta + (alpha + 3.0d0)))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 7.5e-57) {
		tmp = (((beta + 1.0) / (beta + 2.0)) / (beta + 2.0)) / (beta + 3.0);
	} else {
		tmp = ((alpha + 1.0) / (2.0 + (beta + alpha))) * (1.0 / (beta + (alpha + 3.0)));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if alpha <= 7.5e-57:
		tmp = (((beta + 1.0) / (beta + 2.0)) / (beta + 2.0)) / (beta + 3.0)
	else:
		tmp = ((alpha + 1.0) / (2.0 + (beta + alpha))) * (1.0 / (beta + (alpha + 3.0)))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 7.5e-57)
		tmp = Float64(Float64(Float64(Float64(beta + 1.0) / Float64(beta + 2.0)) / Float64(beta + 2.0)) / Float64(beta + 3.0));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(2.0 + Float64(beta + alpha))) * Float64(1.0 / Float64(beta + Float64(alpha + 3.0))));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 7.5e-57)
		tmp = (((beta + 1.0) / (beta + 2.0)) / (beta + 2.0)) / (beta + 3.0);
	else
		tmp = ((alpha + 1.0) / (2.0 + (beta + alpha))) * (1.0 / (beta + (alpha + 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[alpha, 7.5e-57], N[(N[(N[(N[(beta + 1.0), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 7.49999999999999973e-57

   1. Initial program 99.9%

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

   2. Taylor expanded in alpha around 0 99.7%

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

      1. *-un-lft-identity 99.7%

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

      2. associate-/l/ 98.9%

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

      3. metadata-eval 98.9%

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

      4. associate-+l+ 98.9%

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

      5. +-commutative 98.9%

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

      6. metadata-eval 98.9%

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

      7. metadata-eval 98.9%

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

      8. +-commutative 98.9%

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

      9. +-commutative 98.9%

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

   4. Applied egg-rr 98.9%

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

      1. associate-*r/ 98.9%

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

      2. times-frac 99.6%

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

      3. *-lft-identity 99.6%

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

      4. *-commutative 99.6%

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

      5. associate-*r/ 99.7%

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

      6. *-commutative 99.7%

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

      7. times-frac 98.9%

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

      8. associate-*r/ 98.9%

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

      9. *-lft-identity 98.9%

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

      10. *-commutative 98.9%

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

      11. associate-*r/ 98.9%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in alpha around 0 99.9%

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

   8. Taylor expanded in alpha around 0 99.7%

      \[\leadsto \frac{\frac{\frac{\beta + 1}{\beta + 2}}{2 + \beta}}{\color{blue}{\beta + 3}} \]

   if 7.49999999999999973e-57 < alpha

   1. Initial program 83.9%

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

   2. Taylor expanded in beta around inf 25.9%

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

      1. div-inv 25.9%

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

      2. metadata-eval 25.9%

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

      3. +-commutative 25.9%

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

      4. +-commutative 25.9%

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

      5. metadata-eval 25.9%

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

      6. associate-+l+ 25.9%

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

      7. metadata-eval 25.9%

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

      8. +-commutative 25.9%

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

      9. associate-+r+ 25.9%

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

      10. +-commutative 25.9%

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

   4. Applied egg-rr 25.9%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 7.5 \cdot 10^{-57}:\\ \;\;\;\;\frac{\frac{\frac{\beta + 1}{\beta + 2}}{\beta + 2}}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{2 + \left(\beta + \alpha\right)} \cdot \frac{1}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]

Alternative 6: 98.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 1.56 \cdot 10^{+41}:\\ \;\;\;\;\frac{\beta + 1}{t_0 \cdot \left(\left(\beta + 3\right) \cdot \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{1 + t_0}\\ \end{array} \end{array} \]

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 1.56e+41)
     (/ (+ beta 1.0) (* t_0 (* (+ beta 3.0) (+ beta 2.0))))
     (/ (/ (+ alpha 1.0) beta) (+ 1.0 t_0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta + alpha);
	double tmp;
	if (beta <= 1.56e+41) {
		tmp = (beta + 1.0) / (t_0 * ((beta + 3.0) * (beta + 2.0)));
	} else {
		tmp = ((alpha + 1.0) / beta) / (1.0 + 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 <= 1.56d+41) then
        tmp = (beta + 1.0d0) / (t_0 * ((beta + 3.0d0) * (beta + 2.0d0)))
    else
        tmp = ((alpha + 1.0d0) / beta) / (1.0d0 + 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 <= 1.56e+41) {
		tmp = (beta + 1.0) / (t_0 * ((beta + 3.0) * (beta + 2.0)));
	} else {
		tmp = ((alpha + 1.0) / beta) / (1.0 + t_0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = 2.0 + (beta + alpha)
	tmp = 0
	if beta <= 1.56e+41:
		tmp = (beta + 1.0) / (t_0 * ((beta + 3.0) * (beta + 2.0)))
	else:
		tmp = ((alpha + 1.0) / beta) / (1.0 + 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 <= 1.56e+41)
		tmp = Float64(Float64(beta + 1.0) / Float64(t_0 * Float64(Float64(beta + 3.0) * Float64(beta + 2.0))));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / Float64(1.0 + 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 <= 1.56e+41)
		tmp = (beta + 1.0) / (t_0 * ((beta + 3.0) * (beta + 2.0)));
	else
		tmp = ((alpha + 1.0) / beta) / (1.0 + 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, 1.56e+41], N[(N[(beta + 1.0), $MachinePrecision] / N[(t$95$0 * N[(N[(beta + 3.0), $MachinePrecision] * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.56e41

   1. Initial program 99.8%

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

   2. Taylor expanded in alpha around 0 83.1%

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

      1. *-un-lft-identity 83.1%

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

      2. associate-/l/ 83.0%

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

      3. metadata-eval 83.0%

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

      4. associate-+l+ 83.0%

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

      5. +-commutative 83.0%

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

      6. metadata-eval 83.0%

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

      7. metadata-eval 83.0%

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

      8. +-commutative 83.0%

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

      9. +-commutative 83.0%

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

   4. Applied egg-rr 83.0%

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

      1. associate-*r/ 83.0%

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

      2. times-frac 83.1%

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

      3. *-lft-identity 83.1%

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

      4. *-commutative 83.1%

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

      5. associate-*r/ 83.1%

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

      6. *-commutative 83.1%

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

      7. times-frac 83.1%

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

      8. associate-*r/ 83.1%

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

      9. *-lft-identity 83.1%

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

      10. *-commutative 83.1%

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

      11. associate-*r/ 83.1%

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

   6. Simplified 83.1%

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

      1. *-un-lft-identity 83.1%

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

      2. associate-/l/ 83.1%

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

      3. +-commutative 83.1%

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

      4. associate-+r+ 83.1%

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

      5. +-commutative 83.1%

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

      6. +-commutative 83.1%

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

   8. Applied egg-rr 83.1%

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

      1. *-lft-identity 83.1%

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

      2. associate-/l/ 83.1%

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

      3. *-commutative 83.1%

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

      4. +-commutative 83.1%

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

      5. associate-+l+ 83.1%

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

      6. +-commutative 83.1%

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

   10. Simplified 83.1%

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

   11. Taylor expanded in alpha around 0 70.7%

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

   if 1.56e41 < beta

   1. Initial program 81.9%

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

   2. Taylor expanded in beta around inf 89.9%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.56 \cdot 10^{+41}:\\ \;\;\;\;\frac{\beta + 1}{\left(2 + \left(\beta + \alpha\right)\right) \cdot \left(\left(\beta + 3\right) \cdot \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{1 + \left(2 + \left(\beta + \alpha\right)\right)}\\ \end{array} \]

Alternative 7: 98.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 7.5 \cdot 10^{-57}:\\ \;\;\;\;\frac{\frac{\frac{\beta + 1}{\beta + 2}}{\beta + 2}}{3 + \left(\beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{2 + \left(\beta + \alpha\right)} \cdot \frac{1}{\beta + \left(\alpha + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 7.5e-57)
   (/ (/ (/ (+ beta 1.0) (+ beta 2.0)) (+ beta 2.0)) (+ 3.0 (+ beta alpha)))
   (*
    (/ (+ alpha 1.0) (+ 2.0 (+ beta alpha)))
    (/ 1.0 (+ beta (+ alpha 3.0))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 7.5e-57) {
		tmp = (((beta + 1.0) / (beta + 2.0)) / (beta + 2.0)) / (3.0 + (beta + alpha));
	} else {
		tmp = ((alpha + 1.0) / (2.0 + (beta + alpha))) * (1.0 / (beta + (alpha + 3.0)));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 7.5d-57) then
        tmp = (((beta + 1.0d0) / (beta + 2.0d0)) / (beta + 2.0d0)) / (3.0d0 + (beta + alpha))
    else
        tmp = ((alpha + 1.0d0) / (2.0d0 + (beta + alpha))) * (1.0d0 / (beta + (alpha + 3.0d0)))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 7.5e-57) {
		tmp = (((beta + 1.0) / (beta + 2.0)) / (beta + 2.0)) / (3.0 + (beta + alpha));
	} else {
		tmp = ((alpha + 1.0) / (2.0 + (beta + alpha))) * (1.0 / (beta + (alpha + 3.0)));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if alpha <= 7.5e-57:
		tmp = (((beta + 1.0) / (beta + 2.0)) / (beta + 2.0)) / (3.0 + (beta + alpha))
	else:
		tmp = ((alpha + 1.0) / (2.0 + (beta + alpha))) * (1.0 / (beta + (alpha + 3.0)))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 7.5e-57)
		tmp = Float64(Float64(Float64(Float64(beta + 1.0) / Float64(beta + 2.0)) / Float64(beta + 2.0)) / Float64(3.0 + Float64(beta + alpha)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(2.0 + Float64(beta + alpha))) * Float64(1.0 / Float64(beta + Float64(alpha + 3.0))));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 7.5e-57)
		tmp = (((beta + 1.0) / (beta + 2.0)) / (beta + 2.0)) / (3.0 + (beta + alpha));
	else
		tmp = ((alpha + 1.0) / (2.0 + (beta + alpha))) * (1.0 / (beta + (alpha + 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[alpha, 7.5e-57], N[(N[(N[(N[(beta + 1.0), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 7.49999999999999973e-57

   1. Initial program 99.9%

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

   2. Taylor expanded in alpha around 0 99.7%

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

      1. *-un-lft-identity 99.7%

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

      2. associate-/l/ 98.9%

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

      3. metadata-eval 98.9%

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

      4. associate-+l+ 98.9%

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

      5. +-commutative 98.9%

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

      6. metadata-eval 98.9%

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

      7. metadata-eval 98.9%

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

      8. +-commutative 98.9%

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

      9. +-commutative 98.9%

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

   4. Applied egg-rr 98.9%

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

      1. associate-*r/ 98.9%

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

      2. times-frac 99.6%

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

      3. *-lft-identity 99.6%

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

      4. *-commutative 99.6%

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

      5. associate-*r/ 99.7%

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

      6. *-commutative 99.7%

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

      7. times-frac 98.9%

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

      8. associate-*r/ 98.9%

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

      9. *-lft-identity 98.9%

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

      10. *-commutative 98.9%

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

      11. associate-*r/ 98.9%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in alpha around 0 99.9%

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

   if 7.49999999999999973e-57 < alpha

   1. Initial program 83.9%

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

   2. Taylor expanded in beta around inf 25.9%

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

      1. div-inv 25.9%

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

      2. metadata-eval 25.9%

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

      3. +-commutative 25.9%

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

      4. +-commutative 25.9%

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

      5. metadata-eval 25.9%

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

      6. associate-+l+ 25.9%

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

      7. metadata-eval 25.9%

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

      8. +-commutative 25.9%

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

      9. associate-+r+ 25.9%

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

      10. +-commutative 25.9%

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

   4. Applied egg-rr 25.9%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 7.5 \cdot 10^{-57}:\\ \;\;\;\;\frac{\frac{\frac{\beta + 1}{\beta + 2}}{\beta + 2}}{3 + \left(\beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{2 + \left(\beta + \alpha\right)} \cdot \frac{1}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]

Alternative 8: 98.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 5 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{\frac{\beta + 1}{\beta + 2}}{\beta + 2}}{3 + \left(\beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{t_0}}{1 + t_0}\\ \end{array} \end{array} \]

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 5e+15)
     (/ (/ (/ (+ beta 1.0) (+ beta 2.0)) (+ beta 2.0)) (+ 3.0 (+ beta alpha)))
     (/ (/ (+ alpha 1.0) t_0) (+ 1.0 t_0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta + alpha);
	double tmp;
	if (beta <= 5e+15) {
		tmp = (((beta + 1.0) / (beta + 2.0)) / (beta + 2.0)) / (3.0 + (beta + alpha));
	} else {
		tmp = ((alpha + 1.0) / t_0) / (1.0 + 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 <= 5d+15) then
        tmp = (((beta + 1.0d0) / (beta + 2.0d0)) / (beta + 2.0d0)) / (3.0d0 + (beta + alpha))
    else
        tmp = ((alpha + 1.0d0) / t_0) / (1.0d0 + 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 <= 5e+15) {
		tmp = (((beta + 1.0) / (beta + 2.0)) / (beta + 2.0)) / (3.0 + (beta + alpha));
	} else {
		tmp = ((alpha + 1.0) / t_0) / (1.0 + t_0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = 2.0 + (beta + alpha)
	tmp = 0
	if beta <= 5e+15:
		tmp = (((beta + 1.0) / (beta + 2.0)) / (beta + 2.0)) / (3.0 + (beta + alpha))
	else:
		tmp = ((alpha + 1.0) / t_0) / (1.0 + 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 <= 5e+15)
		tmp = Float64(Float64(Float64(Float64(beta + 1.0) / Float64(beta + 2.0)) / Float64(beta + 2.0)) / Float64(3.0 + Float64(beta + alpha)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / t_0) / Float64(1.0 + 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 <= 5e+15)
		tmp = (((beta + 1.0) / (beta + 2.0)) / (beta + 2.0)) / (3.0 + (beta + alpha));
	else
		tmp = ((alpha + 1.0) / t_0) / (1.0 + 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, 5e+15], N[(N[(N[(N[(beta + 1.0), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 5e15

   1. Initial program 99.8%

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

   2. Taylor expanded in alpha around 0 82.6%

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

      1. *-un-lft-identity 82.6%

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

      2. associate-/l/ 82.6%

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

      3. metadata-eval 82.6%

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

      4. associate-+l+ 82.6%

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

      5. +-commutative 82.6%

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

      6. metadata-eval 82.6%

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

      7. metadata-eval 82.6%

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

      8. +-commutative 82.6%

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

      9. +-commutative 82.6%

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

   4. Applied egg-rr 82.6%

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

      1. associate-*r/ 82.6%

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

      2. times-frac 82.6%

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

      3. *-lft-identity 82.6%

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

      4. *-commutative 82.6%

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

      5. associate-*r/ 82.6%

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

      6. *-commutative 82.6%

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

      7. times-frac 82.6%

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

      8. associate-*r/ 82.6%

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

      9. *-lft-identity 82.6%

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

      10. *-commutative 82.6%

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

      11. associate-*r/ 82.6%

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

   6. Simplified 82.6%

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

   7. Taylor expanded in alpha around 0 70.2%

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

   if 5e15 < beta

   1. Initial program 83.0%

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

   2. Taylor expanded in beta around inf 90.7%

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

4. Final simplification 76.8%

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

Alternative 9: 97.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 2.2 \cdot 10^{-80}:\\ \;\;\;\;\frac{\frac{\frac{\beta + 1}{\beta + 2}}{\beta + 2}}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{1 + \left(2 + \left(\beta + \alpha\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 2.2e-80)
   (/ (/ (/ (+ beta 1.0) (+ beta 2.0)) (+ beta 2.0)) (+ beta 3.0))
   (/ (/ (+ alpha 1.0) beta) (+ 1.0 (+ 2.0 (+ beta alpha))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 2.2e-80) {
		tmp = (((beta + 1.0) / (beta + 2.0)) / (beta + 2.0)) / (beta + 3.0);
	} else {
		tmp = ((alpha + 1.0) / beta) / (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) :: tmp
    if (alpha <= 2.2d-80) then
        tmp = (((beta + 1.0d0) / (beta + 2.0d0)) / (beta + 2.0d0)) / (beta + 3.0d0)
    else
        tmp = ((alpha + 1.0d0) / beta) / (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 tmp;
	if (alpha <= 2.2e-80) {
		tmp = (((beta + 1.0) / (beta + 2.0)) / (beta + 2.0)) / (beta + 3.0);
	} else {
		tmp = ((alpha + 1.0) / beta) / (1.0 + (2.0 + (beta + alpha)));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if alpha <= 2.2e-80:
		tmp = (((beta + 1.0) / (beta + 2.0)) / (beta + 2.0)) / (beta + 3.0)
	else:
		tmp = ((alpha + 1.0) / beta) / (1.0 + (2.0 + (beta + alpha)))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 2.2e-80)
		tmp = Float64(Float64(Float64(Float64(beta + 1.0) / Float64(beta + 2.0)) / Float64(beta + 2.0)) / Float64(beta + 3.0));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / 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)
	tmp = 0.0;
	if (alpha <= 2.2e-80)
		tmp = (((beta + 1.0) / (beta + 2.0)) / (beta + 2.0)) / (beta + 3.0);
	else
		tmp = ((alpha + 1.0) / beta) / (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_] := If[LessEqual[alpha, 2.2e-80], N[(N[(N[(N[(beta + 1.0), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(1.0 + N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 2.2000000000000001e-80

   1. Initial program 99.9%

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

   2. Taylor expanded in alpha around 0 99.7%

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

      1. *-un-lft-identity 99.7%

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

      2. associate-/l/ 98.9%

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

      3. metadata-eval 98.9%

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

      4. associate-+l+ 98.9%

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

      5. +-commutative 98.9%

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

      6. metadata-eval 98.9%

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

      7. metadata-eval 98.9%

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

      8. +-commutative 98.9%

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

      9. +-commutative 98.9%

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

   4. Applied egg-rr 98.9%

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

      1. associate-*r/ 98.9%

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

      2. times-frac 99.6%

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

      3. *-lft-identity 99.6%

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

      4. *-commutative 99.6%

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

      5. associate-*r/ 99.7%

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

      6. *-commutative 99.7%

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

      7. times-frac 98.8%

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

      8. associate-*r/ 98.8%

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

      9. *-lft-identity 98.8%

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

      10. *-commutative 98.8%

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

      11. associate-*r/ 98.8%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in alpha around 0 99.9%

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

   8. Taylor expanded in alpha around 0 99.7%

      \[\leadsto \frac{\frac{\frac{\beta + 1}{\beta + 2}}{2 + \beta}}{\color{blue}{\beta + 3}} \]

   if 2.2000000000000001e-80 < alpha

   1. Initial program 84.8%

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

   2. Taylor expanded in beta around inf 24.6%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2.2 \cdot 10^{-80}:\\ \;\;\;\;\frac{\frac{\frac{\beta + 1}{\beta + 2}}{\beta + 2}}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{1 + \left(2 + \left(\beta + \alpha\right)\right)}\\ \end{array} \]

Alternative 10: 97.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.76:\\ \;\;\;\;\frac{0.25 + \left(\beta \cdot \beta\right) \cdot -0.0625}{3 + \left(\beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{1 + \left(\beta + 2\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.76)
   (/ (+ 0.25 (* (* beta beta) -0.0625)) (+ 3.0 (+ beta alpha)))
   (/ (/ (+ alpha 1.0) beta) (+ 1.0 (+ beta 2.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.76) {
		tmp = (0.25 + ((beta * beta) * -0.0625)) / (3.0 + (beta + alpha));
	} else {
		tmp = ((alpha + 1.0) / beta) / (1.0 + (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 <= 1.76d0) then
        tmp = (0.25d0 + ((beta * beta) * (-0.0625d0))) / (3.0d0 + (beta + alpha))
    else
        tmp = ((alpha + 1.0d0) / beta) / (1.0d0 + (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 <= 1.76) {
		tmp = (0.25 + ((beta * beta) * -0.0625)) / (3.0 + (beta + alpha));
	} else {
		tmp = ((alpha + 1.0) / beta) / (1.0 + (beta + 2.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.76:
		tmp = (0.25 + ((beta * beta) * -0.0625)) / (3.0 + (beta + alpha))
	else:
		tmp = ((alpha + 1.0) / beta) / (1.0 + (beta + 2.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.76)
		tmp = Float64(Float64(0.25 + Float64(Float64(beta * beta) * -0.0625)) / Float64(3.0 + Float64(beta + alpha)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / Float64(1.0 + 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 <= 1.76)
		tmp = (0.25 + ((beta * beta) * -0.0625)) / (3.0 + (beta + alpha));
	else
		tmp = ((alpha + 1.0) / beta) / (1.0 + (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, 1.76], N[(N[(0.25 + N[(N[(beta * beta), $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(1.0 + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.76000000000000001

   1. Initial program 99.8%

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

   2. Taylor expanded in alpha around 0 82.4%

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

      1. *-un-lft-identity 82.4%

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

      2. associate-/l/ 82.4%

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

      3. metadata-eval 82.4%

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

      4. associate-+l+ 82.4%

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

      5. +-commutative 82.4%

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

      6. metadata-eval 82.4%

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

      7. metadata-eval 82.4%

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

      8. +-commutative 82.4%

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

      9. +-commutative 82.4%

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

   4. Applied egg-rr 82.4%

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

      1. associate-*r/ 82.4%

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

      2. times-frac 82.4%

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

      3. *-lft-identity 82.4%

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

      4. *-commutative 82.4%

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

      5. associate-*r/ 82.4%

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

      6. *-commutative 82.4%

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

      7. times-frac 82.4%

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

      8. associate-*r/ 82.4%

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

      9. *-lft-identity 82.4%

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

      10. *-commutative 82.4%

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

      11. associate-*r/ 82.4%

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

   6. Simplified 82.4%

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

   7. Taylor expanded in alpha around 0 69.9%

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

   8. Taylor expanded in beta around 0 69.9%

      \[\leadsto \frac{\color{blue}{0.25 + -0.0625 \cdot {\beta}^{2}}}{3 + \left(\alpha + \beta\right)} \]
   9. Step-by-step derivation

      1. *-commutative 69.9%

         \[\leadsto \frac{0.25 + \color{blue}{{\beta}^{2} \cdot -0.0625}}{3 + \left(\alpha + \beta\right)} \]

      2. unpow2 69.9%

         \[\leadsto \frac{0.25 + \color{blue}{\left(\beta \cdot \beta\right)} \cdot -0.0625}{3 + \left(\alpha + \beta\right)} \]

   10. Simplified 69.9%

       \[\leadsto \frac{\color{blue}{0.25 + \left(\beta \cdot \beta\right) \cdot -0.0625}}{3 + \left(\alpha + \beta\right)} \]

   if 1.76000000000000001 < beta

   1. Initial program 83.4%

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

   2. Taylor expanded in alpha around 0 80.2%

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

   3. Taylor expanded in beta around inf 90.0%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.76:\\ \;\;\;\;\frac{0.25 + \left(\beta \cdot \beta\right) \cdot -0.0625}{3 + \left(\beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{1 + \left(\beta + 2\right)}\\ \end{array} \]

Alternative 11: 97.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.76:\\ \;\;\;\;\frac{0.25 + \left(\beta \cdot \beta\right) \cdot -0.0625}{3 + \left(\beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{1 + \left(2 + \left(\beta + \alpha\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.76)
   (/ (+ 0.25 (* (* beta beta) -0.0625)) (+ 3.0 (+ beta alpha)))
   (/ (/ (+ alpha 1.0) beta) (+ 1.0 (+ 2.0 (+ beta alpha))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.76) {
		tmp = (0.25 + ((beta * beta) * -0.0625)) / (3.0 + (beta + alpha));
	} else {
		tmp = ((alpha + 1.0) / beta) / (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) :: tmp
    if (beta <= 1.76d0) then
        tmp = (0.25d0 + ((beta * beta) * (-0.0625d0))) / (3.0d0 + (beta + alpha))
    else
        tmp = ((alpha + 1.0d0) / beta) / (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 tmp;
	if (beta <= 1.76) {
		tmp = (0.25 + ((beta * beta) * -0.0625)) / (3.0 + (beta + alpha));
	} else {
		tmp = ((alpha + 1.0) / beta) / (1.0 + (2.0 + (beta + alpha)));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.76:
		tmp = (0.25 + ((beta * beta) * -0.0625)) / (3.0 + (beta + alpha))
	else:
		tmp = ((alpha + 1.0) / beta) / (1.0 + (2.0 + (beta + alpha)))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.76)
		tmp = Float64(Float64(0.25 + Float64(Float64(beta * beta) * -0.0625)) / Float64(3.0 + Float64(beta + alpha)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / 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)
	tmp = 0.0;
	if (beta <= 1.76)
		tmp = (0.25 + ((beta * beta) * -0.0625)) / (3.0 + (beta + alpha));
	else
		tmp = ((alpha + 1.0) / beta) / (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_] := If[LessEqual[beta, 1.76], N[(N[(0.25 + N[(N[(beta * beta), $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(1.0 + N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.76000000000000001

   1. Initial program 99.8%

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

   2. Taylor expanded in alpha around 0 82.4%

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

      1. *-un-lft-identity 82.4%

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

      2. associate-/l/ 82.4%

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

      3. metadata-eval 82.4%

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

      4. associate-+l+ 82.4%

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

      5. +-commutative 82.4%

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

      6. metadata-eval 82.4%

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

      7. metadata-eval 82.4%

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

      8. +-commutative 82.4%

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

      9. +-commutative 82.4%

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

   4. Applied egg-rr 82.4%

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

      1. associate-*r/ 82.4%

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

      2. times-frac 82.4%

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

      3. *-lft-identity 82.4%

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

      4. *-commutative 82.4%

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

      5. associate-*r/ 82.4%

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

      6. *-commutative 82.4%

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

      7. times-frac 82.4%

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

      8. associate-*r/ 82.4%

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

      9. *-lft-identity 82.4%

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

      10. *-commutative 82.4%

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

      11. associate-*r/ 82.4%

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

   6. Simplified 82.4%

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

   7. Taylor expanded in alpha around 0 69.9%

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

   8. Taylor expanded in beta around 0 69.9%

      \[\leadsto \frac{\color{blue}{0.25 + -0.0625 \cdot {\beta}^{2}}}{3 + \left(\alpha + \beta\right)} \]
   9. Step-by-step derivation

      1. *-commutative 69.9%

         \[\leadsto \frac{0.25 + \color{blue}{{\beta}^{2} \cdot -0.0625}}{3 + \left(\alpha + \beta\right)} \]

      2. unpow2 69.9%

         \[\leadsto \frac{0.25 + \color{blue}{\left(\beta \cdot \beta\right)} \cdot -0.0625}{3 + \left(\alpha + \beta\right)} \]

   10. Simplified 69.9%

       \[\leadsto \frac{\color{blue}{0.25 + \left(\beta \cdot \beta\right) \cdot -0.0625}}{3 + \left(\alpha + \beta\right)} \]

   if 1.76000000000000001 < beta

   1. Initial program 83.4%

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

   2. Taylor expanded in beta around inf 90.1%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.76:\\ \;\;\;\;\frac{0.25 + \left(\beta \cdot \beta\right) \cdot -0.0625}{3 + \left(\beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{1 + \left(2 + \left(\beta + \alpha\right)\right)}\\ \end{array} \]

Alternative 12: 92.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 3 + \left(\beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 2:\\ \;\;\;\;\frac{0.25}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta + 2}}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 3.0 (+ beta alpha))))
   (if (<= beta 2.0) (/ 0.25 t_0) (/ (/ 1.0 (+ beta 2.0)) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 3.0 + (beta + alpha);
	double tmp;
	if (beta <= 2.0) {
		tmp = 0.25 / t_0;
	} else {
		tmp = (1.0 / (beta + 2.0)) / 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 = 3.0d0 + (beta + alpha)
    if (beta <= 2.0d0) then
        tmp = 0.25d0 / t_0
    else
        tmp = (1.0d0 / (beta + 2.0d0)) / t_0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = 3.0 + (beta + alpha);
	double tmp;
	if (beta <= 2.0) {
		tmp = 0.25 / t_0;
	} else {
		tmp = (1.0 / (beta + 2.0)) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = 3.0 + (beta + alpha)
	tmp = 0
	if beta <= 2.0:
		tmp = 0.25 / t_0
	else:
		tmp = (1.0 / (beta + 2.0)) / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(3.0 + Float64(beta + alpha))
	tmp = 0.0
	if (beta <= 2.0)
		tmp = Float64(0.25 / t_0);
	else
		tmp = Float64(Float64(1.0 / Float64(beta + 2.0)) / t_0);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = 3.0 + (beta + alpha);
	tmp = 0.0;
	if (beta <= 2.0)
		tmp = 0.25 / t_0;
	else
		tmp = (1.0 / (beta + 2.0)) / 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[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2.0], N[(0.25 / t$95$0), $MachinePrecision], N[(N[(1.0 / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2

   1. Initial program 99.8%

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

   2. Taylor expanded in alpha around 0 82.4%

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

      1. *-un-lft-identity 82.4%

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

      2. associate-/l/ 82.4%

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

      3. metadata-eval 82.4%

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

      4. associate-+l+ 82.4%

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

      5. +-commutative 82.4%

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

      6. metadata-eval 82.4%

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

      7. metadata-eval 82.4%

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

      8. +-commutative 82.4%

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

      9. +-commutative 82.4%

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

   4. Applied egg-rr 82.4%

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

      1. associate-*r/ 82.4%

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

      2. times-frac 82.4%

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

      3. *-lft-identity 82.4%

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

      4. *-commutative 82.4%

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

      5. associate-*r/ 82.4%

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

      6. *-commutative 82.4%

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

      7. times-frac 82.4%

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

      8. associate-*r/ 82.4%

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

      9. *-lft-identity 82.4%

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

      10. *-commutative 82.4%

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

      11. associate-*r/ 82.4%

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

   6. Simplified 82.4%

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

   7. Taylor expanded in alpha around 0 69.9%

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

   8. Taylor expanded in beta around 0 69.9%

      \[\leadsto \frac{\color{blue}{0.25}}{3 + \left(\alpha + \beta\right)} \]

   if 2 < beta

   1. Initial program 83.4%

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

   2. Taylor expanded in alpha around 0 86.4%

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

      1. *-un-lft-identity 86.4%

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

      2. associate-/l/ 84.8%

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

      3. metadata-eval 84.8%

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

      4. associate-+l+ 84.8%

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

      5. +-commutative 84.8%

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

      6. metadata-eval 84.8%

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

      7. metadata-eval 84.8%

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

      8. +-commutative 84.8%

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

      9. +-commutative 84.8%

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

   4. Applied egg-rr 84.8%

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

      1. associate-*r/ 84.8%

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

      2. times-frac 86.3%

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

      3. *-lft-identity 86.3%

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

      4. *-commutative 86.3%

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

      5. associate-*r/ 86.4%

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

      6. *-commutative 86.4%

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

      7. times-frac 84.8%

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

      8. associate-*r/ 84.8%

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

      9. *-lft-identity 84.8%

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

      10. *-commutative 84.8%

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

      11. associate-*r/ 84.8%

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

   6. Simplified 86.4%

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

   7. Taylor expanded in beta around inf 84.9%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2:\\ \;\;\;\;\frac{0.25}{3 + \left(\beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta + 2}}{3 + \left(\beta + \alpha\right)}\\ \end{array} \]

Alternative 13: 97.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.2:\\ \;\;\;\;\frac{0.25}{3 + \left(\beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{1 + \left(\beta + 2\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.2)
   (/ 0.25 (+ 3.0 (+ beta alpha)))
   (/ (/ (+ alpha 1.0) beta) (+ 1.0 (+ beta 2.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.2) {
		tmp = 0.25 / (3.0 + (beta + alpha));
	} else {
		tmp = ((alpha + 1.0) / beta) / (1.0 + (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 <= 4.2d0) then
        tmp = 0.25d0 / (3.0d0 + (beta + alpha))
    else
        tmp = ((alpha + 1.0d0) / beta) / (1.0d0 + (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 <= 4.2) {
		tmp = 0.25 / (3.0 + (beta + alpha));
	} else {
		tmp = ((alpha + 1.0) / beta) / (1.0 + (beta + 2.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 4.2:
		tmp = 0.25 / (3.0 + (beta + alpha))
	else:
		tmp = ((alpha + 1.0) / beta) / (1.0 + (beta + 2.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.2)
		tmp = Float64(0.25 / Float64(3.0 + Float64(beta + alpha)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / Float64(1.0 + 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 <= 4.2)
		tmp = 0.25 / (3.0 + (beta + alpha));
	else
		tmp = ((alpha + 1.0) / beta) / (1.0 + (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, 4.2], N[(0.25 / N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(1.0 + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.20000000000000018

   1. Initial program 99.8%

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

   2. Taylor expanded in alpha around 0 82.4%

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

      1. *-un-lft-identity 82.4%

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

      2. associate-/l/ 82.4%

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

      3. metadata-eval 82.4%

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

      4. associate-+l+ 82.4%

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

      5. +-commutative 82.4%

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

      6. metadata-eval 82.4%

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

      7. metadata-eval 82.4%

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

      8. +-commutative 82.4%

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

      9. +-commutative 82.4%

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

   4. Applied egg-rr 82.4%

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

      1. associate-*r/ 82.4%

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

      2. times-frac 82.4%

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

      3. *-lft-identity 82.4%

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

      4. *-commutative 82.4%

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

      5. associate-*r/ 82.4%

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

      6. *-commutative 82.4%

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

      7. times-frac 82.4%

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

      8. associate-*r/ 82.4%

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

      9. *-lft-identity 82.4%

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

      10. *-commutative 82.4%

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

      11. associate-*r/ 82.4%

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

   6. Simplified 82.4%

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

   7. Taylor expanded in alpha around 0 69.9%

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

   8. Taylor expanded in beta around 0 69.9%

      \[\leadsto \frac{\color{blue}{0.25}}{3 + \left(\alpha + \beta\right)} \]

   if 4.20000000000000018 < beta

   1. Initial program 83.4%

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

   2. Taylor expanded in alpha around 0 80.2%

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

   3. Taylor expanded in beta around inf 90.0%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.2:\\ \;\;\;\;\frac{0.25}{3 + \left(\beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{1 + \left(\beta + 2\right)}\\ \end{array} \]

Alternative 14: 94.4% accurate, 3.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.2:\\ \;\;\;\;\frac{0.25}{3 + \left(\beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\alpha + 1\right) \cdot \frac{1}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.2)
   (/ 0.25 (+ 3.0 (+ beta alpha)))
   (* (+ alpha 1.0) (/ 1.0 (* beta beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.2) {
		tmp = 0.25 / (3.0 + (beta + alpha));
	} else {
		tmp = (alpha + 1.0) * (1.0 / (beta * beta));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 6.2d0) then
        tmp = 0.25d0 / (3.0d0 + (beta + alpha))
    else
        tmp = (alpha + 1.0d0) * (1.0d0 / (beta * beta))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.2) {
		tmp = 0.25 / (3.0 + (beta + alpha));
	} else {
		tmp = (alpha + 1.0) * (1.0 / (beta * beta));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 6.2:
		tmp = 0.25 / (3.0 + (beta + alpha))
	else:
		tmp = (alpha + 1.0) * (1.0 / (beta * beta))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.2)
		tmp = Float64(0.25 / Float64(3.0 + Float64(beta + alpha)));
	else
		tmp = Float64(Float64(alpha + 1.0) * Float64(1.0 / Float64(beta * beta)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 6.2)
		tmp = 0.25 / (3.0 + (beta + alpha));
	else
		tmp = (alpha + 1.0) * (1.0 / (beta * beta));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 6.2], N[(0.25 / N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(alpha + 1.0), $MachinePrecision] * N[(1.0 / N[(beta * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.20000000000000018

   1. Initial program 99.8%

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

   2. Taylor expanded in alpha around 0 82.4%

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

      1. *-un-lft-identity 82.4%

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

      2. associate-/l/ 82.4%

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

      3. metadata-eval 82.4%

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

      4. associate-+l+ 82.4%

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

      5. +-commutative 82.4%

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

      6. metadata-eval 82.4%

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

      7. metadata-eval 82.4%

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

      8. +-commutative 82.4%

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

      9. +-commutative 82.4%

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

   4. Applied egg-rr 82.4%

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

      1. associate-*r/ 82.4%

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

      2. times-frac 82.4%

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

      3. *-lft-identity 82.4%

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

      4. *-commutative 82.4%

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

      5. associate-*r/ 82.4%

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

      6. *-commutative 82.4%

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

      7. times-frac 82.4%

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

      8. associate-*r/ 82.4%

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

      9. *-lft-identity 82.4%

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

      10. *-commutative 82.4%

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

      11. associate-*r/ 82.4%

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

   6. Simplified 82.4%

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

   7. Taylor expanded in alpha around 0 69.9%

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

   8. Taylor expanded in beta around 0 69.9%

      \[\leadsto \frac{\color{blue}{0.25}}{3 + \left(\alpha + \beta\right)} \]

   if 6.20000000000000018 < beta

   1. Initial program 83.4%

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

   2. Taylor expanded in beta around inf 82.1%

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

      1. unpow2 82.1%

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

   4. Simplified 82.1%

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

      1. div-inv 82.1%

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

   6. Applied egg-rr 82.1%

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

      1. *-commutative 82.1%

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

      2. +-commutative 82.1%

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

   8. Simplified 82.1%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.2:\\ \;\;\;\;\frac{0.25}{3 + \left(\beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\alpha + 1\right) \cdot \frac{1}{\beta \cdot \beta}\\ \end{array} \]

Alternative 15: 91.9% accurate, 3.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 3 + \left(\beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 4:\\ \;\;\;\;\frac{0.25}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 3.0 (+ beta alpha))))
   (if (<= beta 4.0) (/ 0.25 t_0) (/ (/ 1.0 beta) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 3.0 + (beta + alpha);
	double tmp;
	if (beta <= 4.0) {
		tmp = 0.25 / t_0;
	} else {
		tmp = (1.0 / 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 = 3.0d0 + (beta + alpha)
    if (beta <= 4.0d0) then
        tmp = 0.25d0 / t_0
    else
        tmp = (1.0d0 / beta) / t_0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = 3.0 + (beta + alpha);
	double tmp;
	if (beta <= 4.0) {
		tmp = 0.25 / t_0;
	} else {
		tmp = (1.0 / beta) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = 3.0 + (beta + alpha)
	tmp = 0
	if beta <= 4.0:
		tmp = 0.25 / t_0
	else:
		tmp = (1.0 / beta) / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(3.0 + Float64(beta + alpha))
	tmp = 0.0
	if (beta <= 4.0)
		tmp = Float64(0.25 / t_0);
	else
		tmp = Float64(Float64(1.0 / beta) / t_0);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = 3.0 + (beta + alpha);
	tmp = 0.0;
	if (beta <= 4.0)
		tmp = 0.25 / t_0;
	else
		tmp = (1.0 / 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[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 4.0], N[(0.25 / t$95$0), $MachinePrecision], N[(N[(1.0 / beta), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 4

   1. Initial program 99.8%

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

   2. Taylor expanded in alpha around 0 82.4%

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

      1. *-un-lft-identity 82.4%

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

      2. associate-/l/ 82.4%

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

      3. metadata-eval 82.4%

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

      4. associate-+l+ 82.4%

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

      5. +-commutative 82.4%

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

      6. metadata-eval 82.4%

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

      7. metadata-eval 82.4%

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

      8. +-commutative 82.4%

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

      9. +-commutative 82.4%

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

   4. Applied egg-rr 82.4%

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

      1. associate-*r/ 82.4%

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

      2. times-frac 82.4%

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

      3. *-lft-identity 82.4%

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

      4. *-commutative 82.4%

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

      5. associate-*r/ 82.4%

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

      6. *-commutative 82.4%

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

      7. times-frac 82.4%

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

      8. associate-*r/ 82.4%

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

      9. *-lft-identity 82.4%

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

      10. *-commutative 82.4%

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

      11. associate-*r/ 82.4%

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

   6. Simplified 82.4%

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

   7. Taylor expanded in alpha around 0 69.9%

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

   8. Taylor expanded in beta around 0 69.9%

      \[\leadsto \frac{\color{blue}{0.25}}{3 + \left(\alpha + \beta\right)} \]

   if 4 < beta

   1. Initial program 83.4%

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

   2. Taylor expanded in alpha around 0 86.4%

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

      1. *-un-lft-identity 86.4%

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

      2. associate-/l/ 84.8%

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

      3. metadata-eval 84.8%

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

      4. associate-+l+ 84.8%

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

      5. +-commutative 84.8%

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

      6. metadata-eval 84.8%

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

      7. metadata-eval 84.8%

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

      8. +-commutative 84.8%

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

      9. +-commutative 84.8%

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

   4. Applied egg-rr 84.8%

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

      1. associate-*r/ 84.8%

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

      2. times-frac 86.3%

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

      3. *-lft-identity 86.3%

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

      4. *-commutative 86.3%

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

      5. associate-*r/ 86.4%

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

      6. *-commutative 86.4%

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

      7. times-frac 84.8%

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

      8. associate-*r/ 84.8%

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

      9. *-lft-identity 84.8%

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

      10. *-commutative 84.8%

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

      11. associate-*r/ 84.8%

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

   6. Simplified 86.4%

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

   7. Taylor expanded in beta around inf 84.8%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4:\\ \;\;\;\;\frac{0.25}{3 + \left(\beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{3 + \left(\beta + \alpha\right)}\\ \end{array} \]

Alternative 16: 91.5% accurate, 3.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.4:\\ \;\;\;\;\frac{0.25}{3 + \left(\beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.4) (/ 0.25 (+ 3.0 (+ beta alpha))) (/ 1.0 (* beta beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.4) {
		tmp = 0.25 / (3.0 + (beta + alpha));
	} else {
		tmp = 1.0 / (beta * beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 6.4d0) then
        tmp = 0.25d0 / (3.0d0 + (beta + alpha))
    else
        tmp = 1.0d0 / (beta * beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.4) {
		tmp = 0.25 / (3.0 + (beta + alpha));
	} else {
		tmp = 1.0 / (beta * beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 6.4:
		tmp = 0.25 / (3.0 + (beta + alpha))
	else:
		tmp = 1.0 / (beta * beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.4)
		tmp = Float64(0.25 / Float64(3.0 + Float64(beta + alpha)));
	else
		tmp = Float64(1.0 / Float64(beta * beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 6.4)
		tmp = 0.25 / (3.0 + (beta + alpha));
	else
		tmp = 1.0 / (beta * beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 6.4], N[(0.25 / N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.4000000000000004

   1. Initial program 99.8%

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

   2. Taylor expanded in alpha around 0 82.4%

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

      1. *-un-lft-identity 82.4%

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

      2. associate-/l/ 82.4%

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

      3. metadata-eval 82.4%

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

      4. associate-+l+ 82.4%

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

      5. +-commutative 82.4%

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

      6. metadata-eval 82.4%

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

      7. metadata-eval 82.4%

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

      8. +-commutative 82.4%

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

      9. +-commutative 82.4%

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

   4. Applied egg-rr 82.4%

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

      1. associate-*r/ 82.4%

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

      2. times-frac 82.4%

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

      3. *-lft-identity 82.4%

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

      4. *-commutative 82.4%

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

      5. associate-*r/ 82.4%

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

      6. *-commutative 82.4%

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

      7. times-frac 82.4%

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

      8. associate-*r/ 82.4%

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

      9. *-lft-identity 82.4%

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

      10. *-commutative 82.4%

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

      11. associate-*r/ 82.4%

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

   6. Simplified 82.4%

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

   7. Taylor expanded in alpha around 0 69.9%

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

   8. Taylor expanded in beta around 0 69.9%

      \[\leadsto \frac{\color{blue}{0.25}}{3 + \left(\alpha + \beta\right)} \]

   if 6.4000000000000004 < beta

   1. Initial program 83.4%

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

   2. Taylor expanded in alpha around 0 86.4%

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

   3. Taylor expanded in beta around inf 80.9%

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

      1. unpow2 80.9%

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

   5. Simplified 80.9%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.4:\\ \;\;\;\;\frac{0.25}{3 + \left(\beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \]

Alternative 17: 94.4% accurate, 3.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.2:\\ \;\;\;\;\frac{0.25}{3 + \left(\beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.2)
   (/ 0.25 (+ 3.0 (+ beta alpha)))
   (/ (+ alpha 1.0) (* beta beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.2) {
		tmp = 0.25 / (3.0 + (beta + alpha));
	} else {
		tmp = (alpha + 1.0) / (beta * beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 6.2d0) then
        tmp = 0.25d0 / (3.0d0 + (beta + alpha))
    else
        tmp = (alpha + 1.0d0) / (beta * beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.2) {
		tmp = 0.25 / (3.0 + (beta + alpha));
	} else {
		tmp = (alpha + 1.0) / (beta * beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 6.2:
		tmp = 0.25 / (3.0 + (beta + alpha))
	else:
		tmp = (alpha + 1.0) / (beta * beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.2)
		tmp = Float64(0.25 / Float64(3.0 + Float64(beta + alpha)));
	else
		tmp = Float64(Float64(alpha + 1.0) / Float64(beta * beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 6.2)
		tmp = 0.25 / (3.0 + (beta + alpha));
	else
		tmp = (alpha + 1.0) / (beta * beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 6.2], N[(0.25 / N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(alpha + 1.0), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.20000000000000018

   1. Initial program 99.8%

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

   2. Taylor expanded in alpha around 0 82.4%

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

      1. *-un-lft-identity 82.4%

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

      2. associate-/l/ 82.4%

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

      3. metadata-eval 82.4%

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

      4. associate-+l+ 82.4%

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

      5. +-commutative 82.4%

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

      6. metadata-eval 82.4%

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

      7. metadata-eval 82.4%

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

      8. +-commutative 82.4%

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

      9. +-commutative 82.4%

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

   4. Applied egg-rr 82.4%

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

      1. associate-*r/ 82.4%

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

      2. times-frac 82.4%

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

      3. *-lft-identity 82.4%

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

      4. *-commutative 82.4%

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

      5. associate-*r/ 82.4%

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

      6. *-commutative 82.4%

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

      7. times-frac 82.4%

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

      8. associate-*r/ 82.4%

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

      9. *-lft-identity 82.4%

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

      10. *-commutative 82.4%

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

      11. associate-*r/ 82.4%

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

   6. Simplified 82.4%

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

   7. Taylor expanded in alpha around 0 69.9%

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

   8. Taylor expanded in beta around 0 69.9%

      \[\leadsto \frac{\color{blue}{0.25}}{3 + \left(\alpha + \beta\right)} \]

   if 6.20000000000000018 < beta

   1. Initial program 83.4%

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

   2. Taylor expanded in beta around inf 82.1%

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

      1. unpow2 82.1%

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

   4. Simplified 82.1%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.2:\\ \;\;\;\;\frac{0.25}{3 + \left(\beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\beta \cdot \beta}\\ \end{array} \]

Alternative 18: 91.1% accurate, 5.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6:\\ \;\;\;\;\frac{0.25}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.0) (/ 0.25 (+ beta 3.0)) (/ 1.0 (* beta beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.0) {
		tmp = 0.25 / (beta + 3.0);
	} else {
		tmp = 1.0 / (beta * beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 6.0d0) then
        tmp = 0.25d0 / (beta + 3.0d0)
    else
        tmp = 1.0d0 / (beta * beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.0) {
		tmp = 0.25 / (beta + 3.0);
	} else {
		tmp = 1.0 / (beta * beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 6.0:
		tmp = 0.25 / (beta + 3.0)
	else:
		tmp = 1.0 / (beta * beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.0)
		tmp = Float64(0.25 / Float64(beta + 3.0));
	else
		tmp = Float64(1.0 / Float64(beta * beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 6.0)
		tmp = 0.25 / (beta + 3.0);
	else
		tmp = 1.0 / (beta * beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 6.0], N[(0.25 / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6

   1. Initial program 99.8%

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

   2. Taylor expanded in alpha around 0 82.4%

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

      1. *-un-lft-identity 82.4%

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

      2. associate-/l/ 82.4%

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

      3. metadata-eval 82.4%

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

      4. associate-+l+ 82.4%

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

      5. +-commutative 82.4%

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

      6. metadata-eval 82.4%

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

      7. metadata-eval 82.4%

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

      8. +-commutative 82.4%

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

      9. +-commutative 82.4%

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

   4. Applied egg-rr 82.4%

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

      1. associate-*r/ 82.4%

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

      2. times-frac 82.4%

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

      3. *-lft-identity 82.4%

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

      4. *-commutative 82.4%

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

      5. associate-*r/ 82.4%

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

      6. *-commutative 82.4%

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

      7. times-frac 82.4%

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

      8. associate-*r/ 82.4%

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

      9. *-lft-identity 82.4%

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

      10. *-commutative 82.4%

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

      11. associate-*r/ 82.4%

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

   6. Simplified 82.4%

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

   7. Taylor expanded in beta around 0 82.4%

      \[\leadsto \frac{\color{blue}{\frac{0.5}{2 + \alpha}}}{3 + \left(\alpha + \beta\right)} \]

   8. Taylor expanded in alpha around 0 68.7%

      \[\leadsto \color{blue}{\frac{0.25}{\beta + 3}} \]

   if 6 < beta

   1. Initial program 83.4%

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

   2. Taylor expanded in alpha around 0 86.4%

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

   3. Taylor expanded in beta around inf 80.9%

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

      1. unpow2 80.9%

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

   5. Simplified 80.9%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6:\\ \;\;\;\;\frac{0.25}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \]

Alternative 19: 44.6% accurate, 7.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ 0.08333333333333333 + \alpha \cdot -0.027777777777777776 \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (+ 0.08333333333333333 (* alpha -0.027777777777777776)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return 0.08333333333333333 + (alpha * -0.027777777777777776);
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = 0.08333333333333333d0 + (alpha * (-0.027777777777777776d0))
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return 0.08333333333333333 + (alpha * -0.027777777777777776);
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return 0.08333333333333333 + (alpha * -0.027777777777777776)

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return Float64(0.08333333333333333 + Float64(alpha * -0.027777777777777776))
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = 0.08333333333333333 + (alpha * -0.027777777777777776);
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := N[(0.08333333333333333 + N[(alpha * -0.027777777777777776), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.4%

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

2. Taylor expanded in beta around 0 64.9%

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

   1. +-commutative 64.9%

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

4. Simplified 64.9%

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

5. Taylor expanded in alpha around 0 46.8%

   \[\leadsto \color{blue}{0.08333333333333333 + -0.027777777777777776 \cdot \alpha} \]
6. Step-by-step derivation

   1. *-commutative 46.8%

      \[\leadsto 0.08333333333333333 + \color{blue}{\alpha \cdot -0.027777777777777776} \]

7. Simplified 46.8%

   \[\leadsto \color{blue}{0.08333333333333333 + \alpha \cdot -0.027777777777777776} \]

8. Final simplification 46.8%

   \[\leadsto 0.08333333333333333 + \alpha \cdot -0.027777777777777776 \]

Alternative 20: 45.0% accurate, 7.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \frac{0.25}{\alpha + 3} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta) :precision binary64 (/ 0.25 (+ alpha 3.0)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return 0.25 / (alpha + 3.0);
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = 0.25d0 / (alpha + 3.0d0)
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return 0.25 / (alpha + 3.0);
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return 0.25 / (alpha + 3.0)

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return Float64(0.25 / Float64(alpha + 3.0))
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = 0.25 / (alpha + 3.0);
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := N[(0.25 / N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.4%

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

2. Taylor expanded in alpha around 0 83.7%

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

3. Taylor expanded in beta around 0 58.8%

   \[\leadsto \color{blue}{\frac{0.5}{\left(2 + \alpha\right) \cdot \left(3 + \alpha\right)}} \]
4. Step-by-step derivation

   1. associate-/r* 58.8%

      \[\leadsto \color{blue}{\frac{\frac{0.5}{2 + \alpha}}{3 + \alpha}} \]

   2. +-commutative 58.8%

      \[\leadsto \frac{\frac{0.5}{2 + \alpha}}{\color{blue}{\alpha + 3}} \]

5. Simplified 58.8%

   \[\leadsto \color{blue}{\frac{\frac{0.5}{2 + \alpha}}{\alpha + 3}} \]

6. Taylor expanded in alpha around 0 48.2%

   \[\leadsto \frac{\color{blue}{0.25}}{\alpha + 3} \]

7. Final simplification 48.2%

   \[\leadsto \frac{0.25}{\alpha + 3} \]

Alternative 21: 46.5% accurate, 7.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \frac{0.25}{\beta + 3} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta) :precision binary64 (/ 0.25 (+ beta 3.0)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return 0.25 / (beta + 3.0);
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = 0.25d0 / (beta + 3.0d0)
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return 0.25 / (beta + 3.0);
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return 0.25 / (beta + 3.0)

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return Float64(0.25 / Float64(beta + 3.0))
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = 0.25 / (beta + 3.0);
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := N[(0.25 / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.4%

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

2. Taylor expanded in alpha around 0 83.7%

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

   1. *-un-lft-identity 83.7%

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

   2. associate-/l/ 83.2%

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

   3. metadata-eval 83.2%

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

   4. associate-+l+ 83.2%

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

   5. +-commutative 83.2%

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

   6. metadata-eval 83.2%

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

   7. metadata-eval 83.2%

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

   8. +-commutative 83.2%

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

   9. +-commutative 83.2%

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

4. Applied egg-rr 83.2%

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

   1. associate-*r/ 83.2%

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

   2. times-frac 83.7%

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

   3. *-lft-identity 83.7%

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

   4. *-commutative 83.7%

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

   5. associate-*r/ 83.7%

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

   6. *-commutative 83.7%

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

   7. times-frac 83.2%

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

   8. associate-*r/ 83.2%

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

   9. *-lft-identity 83.2%

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

   10. *-commutative 83.2%

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

   11. associate-*r/ 83.2%

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

6. Simplified 83.7%

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

7. Taylor expanded in beta around 0 61.0%

   \[\leadsto \frac{\color{blue}{\frac{0.5}{2 + \alpha}}}{3 + \left(\alpha + \beta\right)} \]

8. Taylor expanded in alpha around 0 48.5%

   \[\leadsto \color{blue}{\frac{0.25}{\beta + 3}} \]

9. Final simplification 48.5%

   \[\leadsto \frac{0.25}{\beta + 3} \]

Alternative 22: 44.4% accurate, 35.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ 0.08333333333333333 \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta) :precision binary64 0.08333333333333333)

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return 0.08333333333333333;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = 0.08333333333333333d0
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return 0.08333333333333333;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return 0.08333333333333333

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return 0.08333333333333333
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = 0.08333333333333333;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := 0.08333333333333333

Derivation

1. Initial program 94.4%

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

2. Taylor expanded in beta around 0 64.9%

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

   1. +-commutative 64.9%

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

4. Simplified 64.9%

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

5. Taylor expanded in alpha around 0 47.1%

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

6. Final simplification 47.1%

   \[\leadsto 0.08333333333333333 \]

Reproduce

herbie shell --seed 2023174 
(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)))