Octave 3.8, jcobi/3

Percentage Accurate: 94.5% → 99.7%

Time: 13.8s

Alternatives: 19

Speedup: 2.5×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.5% 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: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 5 \cdot 10^{+144}:\\ \;\;\;\;\frac{\frac{1}{\frac{t\_0}{\left(\beta + 1\right) \cdot \left(1 + \alpha\right)}}}{\beta \cdot \left(5 + \left(\beta + 2 \cdot \alpha\right)\right) + \left(2 + \alpha\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{t\_0} \cdot \frac{1 - \frac{2 \cdot \alpha + 4}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 5e+144)
     (/
      (/ 1.0 (/ t_0 (* (+ beta 1.0) (+ 1.0 alpha))))
      (+
       (* beta (+ 5.0 (+ beta (* 2.0 alpha))))
       (* (+ 2.0 alpha) (+ alpha 3.0))))
     (*
      (/ (+ 1.0 alpha) t_0)
      (/ (- 1.0 (/ (+ (* 2.0 alpha) 4.0) beta)) beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 5e+144) {
		tmp = (1.0 / (t_0 / ((beta + 1.0) * (1.0 + alpha)))) / ((beta * (5.0 + (beta + (2.0 * alpha)))) + ((2.0 + alpha) * (alpha + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / t_0) * ((1.0 - (((2.0 * alpha) + 4.0) / 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) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    if (beta <= 5d+144) then
        tmp = (1.0d0 / (t_0 / ((beta + 1.0d0) * (1.0d0 + alpha)))) / ((beta * (5.0d0 + (beta + (2.0d0 * alpha)))) + ((2.0d0 + alpha) * (alpha + 3.0d0)))
    else
        tmp = ((1.0d0 + alpha) / t_0) * ((1.0d0 - (((2.0d0 * alpha) + 4.0d0) / beta)) / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 5e+144) {
		tmp = (1.0 / (t_0 / ((beta + 1.0) * (1.0 + alpha)))) / ((beta * (5.0 + (beta + (2.0 * alpha)))) + ((2.0 + alpha) * (alpha + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / t_0) * ((1.0 - (((2.0 * alpha) + 4.0) / beta)) / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 5e+144:
		tmp = (1.0 / (t_0 / ((beta + 1.0) * (1.0 + alpha)))) / ((beta * (5.0 + (beta + (2.0 * alpha)))) + ((2.0 + alpha) * (alpha + 3.0)))
	else:
		tmp = ((1.0 + alpha) / t_0) * ((1.0 - (((2.0 * alpha) + 4.0) / beta)) / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 5e+144)
		tmp = Float64(Float64(1.0 / Float64(t_0 / Float64(Float64(beta + 1.0) * Float64(1.0 + alpha)))) / Float64(Float64(beta * Float64(5.0 + Float64(beta + Float64(2.0 * alpha)))) + Float64(Float64(2.0 + alpha) * Float64(alpha + 3.0))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / t_0) * Float64(Float64(1.0 - Float64(Float64(Float64(2.0 * alpha) + 4.0) / beta)) / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 5e+144)
		tmp = (1.0 / (t_0 / ((beta + 1.0) * (1.0 + alpha)))) / ((beta * (5.0 + (beta + (2.0 * alpha)))) + ((2.0 + alpha) * (alpha + 3.0)));
	else
		tmp = ((1.0 + alpha) / t_0) * ((1.0 - (((2.0 * alpha) + 4.0) / 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_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 5e+144], N[(N[(1.0 / N[(t$95$0 / N[(N[(beta + 1.0), $MachinePrecision] * N[(1.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(beta * N[(5.0 + N[(beta + N[(2.0 * alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 + alpha), $MachinePrecision] * N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(N[(1.0 - N[(N[(N[(2.0 * alpha), $MachinePrecision] + 4.0), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.9999999999999999e144

   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/ 97.7%

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

      2. +-commutative 97.7%

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

      3. associate-+l+ 97.7%

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

      4. *-commutative 97.7%

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

      5. metadata-eval 97.7%

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

      6. associate-+l+ 97.7%

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

      7. metadata-eval 97.7%

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

      8. +-commutative 97.7%

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

      9. +-commutative 97.7%

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

      10. +-commutative 97.7%

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

      11. metadata-eval 97.7%

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

      12. metadata-eval 97.7%

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

      13. associate-+l+ 97.7%

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

   3. Simplified 97.7%

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

      1. clear-num 97.7%

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

      2. inv-pow 97.7%

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

      3. +-commutative 97.7%

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

      4. associate-+r+ 97.7%

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

      5. *-commutative 97.7%

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

      6. +-commutative 97.7%

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

      7. *-commutative 97.7%

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

      8. associate-+r+ 97.7%

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

      9. +-commutative 97.7%

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

      10. distribute-rgt1-in 97.7%

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

      11. fma-define 97.7%

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

   6. Applied egg-rr 97.7%

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

      1. unpow-1 97.7%

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

      2. +-commutative 97.7%

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

      3. +-commutative 97.7%

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

      4. +-commutative 97.7%

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

      5. +-commutative 97.7%

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

      6. fma-undefine 97.7%

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

      7. +-commutative 97.7%

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

      8. *-commutative 97.7%

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

      9. +-commutative 97.7%

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

      10. associate-+r+ 97.7%

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

      11. distribute-lft1-in 97.7%

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

      12. +-commutative 97.7%

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

      13. +-commutative 97.7%

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

   8. Simplified 97.7%

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

   9. Taylor expanded in beta around 0 97.7%

      \[\leadsto \frac{\frac{1}{\frac{\left(\beta + 2\right) + \alpha}{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}}}{\color{blue}{\beta \cdot \left(5 + \left(\beta + 2 \cdot \alpha\right)\right) + \left(2 + \alpha\right) \cdot \left(3 + \alpha\right)}} \]

   if 4.9999999999999999e144 < beta

   1. Initial program 78.6%

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

   3. Simplified 75.8%

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

      1. times-frac 89.7%

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

      2. +-commutative 89.7%

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

   6. Applied egg-rr 89.7%

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

   7. Taylor expanded in beta around inf 91.8%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5 \cdot 10^{+144}:\\ \;\;\;\;\frac{\frac{1}{\frac{\alpha + \left(\beta + 2\right)}{\left(\beta + 1\right) \cdot \left(1 + \alpha\right)}}}{\beta \cdot \left(5 + \left(\beta + 2 \cdot \alpha\right)\right) + \left(2 + \alpha\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 - \frac{2 \cdot \alpha + 4}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 2 \cdot 10^{+145}:\\ \;\;\;\;\frac{\frac{1}{\frac{t\_0}{\left(\beta + 1\right) \cdot \left(1 + \alpha\right)}}}{t\_0 \cdot \left(3 + \left(\beta + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{t\_0} \cdot \frac{1 - \frac{2 \cdot \alpha + 4}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 2e+145)
     (/
      (/ 1.0 (/ t_0 (* (+ beta 1.0) (+ 1.0 alpha))))
      (* t_0 (+ 3.0 (+ beta alpha))))
     (*
      (/ (+ 1.0 alpha) t_0)
      (/ (- 1.0 (/ (+ (* 2.0 alpha) 4.0) beta)) beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 2e+145) {
		tmp = (1.0 / (t_0 / ((beta + 1.0) * (1.0 + alpha)))) / (t_0 * (3.0 + (beta + alpha)));
	} else {
		tmp = ((1.0 + alpha) / t_0) * ((1.0 - (((2.0 * alpha) + 4.0) / 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) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    if (beta <= 2d+145) then
        tmp = (1.0d0 / (t_0 / ((beta + 1.0d0) * (1.0d0 + alpha)))) / (t_0 * (3.0d0 + (beta + alpha)))
    else
        tmp = ((1.0d0 + alpha) / t_0) * ((1.0d0 - (((2.0d0 * alpha) + 4.0d0) / beta)) / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 2e+145) {
		tmp = (1.0 / (t_0 / ((beta + 1.0) * (1.0 + alpha)))) / (t_0 * (3.0 + (beta + alpha)));
	} else {
		tmp = ((1.0 + alpha) / t_0) * ((1.0 - (((2.0 * alpha) + 4.0) / beta)) / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 2e+145:
		tmp = (1.0 / (t_0 / ((beta + 1.0) * (1.0 + alpha)))) / (t_0 * (3.0 + (beta + alpha)))
	else:
		tmp = ((1.0 + alpha) / t_0) * ((1.0 - (((2.0 * alpha) + 4.0) / beta)) / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 2e+145)
		tmp = Float64(Float64(1.0 / Float64(t_0 / Float64(Float64(beta + 1.0) * Float64(1.0 + alpha)))) / Float64(t_0 * Float64(3.0 + Float64(beta + alpha))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / t_0) * Float64(Float64(1.0 - Float64(Float64(Float64(2.0 * alpha) + 4.0) / beta)) / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 2e+145)
		tmp = (1.0 / (t_0 / ((beta + 1.0) * (1.0 + alpha)))) / (t_0 * (3.0 + (beta + alpha)));
	else
		tmp = ((1.0 + alpha) / t_0) * ((1.0 - (((2.0 * alpha) + 4.0) / 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_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2e+145], N[(N[(1.0 / N[(t$95$0 / N[(N[(beta + 1.0), $MachinePrecision] * N[(1.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(N[(1.0 - N[(N[(N[(2.0 * alpha), $MachinePrecision] + 4.0), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2e145

   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/ 97.7%

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

      2. +-commutative 97.7%

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

      3. associate-+l+ 97.7%

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

      4. *-commutative 97.7%

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

      5. metadata-eval 97.7%

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

      6. associate-+l+ 97.7%

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

      7. metadata-eval 97.7%

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

      8. +-commutative 97.7%

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

      9. +-commutative 97.7%

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

      10. +-commutative 97.7%

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

      11. metadata-eval 97.7%

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

      12. metadata-eval 97.7%

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

      13. associate-+l+ 97.7%

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

   3. Simplified 97.7%

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

      1. clear-num 97.7%

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

      2. inv-pow 97.7%

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

      3. +-commutative 97.7%

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

      4. associate-+r+ 97.7%

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

      5. *-commutative 97.7%

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

      6. +-commutative 97.7%

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

      7. *-commutative 97.7%

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

      8. associate-+r+ 97.7%

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

      9. +-commutative 97.7%

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

      10. distribute-rgt1-in 97.7%

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

      11. fma-define 97.7%

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

   6. Applied egg-rr 97.7%

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

      1. unpow-1 97.7%

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

      2. +-commutative 97.7%

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

      3. +-commutative 97.7%

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

      4. +-commutative 97.7%

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

      5. +-commutative 97.7%

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

      6. fma-undefine 97.7%

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

      7. +-commutative 97.7%

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

      8. *-commutative 97.7%

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

      9. +-commutative 97.7%

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

      10. associate-+r+ 97.7%

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

      11. distribute-lft1-in 97.7%

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

      12. +-commutative 97.7%

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

      13. +-commutative 97.7%

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

   8. Simplified 97.7%

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

   if 2e145 < beta

   1. Initial program 78.6%

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

   3. Simplified 75.8%

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

      1. times-frac 89.7%

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

      2. +-commutative 89.7%

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

   6. Applied egg-rr 89.7%

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

   7. Taylor expanded in beta around inf 91.8%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2 \cdot 10^{+145}:\\ \;\;\;\;\frac{\frac{1}{\frac{\alpha + \left(\beta + 2\right)}{\left(\beta + 1\right) \cdot \left(1 + \alpha\right)}}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(3 + \left(\beta + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 - \frac{2 \cdot \alpha + 4}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.4% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.2%

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

3. Simplified 84.0%

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

   1. times-frac 97.0%

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

   2. +-commutative 97.0%

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

6. Applied egg-rr 97.0%

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

7. Taylor expanded in alpha around 0 68.3%

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

   1. associate-/r* 68.7%

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

   2. +-commutative 68.7%

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

   3. +-commutative 68.7%

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

9. Simplified 68.7%

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

    1. clear-num 68.7%

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

    2. frac-2neg 68.7%

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

    3. frac-times 68.6%

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

11. Applied egg-rr 68.6%

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

    1. *-lft-identity 68.6%

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

    2. *-commutative 68.6%

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

    3. associate-/r* 68.7%

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

    4. distribute-neg-frac2 68.7%

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

    5. +-commutative 68.7%

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

    6. distribute-neg-in 68.7%

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

    7. metadata-eval 68.7%

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

    8. unsub-neg 68.7%

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

    9. +-commutative 68.7%

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

    10. distribute-neg-in 68.7%

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

    11. metadata-eval 68.7%

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

    12. unsub-neg 68.7%

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

    13. +-commutative 68.7%

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

13. Simplified 68.7%

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

14. Final simplification 68.7%

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

Alternative 4: 98.3% accurate, 1.7× speedup

Math

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

FPCore

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

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return ((1.0 + alpha) / (alpha + (beta + 2.0))) * (((beta + 1.0) / (beta + 2.0)) / (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 = ((1.0d0 + alpha) / (alpha + (beta + 2.0d0))) * (((beta + 1.0d0) / (beta + 2.0d0)) / (beta + 3.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.2%

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

3. Simplified 84.0%

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

   1. times-frac 97.0%

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

   2. +-commutative 97.0%

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

6. Applied egg-rr 97.0%

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

7. Taylor expanded in alpha around 0 68.3%

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

   1. associate-/r* 68.7%

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

   2. +-commutative 68.7%

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

   3. +-commutative 68.7%

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

9. Simplified 68.7%

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

10. Final simplification 68.7%

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

Alternative 5: 98.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 9.7 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{\frac{\beta + 1}{-2 - \beta}}{-3 - \beta}}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 9.7e+14)
   (/ (/ (/ (+ beta 1.0) (- -2.0 beta)) (- -3.0 beta)) (+ beta 2.0))
   (/ (/ (+ 1.0 alpha) (+ alpha (+ beta 2.0))) (+ alpha (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 9.7e+14) {
		tmp = (((beta + 1.0) / (-2.0 - beta)) / (-3.0 - beta)) / (beta + 2.0);
	} else {
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 9.7d+14) then
        tmp = (((beta + 1.0d0) / ((-2.0d0) - beta)) / ((-3.0d0) - beta)) / (beta + 2.0d0)
    else
        tmp = ((1.0d0 + alpha) / (alpha + (beta + 2.0d0))) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 9.7e+14) {
		tmp = (((beta + 1.0) / (-2.0 - beta)) / (-3.0 - beta)) / (beta + 2.0);
	} else {
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 9.7e+14:
		tmp = (((beta + 1.0) / (-2.0 - beta)) / (-3.0 - beta)) / (beta + 2.0)
	else:
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) / (alpha + (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 9.7e+14)
		tmp = Float64(Float64(Float64(Float64(beta + 1.0) / Float64(-2.0 - beta)) / Float64(-3.0 - beta)) / Float64(beta + 2.0));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(beta + 2.0))) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 9.7e+14)
		tmp = (((beta + 1.0) / (-2.0 - beta)) / (-3.0 - beta)) / (beta + 2.0);
	else
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 9.7e+14], N[(N[(N[(N[(beta + 1.0), $MachinePrecision] / N[(-2.0 - beta), $MachinePrecision]), $MachinePrecision] / N[(-3.0 - beta), $MachinePrecision]), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 9.7e14

   1. Initial program 99.8%

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

   3. Simplified 91.3%

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

      1. times-frac 99.5%

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

      2. +-commutative 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in alpha around 0 63.1%

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

      1. associate-/r* 63.1%

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

      2. +-commutative 63.1%

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

      3. +-commutative 63.1%

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

   9. Simplified 63.1%

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

       1. clear-num 63.1%

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

       2. frac-2neg 63.1%

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

       3. frac-times 63.1%

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

   11. Applied egg-rr 63.1%

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

       1. *-lft-identity 63.1%

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

       2. *-commutative 63.1%

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

       3. associate-/r* 63.1%

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

       4. distribute-neg-frac2 63.1%

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

       5. +-commutative 63.1%

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

       6. distribute-neg-in 63.1%

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

       7. metadata-eval 63.1%

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

       8. unsub-neg 63.1%

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

       9. +-commutative 63.1%

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

       10. distribute-neg-in 63.1%

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

       11. metadata-eval 63.1%

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

       12. unsub-neg 63.1%

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

       13. +-commutative 63.1%

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

   13. Simplified 63.1%

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

   14. Taylor expanded in alpha around 0 63.2%

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

   if 9.7e14 < beta

   1. Initial program 84.5%

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

      1. associate-/l/ 80.3%

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

      2. +-commutative 80.3%

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

      3. associate-+l+ 80.3%

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

      4. *-commutative 80.3%

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

      5. metadata-eval 80.3%

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

      6. associate-+l+ 80.3%

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

      7. metadata-eval 80.3%

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

      8. +-commutative 80.3%

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

      9. +-commutative 80.3%

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

      10. +-commutative 80.3%

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

      11. metadata-eval 80.3%

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

      12. metadata-eval 80.3%

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

      13. associate-+l+ 80.3%

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

   3. Simplified 80.3%

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

      1. clear-num 80.3%

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

      2. inv-pow 80.3%

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

      3. +-commutative 80.3%

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

      4. associate-+r+ 80.3%

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

      5. *-commutative 80.3%

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

      6. +-commutative 80.3%

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

      7. *-commutative 80.3%

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

      8. associate-+r+ 80.3%

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

      9. +-commutative 80.3%

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

      10. distribute-rgt1-in 80.3%

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

      11. fma-define 80.3%

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

   6. Applied egg-rr 80.3%

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

      1. unpow-1 80.3%

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

      2. +-commutative 80.3%

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

      3. +-commutative 80.3%

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

      4. +-commutative 80.3%

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

      5. +-commutative 80.3%

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

      6. fma-undefine 80.3%

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

      7. +-commutative 80.3%

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

      8. *-commutative 80.3%

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

      9. +-commutative 80.3%

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

      10. associate-+r+ 80.3%

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

      11. distribute-lft1-in 80.3%

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

      12. +-commutative 80.3%

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

      13. +-commutative 80.3%

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

   8. Simplified 80.3%

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

   9. Taylor expanded in beta around inf 84.6%

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

       1. *-un-lft-identity 84.6%

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

       2. *-commutative 84.6%

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

       3. associate-+l+ 84.6%

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

   11. Applied egg-rr 84.6%

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

       1. *-lft-identity 84.6%

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

       2. associate-/r* 82.2%

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

       3. +-commutative 82.2%

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

   13. Simplified 82.2%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 9.7 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{\frac{\beta + 1}{-2 - \beta}}{-3 - \beta}}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.6% accurate, 1.7× speedup

Math

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

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5.0)
   (/ (/ (+ 1.0 alpha) (+ 2.0 alpha)) (* (+ 2.0 alpha) (+ alpha 3.0)))
   (/ (/ (+ 1.0 alpha) (+ alpha (+ beta 2.0))) (+ alpha (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.0) {
		tmp = ((1.0 + alpha) / (2.0 + alpha)) / ((2.0 + alpha) * (alpha + 3.0));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 5.0d0) then
        tmp = ((1.0d0 + alpha) / (2.0d0 + alpha)) / ((2.0d0 + alpha) * (alpha + 3.0d0))
    else
        tmp = ((1.0d0 + alpha) / (alpha + (beta + 2.0d0))) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.0) {
		tmp = ((1.0 + alpha) / (2.0 + alpha)) / ((2.0 + alpha) * (alpha + 3.0));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 5.0:
		tmp = ((1.0 + alpha) / (2.0 + alpha)) / ((2.0 + alpha) * (alpha + 3.0))
	else:
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) / (alpha + (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 5.0)
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(2.0 + alpha)) / Float64(Float64(2.0 + alpha) * Float64(alpha + 3.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(beta + 2.0))) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 5.0)
		tmp = ((1.0 + alpha) / (2.0 + alpha)) / ((2.0 + alpha) * (alpha + 3.0));
	else
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 5.0], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(2.0 + alpha), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 + alpha), $MachinePrecision] * N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5

   1. Initial program 99.8%

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

      1. associate-/l/ 99.5%

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

      2. +-commutative 99.5%

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

      3. associate-+l+ 99.5%

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

      4. *-commutative 99.5%

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

      5. metadata-eval 99.5%

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

      6. associate-+l+ 99.5%

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

      7. metadata-eval 99.5%

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

      8. +-commutative 99.5%

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

      9. +-commutative 99.5%

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

      10. +-commutative 99.5%

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

      11. metadata-eval 99.5%

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

      12. metadata-eval 99.5%

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

      13. associate-+l+ 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in beta around 0 97.2%

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

   6. Taylor expanded in beta around 0 97.2%

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

   if 5 < beta

   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. Step-by-step derivation

      1. associate-/l/ 80.8%

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

      2. +-commutative 80.8%

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

      3. associate-+l+ 80.8%

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

      4. *-commutative 80.8%

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

      5. metadata-eval 80.8%

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

      6. associate-+l+ 80.8%

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

      7. metadata-eval 80.8%

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

      8. +-commutative 80.8%

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

      9. +-commutative 80.8%

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

      10. +-commutative 80.8%

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

      11. metadata-eval 80.8%

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

      12. metadata-eval 80.8%

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

      13. associate-+l+ 80.8%

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

   3. Simplified 80.8%

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

      1. clear-num 80.8%

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

      2. inv-pow 80.8%

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

      3. +-commutative 80.8%

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

      4. associate-+r+ 80.8%

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

      5. *-commutative 80.8%

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

      6. +-commutative 80.8%

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

      7. *-commutative 80.8%

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

      8. associate-+r+ 80.8%

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

      9. +-commutative 80.8%

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

      10. distribute-rgt1-in 80.8%

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

      11. fma-define 80.8%

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

   6. Applied egg-rr 80.8%

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

      1. unpow-1 80.8%

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

      2. +-commutative 80.8%

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

      3. +-commutative 80.8%

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

      4. +-commutative 80.8%

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

      5. +-commutative 80.8%

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

      6. fma-undefine 80.8%

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

      7. +-commutative 80.8%

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

      8. *-commutative 80.8%

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

      9. +-commutative 80.8%

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

      10. associate-+r+ 80.8%

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

      11. distribute-lft1-in 80.8%

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

      12. +-commutative 80.8%

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

      13. +-commutative 80.8%

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

   8. Simplified 80.8%

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

   9. Taylor expanded in beta around inf 83.1%

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

       1. *-un-lft-identity 83.1%

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

       2. *-commutative 83.1%

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

       3. associate-+l+ 83.1%

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

   11. Applied egg-rr 83.1%

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

       1. *-lft-identity 83.1%

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

       2. associate-/r* 80.9%

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

       3. +-commutative 80.9%

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

   13. Simplified 80.9%

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

4. Final simplification 92.2%

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

Alternative 7: 97.5% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if beta < 5.4000000000000004

   1. Initial program 99.8%

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

      1. associate-/l/ 99.5%

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

      2. +-commutative 99.5%

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

      3. associate-+l+ 99.5%

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

      4. *-commutative 99.5%

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

      5. metadata-eval 99.5%

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

      6. associate-+l+ 99.5%

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

      7. metadata-eval 99.5%

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

      8. +-commutative 99.5%

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

      9. +-commutative 99.5%

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

      10. +-commutative 99.5%

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

      11. metadata-eval 99.5%

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

      12. metadata-eval 99.5%

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

      13. associate-+l+ 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in beta around 0 97.2%

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

   6. Taylor expanded in beta around 0 97.2%

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

   if 5.4000000000000004 < beta

   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. Add Preprocessing

   3. Taylor expanded in beta around inf 80.3%

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

      1. div-inv 80.2%

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

      2. +-commutative 80.2%

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

      3. metadata-eval 80.2%

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

      4. associate-+l+ 80.2%

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

      5. metadata-eval 80.2%

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

      6. associate-+r+ 80.2%

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

   5. Applied egg-rr 80.2%

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

      1. associate-*r/ 80.3%

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

      2. *-commutative 80.3%

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

      3. *-lft-identity 80.3%

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

      4. +-commutative 80.3%

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

      5. +-commutative 80.3%

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

      6. +-commutative 80.3%

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

      7. +-commutative 80.3%

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

   7. Simplified 80.3%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.4:\\ \;\;\;\;\frac{\frac{1 + \alpha}{2 + \alpha}}{\left(2 + \alpha\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.0% accurate, 1.7× speedup

Math

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

FPCore

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

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.3) {
		tmp = (0.16666666666666666 + (beta * 0.027777777777777776)) / ((alpha + (beta + 2.0)) / (1.0 + alpha));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 3.3d0) then
        tmp = (0.16666666666666666d0 + (beta * 0.027777777777777776d0)) / ((alpha + (beta + 2.0d0)) / (1.0d0 + alpha))
    else
        tmp = ((1.0d0 + alpha) / beta) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.3) {
		tmp = (0.16666666666666666 + (beta * 0.027777777777777776)) / ((alpha + (beta + 2.0)) / (1.0 + alpha));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 3.3:
		tmp = (0.16666666666666666 + (beta * 0.027777777777777776)) / ((alpha + (beta + 2.0)) / (1.0 + alpha))
	else:
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.3)
		tmp = Float64(Float64(0.16666666666666666 + Float64(beta * 0.027777777777777776)) / Float64(Float64(alpha + Float64(beta + 2.0)) / Float64(1.0 + alpha)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.3)
		tmp = (0.16666666666666666 + (beta * 0.027777777777777776)) / ((alpha + (beta + 2.0)) / (1.0 + alpha));
	else
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 3.3], N[(N[(0.16666666666666666 + N[(beta * 0.027777777777777776), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.2999999999999998

   1. Initial program 99.8%

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

   3. Simplified 91.2%

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

      1. times-frac 99.5%

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

      2. +-commutative 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in alpha around 0 62.7%

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

      1. associate-/r* 62.7%

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

      2. +-commutative 62.7%

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

      3. +-commutative 62.7%

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

   9. Simplified 62.7%

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

       1. clear-num 62.7%

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

       2. frac-2neg 62.7%

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

       3. frac-times 62.7%

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

   11. Applied egg-rr 62.7%

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

       1. *-lft-identity 62.7%

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

       2. *-commutative 62.7%

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

       3. associate-/r* 62.7%

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

       4. distribute-neg-frac2 62.7%

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

       5. +-commutative 62.7%

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

       6. distribute-neg-in 62.7%

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

       7. metadata-eval 62.7%

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

       8. unsub-neg 62.7%

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

       9. +-commutative 62.7%

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

       10. distribute-neg-in 62.7%

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

       11. metadata-eval 62.7%

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

       12. unsub-neg 62.7%

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

       13. +-commutative 62.7%

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

   13. Simplified 62.7%

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

   14. Taylor expanded in beta around 0 62.1%

       \[\leadsto \frac{\color{blue}{0.16666666666666666 + 0.027777777777777776 \cdot \beta}}{\frac{\alpha + \left(\beta + 2\right)}{1 + \alpha}} \]
   15. Step-by-step derivation

       1. *-commutative 62.1%

          \[\leadsto \frac{0.16666666666666666 + \color{blue}{\beta \cdot 0.027777777777777776}}{\frac{\alpha + \left(\beta + 2\right)}{1 + \alpha}} \]

   16. Simplified 62.1%

       \[\leadsto \frac{\color{blue}{0.16666666666666666 + \beta \cdot 0.027777777777777776}}{\frac{\alpha + \left(\beta + 2\right)}{1 + \alpha}} \]

   if 3.2999999999999998 < beta

   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. Add Preprocessing

   3. Taylor expanded in beta around inf 80.3%

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

      1. div-inv 80.2%

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

      2. +-commutative 80.2%

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

      3. metadata-eval 80.2%

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

      4. associate-+l+ 80.2%

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

      5. metadata-eval 80.2%

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

      6. associate-+r+ 80.2%

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

   5. Applied egg-rr 80.2%

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

      1. associate-*r/ 80.3%

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

      2. *-commutative 80.3%

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

      3. *-lft-identity 80.3%

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

      4. +-commutative 80.3%

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

      5. +-commutative 80.3%

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

      6. +-commutative 80.3%

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

      7. +-commutative 80.3%

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

   7. Simplified 80.3%

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

4. Final simplification 67.7%

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

Alternative 9: 97.0% accurate, 1.7× speedup

Math

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

FPCore

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

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.3) {
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) * (0.16666666666666666 + (beta * 0.027777777777777776));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 3.3d0) then
        tmp = ((1.0d0 + alpha) / (alpha + (beta + 2.0d0))) * (0.16666666666666666d0 + (beta * 0.027777777777777776d0))
    else
        tmp = ((1.0d0 + alpha) / beta) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.3) {
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) * (0.16666666666666666 + (beta * 0.027777777777777776));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 3.3:
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) * (0.16666666666666666 + (beta * 0.027777777777777776))
	else:
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.3)
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(beta + 2.0))) * Float64(0.16666666666666666 + Float64(beta * 0.027777777777777776)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.3)
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) * (0.16666666666666666 + (beta * 0.027777777777777776));
	else
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 3.3], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.16666666666666666 + N[(beta * 0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.2999999999999998

   1. Initial program 99.8%

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

   3. Simplified 91.2%

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

      1. times-frac 99.5%

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

      2. +-commutative 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in alpha around 0 62.7%

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

      1. associate-/r* 62.7%

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

      2. +-commutative 62.7%

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

      3. +-commutative 62.7%

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

   9. Simplified 62.7%

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

   10. Taylor expanded in beta around 0 62.1%

       \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \color{blue}{\left(0.16666666666666666 + 0.027777777777777776 \cdot \beta\right)} \]
   11. Step-by-step derivation

       1. *-commutative 62.1%

          \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \left(0.16666666666666666 + \color{blue}{\beta \cdot 0.027777777777777776}\right) \]

   12. Simplified 62.1%

       \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \color{blue}{\left(0.16666666666666666 + \beta \cdot 0.027777777777777776\right)} \]

   if 3.2999999999999998 < beta

   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. Add Preprocessing

   3. Taylor expanded in beta around inf 80.3%

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

      1. div-inv 80.2%

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

      2. +-commutative 80.2%

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

      3. metadata-eval 80.2%

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

      4. associate-+l+ 80.2%

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

      5. metadata-eval 80.2%

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

      6. associate-+r+ 80.2%

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

   5. Applied egg-rr 80.2%

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

      1. associate-*r/ 80.3%

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

      2. *-commutative 80.3%

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

      3. *-lft-identity 80.3%

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

      4. +-commutative 80.3%

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

      5. +-commutative 80.3%

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

      6. +-commutative 80.3%

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

      7. +-commutative 80.3%

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

   7. Simplified 80.3%

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

4. Final simplification 67.7%

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

Alternative 10: 96.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.4:\\ \;\;\;\;\frac{0.16666666666666666}{\frac{\alpha + \left(\beta + 2\right)}{1 + \alpha}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5.4)
   (/ 0.16666666666666666 (/ (+ alpha (+ beta 2.0)) (+ 1.0 alpha)))
   (/ (/ (+ 1.0 alpha) beta) (+ alpha (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.4) {
		tmp = 0.16666666666666666 / ((alpha + (beta + 2.0)) / (1.0 + alpha));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 5.4d0) then
        tmp = 0.16666666666666666d0 / ((alpha + (beta + 2.0d0)) / (1.0d0 + alpha))
    else
        tmp = ((1.0d0 + alpha) / beta) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.4) {
		tmp = 0.16666666666666666 / ((alpha + (beta + 2.0)) / (1.0 + alpha));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 5.4:
		tmp = 0.16666666666666666 / ((alpha + (beta + 2.0)) / (1.0 + alpha))
	else:
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 5.4)
		tmp = Float64(0.16666666666666666 / Float64(Float64(alpha + Float64(beta + 2.0)) / Float64(1.0 + alpha)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 5.4)
		tmp = 0.16666666666666666 / ((alpha + (beta + 2.0)) / (1.0 + alpha));
	else
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 5.4], N[(0.16666666666666666 / N[(N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.4000000000000004

   1. Initial program 99.8%

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

   3. Simplified 91.2%

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

      1. times-frac 99.5%

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

      2. +-commutative 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in alpha around 0 62.7%

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

      1. associate-/r* 62.7%

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

      2. +-commutative 62.7%

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

      3. +-commutative 62.7%

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

   9. Simplified 62.7%

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

       1. clear-num 62.7%

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

       2. frac-2neg 62.7%

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

       3. frac-times 62.7%

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

   11. Applied egg-rr 62.7%

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

       1. *-lft-identity 62.7%

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

       2. *-commutative 62.7%

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

       3. associate-/r* 62.7%

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

       4. distribute-neg-frac2 62.7%

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

       5. +-commutative 62.7%

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

       6. distribute-neg-in 62.7%

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

       7. metadata-eval 62.7%

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

       8. unsub-neg 62.7%

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

       9. +-commutative 62.7%

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

       10. distribute-neg-in 62.7%

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

       11. metadata-eval 62.7%

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

       12. unsub-neg 62.7%

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

       13. +-commutative 62.7%

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

   13. Simplified 62.7%

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

   14. Taylor expanded in beta around 0 62.0%

       \[\leadsto \frac{\color{blue}{0.16666666666666666}}{\frac{\alpha + \left(\beta + 2\right)}{1 + \alpha}} \]

   if 5.4000000000000004 < beta

   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. Add Preprocessing

   3. Taylor expanded in beta around inf 80.3%

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

      1. div-inv 80.2%

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

      2. +-commutative 80.2%

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

      3. metadata-eval 80.2%

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

      4. associate-+l+ 80.2%

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

      5. metadata-eval 80.2%

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

      6. associate-+r+ 80.2%

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

   5. Applied egg-rr 80.2%

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

      1. associate-*r/ 80.3%

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

      2. *-commutative 80.3%

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

      3. *-lft-identity 80.3%

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

      4. +-commutative 80.3%

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

      5. +-commutative 80.3%

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

      6. +-commutative 80.3%

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

      7. +-commutative 80.3%

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

   7. Simplified 80.3%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.4:\\ \;\;\;\;\frac{0.16666666666666666}{\frac{\alpha + \left(\beta + 2\right)}{1 + \alpha}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 96.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.4:\\ \;\;\;\;\frac{0.16666666666666666}{\frac{\alpha + \left(\beta + 2\right)}{1 + \alpha}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5.4)
   (/ 0.16666666666666666 (/ (+ alpha (+ beta 2.0)) (+ 1.0 alpha)))
   (/ (/ (+ 1.0 alpha) beta) (+ beta 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.4) {
		tmp = 0.16666666666666666 / ((alpha + (beta + 2.0)) / (1.0 + alpha));
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 5.4d0) then
        tmp = 0.16666666666666666d0 / ((alpha + (beta + 2.0d0)) / (1.0d0 + alpha))
    else
        tmp = ((1.0d0 + alpha) / beta) / (beta + 3.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.4) {
		tmp = 0.16666666666666666 / ((alpha + (beta + 2.0)) / (1.0 + alpha));
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 5.4:
		tmp = 0.16666666666666666 / ((alpha + (beta + 2.0)) / (1.0 + alpha))
	else:
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 5.4)
		tmp = Float64(0.16666666666666666 / Float64(Float64(alpha + Float64(beta + 2.0)) / Float64(1.0 + alpha)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(beta + 3.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 5.4)
		tmp = 0.16666666666666666 / ((alpha + (beta + 2.0)) / (1.0 + alpha));
	else
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 5.4], N[(0.16666666666666666 / N[(N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.4000000000000004

   1. Initial program 99.8%

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

   3. Simplified 91.2%

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

      1. times-frac 99.5%

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

      2. +-commutative 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in alpha around 0 62.7%

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

      1. associate-/r* 62.7%

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

      2. +-commutative 62.7%

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

      3. +-commutative 62.7%

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

   9. Simplified 62.7%

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

       1. clear-num 62.7%

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

       2. frac-2neg 62.7%

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

       3. frac-times 62.7%

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

   11. Applied egg-rr 62.7%

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

       1. *-lft-identity 62.7%

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

       2. *-commutative 62.7%

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

       3. associate-/r* 62.7%

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

       4. distribute-neg-frac2 62.7%

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

       5. +-commutative 62.7%

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

       6. distribute-neg-in 62.7%

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

       7. metadata-eval 62.7%

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

       8. unsub-neg 62.7%

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

       9. +-commutative 62.7%

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

       10. distribute-neg-in 62.7%

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

       11. metadata-eval 62.7%

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

       12. unsub-neg 62.7%

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

       13. +-commutative 62.7%

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

   13. Simplified 62.7%

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

   14. Taylor expanded in beta around 0 62.0%

       \[\leadsto \frac{\color{blue}{0.16666666666666666}}{\frac{\alpha + \left(\beta + 2\right)}{1 + \alpha}} \]

   if 5.4000000000000004 < beta

   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. Add Preprocessing

   3. Taylor expanded in beta around inf 80.3%

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

   4. Taylor expanded in alpha around 0 80.1%

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

      1. +-commutative 80.1%

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

   6. Simplified 80.1%

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

Alternative 12: 96.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.4:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot 0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5.4)
   (* (/ (+ 1.0 alpha) (+ alpha (+ beta 2.0))) 0.16666666666666666)
   (/ (/ (+ 1.0 alpha) beta) (+ beta 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.4) {
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) * 0.16666666666666666;
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 5.4d0) then
        tmp = ((1.0d0 + alpha) / (alpha + (beta + 2.0d0))) * 0.16666666666666666d0
    else
        tmp = ((1.0d0 + alpha) / beta) / (beta + 3.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.4) {
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) * 0.16666666666666666;
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 5.4:
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) * 0.16666666666666666
	else:
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 5.4)
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(beta + 2.0))) * 0.16666666666666666);
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(beta + 3.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 5.4)
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) * 0.16666666666666666;
	else
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 5.4], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.16666666666666666), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.4000000000000004

   1. Initial program 99.8%

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

   3. Simplified 91.2%

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

      1. times-frac 99.5%

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

      2. +-commutative 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in alpha around 0 62.7%

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

      1. associate-/r* 62.7%

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

      2. +-commutative 62.7%

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

      3. +-commutative 62.7%

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

   9. Simplified 62.7%

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

   10. Taylor expanded in beta around 0 62.0%

       \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \color{blue}{0.16666666666666666} \]

   if 5.4000000000000004 < beta

   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. Add Preprocessing

   3. Taylor expanded in beta around inf 80.3%

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

   4. Taylor expanded in alpha around 0 80.1%

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

      1. +-commutative 80.1%

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

   6. Simplified 80.1%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.4:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot 0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 96.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.3:\\ \;\;\;\;\frac{0.16666666666666666 + \alpha \cdot 0.16666666666666666}{2 + \alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.3)
   (/ (+ 0.16666666666666666 (* alpha 0.16666666666666666)) (+ 2.0 alpha))
   (/ (/ (+ 1.0 alpha) beta) (+ beta 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.3) {
		tmp = (0.16666666666666666 + (alpha * 0.16666666666666666)) / (2.0 + alpha);
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.3d0) then
        tmp = (0.16666666666666666d0 + (alpha * 0.16666666666666666d0)) / (2.0d0 + alpha)
    else
        tmp = ((1.0d0 + alpha) / beta) / (beta + 3.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.3) {
		tmp = (0.16666666666666666 + (alpha * 0.16666666666666666)) / (2.0 + alpha);
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.3:
		tmp = (0.16666666666666666 + (alpha * 0.16666666666666666)) / (2.0 + alpha)
	else:
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.3)
		tmp = Float64(Float64(0.16666666666666666 + Float64(alpha * 0.16666666666666666)) / Float64(2.0 + alpha));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(beta + 3.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.3)
		tmp = (0.16666666666666666 + (alpha * 0.16666666666666666)) / (2.0 + alpha);
	else
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.3], N[(N[(0.16666666666666666 + N[(alpha * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] / N[(2.0 + alpha), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.2999999999999998

   1. Initial program 99.8%

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

   3. Simplified 91.2%

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

      1. times-frac 99.5%

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

      2. +-commutative 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in alpha around 0 62.7%

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

      1. associate-/r* 62.7%

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

      2. +-commutative 62.7%

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

      3. +-commutative 62.7%

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

   9. Simplified 62.7%

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

       1. clear-num 62.7%

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

       2. frac-2neg 62.7%

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

       3. frac-times 62.7%

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

   11. Applied egg-rr 62.7%

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

       1. *-lft-identity 62.7%

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

       2. *-commutative 62.7%

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

       3. associate-/r* 62.7%

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

       4. distribute-neg-frac2 62.7%

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

       5. +-commutative 62.7%

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

       6. distribute-neg-in 62.7%

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

       7. metadata-eval 62.7%

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

       8. unsub-neg 62.7%

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

       9. +-commutative 62.7%

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

       10. distribute-neg-in 62.7%

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

       11. metadata-eval 62.7%

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

       12. unsub-neg 62.7%

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

       13. +-commutative 62.7%

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

   13. Simplified 62.7%

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

   14. Taylor expanded in beta around 0 61.9%

       \[\leadsto \color{blue}{0.16666666666666666 \cdot \frac{1 + \alpha}{2 + \alpha}} \]
   15. Step-by-step derivation

       1. associate-*r/ 61.9%

          \[\leadsto \color{blue}{\frac{0.16666666666666666 \cdot \left(1 + \alpha\right)}{2 + \alpha}} \]

       2. distribute-lft-in 61.9%

          \[\leadsto \frac{\color{blue}{0.16666666666666666 \cdot 1 + 0.16666666666666666 \cdot \alpha}}{2 + \alpha} \]

       3. metadata-eval 61.9%

          \[\leadsto \frac{\color{blue}{0.16666666666666666} + 0.16666666666666666 \cdot \alpha}{2 + \alpha} \]

       4. +-commutative 61.9%

          \[\leadsto \frac{0.16666666666666666 + 0.16666666666666666 \cdot \alpha}{\color{blue}{\alpha + 2}} \]

   16. Simplified 61.9%

       \[\leadsto \color{blue}{\frac{0.16666666666666666 + 0.16666666666666666 \cdot \alpha}{\alpha + 2}} \]

   if 2.2999999999999998 < beta

   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. Add Preprocessing

   3. Taylor expanded in beta around inf 80.3%

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

   4. Taylor expanded in alpha around 0 80.1%

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

      1. +-commutative 80.1%

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

   6. Simplified 80.1%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.3:\\ \;\;\;\;\frac{0.16666666666666666 + \alpha \cdot 0.16666666666666666}{2 + \alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 91.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.3:\\ \;\;\;\;\frac{0.16666666666666666 + \alpha \cdot 0.16666666666666666}{2 + \alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta + 3}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.3)
   (/ (+ 0.16666666666666666 (* alpha 0.16666666666666666)) (+ 2.0 alpha))
   (/ (/ 1.0 (+ beta 3.0)) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.3) {
		tmp = (0.16666666666666666 + (alpha * 0.16666666666666666)) / (2.0 + alpha);
	} else {
		tmp = (1.0 / (beta + 3.0)) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.3d0) then
        tmp = (0.16666666666666666d0 + (alpha * 0.16666666666666666d0)) / (2.0d0 + alpha)
    else
        tmp = (1.0d0 / (beta + 3.0d0)) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.3) {
		tmp = (0.16666666666666666 + (alpha * 0.16666666666666666)) / (2.0 + alpha);
	} else {
		tmp = (1.0 / (beta + 3.0)) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.3:
		tmp = (0.16666666666666666 + (alpha * 0.16666666666666666)) / (2.0 + alpha)
	else:
		tmp = (1.0 / (beta + 3.0)) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.3)
		tmp = Float64(Float64(0.16666666666666666 + Float64(alpha * 0.16666666666666666)) / Float64(2.0 + alpha));
	else
		tmp = Float64(Float64(1.0 / Float64(beta + 3.0)) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.3)
		tmp = (0.16666666666666666 + (alpha * 0.16666666666666666)) / (2.0 + alpha);
	else
		tmp = (1.0 / (beta + 3.0)) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.3], N[(N[(0.16666666666666666 + N[(alpha * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] / N[(2.0 + alpha), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.2999999999999998

   1. Initial program 99.8%

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

   3. Simplified 91.2%

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

      1. times-frac 99.5%

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

      2. +-commutative 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in alpha around 0 62.7%

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

      1. associate-/r* 62.7%

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

      2. +-commutative 62.7%

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

      3. +-commutative 62.7%

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

   9. Simplified 62.7%

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

       1. clear-num 62.7%

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

       2. frac-2neg 62.7%

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

       3. frac-times 62.7%

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

   11. Applied egg-rr 62.7%

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

       1. *-lft-identity 62.7%

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

       2. *-commutative 62.7%

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

       3. associate-/r* 62.7%

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

       4. distribute-neg-frac2 62.7%

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

       5. +-commutative 62.7%

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

       6. distribute-neg-in 62.7%

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

       7. metadata-eval 62.7%

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

       8. unsub-neg 62.7%

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

       9. +-commutative 62.7%

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

       10. distribute-neg-in 62.7%

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

       11. metadata-eval 62.7%

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

       12. unsub-neg 62.7%

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

       13. +-commutative 62.7%

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

   13. Simplified 62.7%

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

   14. Taylor expanded in beta around 0 61.9%

       \[\leadsto \color{blue}{0.16666666666666666 \cdot \frac{1 + \alpha}{2 + \alpha}} \]
   15. Step-by-step derivation

       1. associate-*r/ 61.9%

          \[\leadsto \color{blue}{\frac{0.16666666666666666 \cdot \left(1 + \alpha\right)}{2 + \alpha}} \]

       2. distribute-lft-in 61.9%

          \[\leadsto \frac{\color{blue}{0.16666666666666666 \cdot 1 + 0.16666666666666666 \cdot \alpha}}{2 + \alpha} \]

       3. metadata-eval 61.9%

          \[\leadsto \frac{\color{blue}{0.16666666666666666} + 0.16666666666666666 \cdot \alpha}{2 + \alpha} \]

       4. +-commutative 61.9%

          \[\leadsto \frac{0.16666666666666666 + 0.16666666666666666 \cdot \alpha}{\color{blue}{\alpha + 2}} \]

   16. Simplified 61.9%

       \[\leadsto \color{blue}{\frac{0.16666666666666666 + 0.16666666666666666 \cdot \alpha}{\alpha + 2}} \]

   if 2.2999999999999998 < beta

   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. Add Preprocessing

   3. Taylor expanded in beta around inf 80.3%

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

   4. Taylor expanded in alpha around 0 74.2%

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

      1. inv-pow 74.2%

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

      2. +-commutative 74.2%

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

      3. unpow-prod-down 74.4%

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

      4. inv-pow 74.4%

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

   6. Applied egg-rr 74.4%

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

      1. associate-*l/ 74.5%

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

      2. *-lft-identity 74.5%

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

      3. unpow-1 74.5%

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

   8. Simplified 74.5%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.3:\\ \;\;\;\;\frac{0.16666666666666666 + \alpha \cdot 0.16666666666666666}{2 + \alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta + 3}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 91.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.3:\\ \;\;\;\;\frac{1 + \alpha}{2 + \alpha} \cdot 0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta + 3}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.3)
   (* (/ (+ 1.0 alpha) (+ 2.0 alpha)) 0.16666666666666666)
   (/ (/ 1.0 (+ beta 3.0)) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.3) {
		tmp = ((1.0 + alpha) / (2.0 + alpha)) * 0.16666666666666666;
	} else {
		tmp = (1.0 / (beta + 3.0)) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.3d0) then
        tmp = ((1.0d0 + alpha) / (2.0d0 + alpha)) * 0.16666666666666666d0
    else
        tmp = (1.0d0 / (beta + 3.0d0)) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.3) {
		tmp = ((1.0 + alpha) / (2.0 + alpha)) * 0.16666666666666666;
	} else {
		tmp = (1.0 / (beta + 3.0)) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.3:
		tmp = ((1.0 + alpha) / (2.0 + alpha)) * 0.16666666666666666
	else:
		tmp = (1.0 / (beta + 3.0)) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.3)
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(2.0 + alpha)) * 0.16666666666666666);
	else
		tmp = Float64(Float64(1.0 / Float64(beta + 3.0)) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.3)
		tmp = ((1.0 + alpha) / (2.0 + alpha)) * 0.16666666666666666;
	else
		tmp = (1.0 / (beta + 3.0)) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.3], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(2.0 + alpha), $MachinePrecision]), $MachinePrecision] * 0.16666666666666666), $MachinePrecision], N[(N[(1.0 / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.2999999999999998

   1. Initial program 99.8%

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

   3. Simplified 91.2%

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

      1. times-frac 99.5%

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

      2. +-commutative 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in alpha around 0 62.7%

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

      1. associate-/r* 62.7%

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

      2. +-commutative 62.7%

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

      3. +-commutative 62.7%

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

   9. Simplified 62.7%

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

   10. Taylor expanded in beta around 0 61.9%

       \[\leadsto \color{blue}{0.16666666666666666 \cdot \frac{1 + \alpha}{2 + \alpha}} \]

   if 2.2999999999999998 < beta

   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. Add Preprocessing

   3. Taylor expanded in beta around inf 80.3%

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

   4. Taylor expanded in alpha around 0 74.2%

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

      1. inv-pow 74.2%

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

      2. +-commutative 74.2%

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

      3. unpow-prod-down 74.4%

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

      4. inv-pow 74.4%

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

   6. Applied egg-rr 74.4%

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

      1. associate-*l/ 74.5%

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

      2. *-lft-identity 74.5%

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

      3. unpow-1 74.5%

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

   8. Simplified 74.5%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.3:\\ \;\;\;\;\frac{1 + \alpha}{2 + \alpha} \cdot 0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta + 3}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 51.0% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.2%

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

3. Taylor expanded in beta around inf 26.8%

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

4. Taylor expanded in alpha around 0 25.0%

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

   1. inv-pow 25.0%

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

   2. +-commutative 25.0%

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

   3. unpow-prod-down 25.1%

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

   4. inv-pow 25.1%

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

6. Applied egg-rr 25.1%

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

   1. associate-*l/ 25.1%

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

   2. *-lft-identity 25.1%

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

   3. unpow-1 25.1%

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

8. Simplified 25.1%

   \[\leadsto \color{blue}{\frac{\frac{1}{\beta + 3}}{\beta}} \]
9. Add Preprocessing

Alternative 17: 51.0% accurate, 5.0× speedup

Math

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

FPCore

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

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return (1.0 / beta) / (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 = (1.0d0 / beta) / (beta + 3.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.2%

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

3. Taylor expanded in beta around inf 26.8%

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

4. Taylor expanded in alpha around 0 25.0%

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

   1. associate-/r* 25.1%

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

   2. +-commutative 25.1%

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

6. Simplified 25.1%

   \[\leadsto \color{blue}{\frac{\frac{1}{\beta}}{\beta + 3}} \]
7. Add Preprocessing

Alternative 18: 50.6% accurate, 5.0× speedup

Math

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

FPCore

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

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return 1.0 / (beta * (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 = 1.0d0 / (beta * (beta + 3.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.2%

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

3. Taylor expanded in beta around inf 26.8%

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

4. Taylor expanded in alpha around 0 25.0%

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

5. Final simplification 25.0%

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

Alternative 19: 6.0% accurate, 11.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \frac{0.3333333333333333}{\beta} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta) :precision binary64 (/ 0.3333333333333333 beta))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return 0.3333333333333333 / beta;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = 0.3333333333333333d0 / beta
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return 0.3333333333333333 / beta;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return 0.3333333333333333 / beta

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return Float64(0.3333333333333333 / beta)
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = 0.3333333333333333 / beta;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := N[(0.3333333333333333 / beta), $MachinePrecision]

Derivation

1. Initial program 95.2%

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

3. Taylor expanded in beta around inf 26.8%

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

4. Taylor expanded in alpha around 0 25.0%

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

5. Taylor expanded in beta around 0 4.2%

   \[\leadsto \color{blue}{\frac{0.3333333333333333}{\beta}} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024163 
(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)))