Octave 3.8, jcobi/3

Percentage Accurate: 94.1% → 99.6%

Time: 14.1s

Alternatives: 17

Speedup: 3.9×

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: 94.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\beta + \alpha\right)\\ t_1 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 4 \cdot 10^{+34}:\\ \;\;\;\;\frac{\alpha + \left(1 - \beta \cdot \left(-1 - \alpha\right)\right)}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \left(t_1 \cdot t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(\alpha + 1\right) + \left(\frac{1}{\beta} + \frac{\alpha}{\beta}\right)\right) + \frac{-1 - \alpha}{\frac{\beta}{\alpha + 2}}}{t_0}}{1 + t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (+ beta alpha))) (t_1 (+ alpha (+ beta 2.0))))
   (if (<= beta 4e+34)
     (/
      (+ alpha (- 1.0 (* beta (- -1.0 alpha))))
      (* (+ alpha (+ beta 3.0)) (* t_1 t_1)))
     (/
      (/
       (+
        (+ (+ alpha 1.0) (+ (/ 1.0 beta) (/ alpha beta)))
        (/ (- -1.0 alpha) (/ beta (+ alpha 2.0))))
       t_0)
      (+ 1.0 t_0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta + alpha);
	double t_1 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 4e+34) {
		tmp = (alpha + (1.0 - (beta * (-1.0 - alpha)))) / ((alpha + (beta + 3.0)) * (t_1 * t_1));
	} else {
		tmp = ((((alpha + 1.0) + ((1.0 / beta) + (alpha / beta))) + ((-1.0 - alpha) / (beta / (alpha + 2.0)))) / t_0) / (1.0 + t_0);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 2.0d0 + (beta + alpha)
    t_1 = alpha + (beta + 2.0d0)
    if (beta <= 4d+34) then
        tmp = (alpha + (1.0d0 - (beta * ((-1.0d0) - alpha)))) / ((alpha + (beta + 3.0d0)) * (t_1 * t_1))
    else
        tmp = ((((alpha + 1.0d0) + ((1.0d0 / beta) + (alpha / beta))) + (((-1.0d0) - alpha) / (beta / (alpha + 2.0d0)))) / t_0) / (1.0d0 + t_0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta + alpha);
	double t_1 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 4e+34) {
		tmp = (alpha + (1.0 - (beta * (-1.0 - alpha)))) / ((alpha + (beta + 3.0)) * (t_1 * t_1));
	} else {
		tmp = ((((alpha + 1.0) + ((1.0 / beta) + (alpha / beta))) + ((-1.0 - alpha) / (beta / (alpha + 2.0)))) / t_0) / (1.0 + t_0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = 2.0 + (beta + alpha)
	t_1 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 4e+34:
		tmp = (alpha + (1.0 - (beta * (-1.0 - alpha)))) / ((alpha + (beta + 3.0)) * (t_1 * t_1))
	else:
		tmp = ((((alpha + 1.0) + ((1.0 / beta) + (alpha / beta))) + ((-1.0 - alpha) / (beta / (alpha + 2.0)))) / t_0) / (1.0 + t_0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(beta + alpha))
	t_1 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 4e+34)
		tmp = Float64(Float64(alpha + Float64(1.0 - Float64(beta * Float64(-1.0 - alpha)))) / Float64(Float64(alpha + Float64(beta + 3.0)) * Float64(t_1 * t_1)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(alpha + 1.0) + Float64(Float64(1.0 / beta) + Float64(alpha / beta))) + Float64(Float64(-1.0 - alpha) / Float64(beta / Float64(alpha + 2.0)))) / t_0) / Float64(1.0 + t_0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = 2.0 + (beta + alpha);
	t_1 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 4e+34)
		tmp = (alpha + (1.0 - (beta * (-1.0 - alpha)))) / ((alpha + (beta + 3.0)) * (t_1 * t_1));
	else
		tmp = ((((alpha + 1.0) + ((1.0 / beta) + (alpha / beta))) + ((-1.0 - alpha) / (beta / (alpha + 2.0)))) / t_0) / (1.0 + t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 4e+34], N[(N[(alpha + N[(1.0 - N[(beta * N[(-1.0 - alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(alpha + 1.0), $MachinePrecision] + N[(N[(1.0 / beta), $MachinePrecision] + N[(alpha / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 - alpha), $MachinePrecision] / N[(beta / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.99999999999999978e34

   1. Initial program 99.8%

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

      1. associate-/l/ 99.1%

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

      2. associate-/l/ 95.9%

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

      3. associate-+l+ 95.9%

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

      4. associate-+l+ 95.9%

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

      5. *-commutative 95.9%

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

      6. fma-def 95.9%

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

      7. metadata-eval 95.9%

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

      8. associate-+l+ 95.9%

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

      9. associate-+l+ 95.9%

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

      10. metadata-eval 95.9%

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

      11. metadata-eval 95.9%

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

      12. associate-+l+ 95.9%

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

      13. metadata-eval 95.9%

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

      14. associate-+l+ 95.9%

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

   3. Simplified 95.9%

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

   4. Taylor expanded in beta around 0 95.9%

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

   if 3.99999999999999978e34 < beta

   1. Initial program 71.7%

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

   2. Taylor expanded in beta around inf 70.0%

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

      1. associate-+r+ 70.0%

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

      2. associate-/l* 75.5%

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

   4. Simplified 75.5%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4 \cdot 10^{+34}:\\ \;\;\;\;\frac{\alpha + \left(1 - \beta \cdot \left(-1 - \alpha\right)\right)}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(\alpha + 1\right) + \left(\frac{1}{\beta} + \frac{\alpha}{\beta}\right)\right) + \frac{-1 - \alpha}{\frac{\beta}{\alpha + 2}}}{2 + \left(\beta + \alpha\right)}}{1 + \left(2 + \left(\beta + \alpha\right)\right)}\\ \end{array} \]

Alternative 2: 99.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 10^{+141}:\\ \;\;\;\;\frac{\frac{\frac{1 + \left(\left(\beta + \alpha\right) + \beta \cdot \alpha\right)}{t_0}}{t_0}}{1 + t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\beta + \left(\alpha + 3\right)} \cdot \frac{1 + \frac{-1 - \alpha}{\beta}}{\beta + \left(\alpha + 2\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (+ beta alpha))))
   (if (<= beta 1e+141)
     (/ (/ (/ (+ 1.0 (+ (+ beta alpha) (* beta alpha))) t_0) t_0) (+ 1.0 t_0))
     (*
      (/ (+ alpha 1.0) (+ beta (+ alpha 3.0)))
      (/ (+ 1.0 (/ (- -1.0 alpha) beta)) (+ beta (+ alpha 2.0)))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta + alpha);
	double tmp;
	if (beta <= 1e+141) {
		tmp = (((1.0 + ((beta + alpha) + (beta * alpha))) / t_0) / t_0) / (1.0 + t_0);
	} else {
		tmp = ((alpha + 1.0) / (beta + (alpha + 3.0))) * ((1.0 + ((-1.0 - alpha) / beta)) / (beta + (alpha + 2.0)));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 + (beta + alpha)
    if (beta <= 1d+141) then
        tmp = (((1.0d0 + ((beta + alpha) + (beta * alpha))) / t_0) / t_0) / (1.0d0 + t_0)
    else
        tmp = ((alpha + 1.0d0) / (beta + (alpha + 3.0d0))) * ((1.0d0 + (((-1.0d0) - alpha) / beta)) / (beta + (alpha + 2.0d0)))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta + alpha);
	double tmp;
	if (beta <= 1e+141) {
		tmp = (((1.0 + ((beta + alpha) + (beta * alpha))) / t_0) / t_0) / (1.0 + t_0);
	} else {
		tmp = ((alpha + 1.0) / (beta + (alpha + 3.0))) * ((1.0 + ((-1.0 - alpha) / beta)) / (beta + (alpha + 2.0)));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = 2.0 + (beta + alpha)
	tmp = 0
	if beta <= 1e+141:
		tmp = (((1.0 + ((beta + alpha) + (beta * alpha))) / t_0) / t_0) / (1.0 + t_0)
	else:
		tmp = ((alpha + 1.0) / (beta + (alpha + 3.0))) * ((1.0 + ((-1.0 - alpha) / beta)) / (beta + (alpha + 2.0)))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(beta + alpha))
	tmp = 0.0
	if (beta <= 1e+141)
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(Float64(beta + alpha) + Float64(beta * alpha))) / t_0) / t_0) / Float64(1.0 + t_0));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(beta + Float64(alpha + 3.0))) * Float64(Float64(1.0 + Float64(Float64(-1.0 - alpha) / beta)) / Float64(beta + Float64(alpha + 2.0))));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = 2.0 + (beta + alpha);
	tmp = 0.0;
	if (beta <= 1e+141)
		tmp = (((1.0 + ((beta + alpha) + (beta * alpha))) / t_0) / t_0) / (1.0 + t_0);
	else
		tmp = ((alpha + 1.0) / (beta + (alpha + 3.0))) * ((1.0 + ((-1.0 - alpha) / beta)) / (beta + (alpha + 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1e+141], N[(N[(N[(N[(1.0 + N[(N[(beta + alpha), $MachinePrecision] + N[(beta * alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(N[(-1.0 - alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.00000000000000002e141

   1. Initial program 98.5%

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

   if 1.00000000000000002e141 < beta

   1. Initial program 63.4%

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

      1. associate-/l/ 61.6%

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

      2. associate-+l+ 61.6%

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

      3. +-commutative 61.6%

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

      4. associate-+r+ 61.6%

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

      5. associate-+l+ 61.6%

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

      6. distribute-rgt1-in 61.6%

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

      7. *-rgt-identity 61.6%

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

      8. distribute-lft-out 61.6%

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

      9. +-commutative 61.6%

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

      10. associate-*l/ 90.8%

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

      11. *-commutative 90.8%

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

      12. associate-*r/ 90.8%

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

   3. Simplified 90.8%

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

   4. Taylor expanded in beta around inf 90.8%

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

      1. mul-1-neg 90.8%

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

      2. unsub-neg 90.8%

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

   6. Simplified 90.8%

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

      1. expm1-log1p-u 90.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\alpha + 1\right) \cdot \frac{1 - \frac{1 + \alpha}{\beta}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}\right)\right)} \]

      2. expm1-udef 88.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\alpha + 1\right) \cdot \frac{1 - \frac{1 + \alpha}{\beta}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}\right)} - 1} \]

      3. +-commutative 88.3%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(1 + \alpha\right)} \cdot \frac{1 - \frac{1 + \alpha}{\beta}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}\right)} - 1 \]

      4. *-commutative 88.3%

         \[\leadsto e^{\mathsf{log1p}\left(\left(1 + \alpha\right) \cdot \frac{1 - \frac{1 + \alpha}{\beta}}{\color{blue}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}}\right)} - 1 \]

      5. +-commutative 88.3%

         \[\leadsto e^{\mathsf{log1p}\left(\left(1 + \alpha\right) \cdot \frac{1 - \frac{1 + \alpha}{\beta}}{\color{blue}{\left(\left(\beta + 3\right) + \alpha\right)} \cdot \left(\alpha + \left(\beta + 2\right)\right)}\right)} - 1 \]

      6. +-commutative 88.3%

         \[\leadsto e^{\mathsf{log1p}\left(\left(1 + \alpha\right) \cdot \frac{1 - \frac{1 + \alpha}{\beta}}{\left(\left(\beta + 3\right) + \alpha\right) \cdot \color{blue}{\left(\left(\beta + 2\right) + \alpha\right)}}\right)} - 1 \]

   8. Applied egg-rr 88.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(1 + \alpha\right) \cdot \frac{1 - \frac{1 + \alpha}{\beta}}{\left(\left(\beta + 3\right) + \alpha\right) \cdot \left(\left(\beta + 2\right) + \alpha\right)}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 90.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(1 + \alpha\right) \cdot \frac{1 - \frac{1 + \alpha}{\beta}}{\left(\left(\beta + 3\right) + \alpha\right) \cdot \left(\left(\beta + 2\right) + \alpha\right)}\right)\right)} \]

      2. expm1-log1p 90.8%

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

      3. associate-*r/ 76.7%

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

      4. times-frac 81.0%

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

      5. associate-+l+ 81.0%

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

      6. sub-neg 81.0%

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

      7. distribute-neg-frac 81.0%

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

      8. distribute-neg-in 81.0%

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

      9. metadata-eval 81.0%

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

      10. sub-neg 81.0%

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

      11. associate-+l+ 81.0%

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

   10. Simplified 81.0%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 10^{+141}:\\ \;\;\;\;\frac{\frac{\frac{1 + \left(\left(\beta + \alpha\right) + \beta \cdot \alpha\right)}{2 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}}{1 + \left(2 + \left(\beta + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\beta + \left(\alpha + 3\right)} \cdot \frac{1 + \frac{-1 - \alpha}{\beta}}{\beta + \left(\alpha + 2\right)}\\ \end{array} \]

Alternative 3: 99.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 5 \cdot 10^{+139}:\\ \;\;\;\;\left(\alpha + 1\right) \cdot \frac{\frac{\beta + 1}{t_0}}{\left(\alpha + \left(\beta + 3\right)\right) \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\beta + \left(\alpha + 3\right)} \cdot \frac{1 + \frac{-1 - \alpha}{\beta}}{\beta + \left(\alpha + 2\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 5e+139)
     (* (+ alpha 1.0) (/ (/ (+ beta 1.0) t_0) (* (+ alpha (+ beta 3.0)) t_0)))
     (*
      (/ (+ alpha 1.0) (+ beta (+ alpha 3.0)))
      (/ (+ 1.0 (/ (- -1.0 alpha) beta)) (+ beta (+ alpha 2.0)))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 5e+139) {
		tmp = (alpha + 1.0) * (((beta + 1.0) / t_0) / ((alpha + (beta + 3.0)) * t_0));
	} else {
		tmp = ((alpha + 1.0) / (beta + (alpha + 3.0))) * ((1.0 + ((-1.0 - alpha) / beta)) / (beta + (alpha + 2.0)));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    if (beta <= 5d+139) then
        tmp = (alpha + 1.0d0) * (((beta + 1.0d0) / t_0) / ((alpha + (beta + 3.0d0)) * t_0))
    else
        tmp = ((alpha + 1.0d0) / (beta + (alpha + 3.0d0))) * ((1.0d0 + (((-1.0d0) - alpha) / beta)) / (beta + (alpha + 2.0d0)))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 5e+139) {
		tmp = (alpha + 1.0) * (((beta + 1.0) / t_0) / ((alpha + (beta + 3.0)) * t_0));
	} else {
		tmp = ((alpha + 1.0) / (beta + (alpha + 3.0))) * ((1.0 + ((-1.0 - alpha) / beta)) / (beta + (alpha + 2.0)));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 5e+139:
		tmp = (alpha + 1.0) * (((beta + 1.0) / t_0) / ((alpha + (beta + 3.0)) * t_0))
	else:
		tmp = ((alpha + 1.0) / (beta + (alpha + 3.0))) * ((1.0 + ((-1.0 - alpha) / beta)) / (beta + (alpha + 2.0)))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 5e+139)
		tmp = Float64(Float64(alpha + 1.0) * Float64(Float64(Float64(beta + 1.0) / t_0) / Float64(Float64(alpha + Float64(beta + 3.0)) * t_0)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(beta + Float64(alpha + 3.0))) * Float64(Float64(1.0 + Float64(Float64(-1.0 - alpha) / beta)) / Float64(beta + Float64(alpha + 2.0))));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 5e+139)
		tmp = (alpha + 1.0) * (((beta + 1.0) / t_0) / ((alpha + (beta + 3.0)) * t_0));
	else
		tmp = ((alpha + 1.0) / (beta + (alpha + 3.0))) * ((1.0 + ((-1.0 - alpha) / beta)) / (beta + (alpha + 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 5e+139], N[(N[(alpha + 1.0), $MachinePrecision] * N[(N[(N[(beta + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(N[(-1.0 - alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.0000000000000003e139

   1. Initial program 98.5%

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

      1. associate-/l/ 96.5%

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

      2. associate-+l+ 96.5%

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

      3. +-commutative 96.5%

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

      4. associate-+r+ 96.5%

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

      5. associate-+l+ 96.5%

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

      6. distribute-rgt1-in 96.5%

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

      7. *-rgt-identity 96.5%

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

      8. distribute-lft-out 96.5%

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

      9. +-commutative 96.5%

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

      10. associate-*l/ 97.9%

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

      11. *-commutative 97.9%

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

      12. associate-*r/ 94.9%

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

   3. Simplified 94.9%

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

   if 5.0000000000000003e139 < beta

   1. Initial program 63.4%

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

      1. associate-/l/ 61.6%

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

      2. associate-+l+ 61.6%

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

      3. +-commutative 61.6%

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

      4. associate-+r+ 61.6%

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

      5. associate-+l+ 61.6%

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

      6. distribute-rgt1-in 61.6%

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

      7. *-rgt-identity 61.6%

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

      8. distribute-lft-out 61.6%

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

      9. +-commutative 61.6%

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

      10. associate-*l/ 90.8%

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

      11. *-commutative 90.8%

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

      12. associate-*r/ 90.8%

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

   3. Simplified 90.8%

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

   4. Taylor expanded in beta around inf 90.8%

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

      1. mul-1-neg 90.8%

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

      2. unsub-neg 90.8%

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

   6. Simplified 90.8%

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

      1. expm1-log1p-u 90.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\alpha + 1\right) \cdot \frac{1 - \frac{1 + \alpha}{\beta}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}\right)\right)} \]

      2. expm1-udef 88.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\alpha + 1\right) \cdot \frac{1 - \frac{1 + \alpha}{\beta}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}\right)} - 1} \]

      3. +-commutative 88.3%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(1 + \alpha\right)} \cdot \frac{1 - \frac{1 + \alpha}{\beta}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}\right)} - 1 \]

      4. *-commutative 88.3%

         \[\leadsto e^{\mathsf{log1p}\left(\left(1 + \alpha\right) \cdot \frac{1 - \frac{1 + \alpha}{\beta}}{\color{blue}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}}\right)} - 1 \]

      5. +-commutative 88.3%

         \[\leadsto e^{\mathsf{log1p}\left(\left(1 + \alpha\right) \cdot \frac{1 - \frac{1 + \alpha}{\beta}}{\color{blue}{\left(\left(\beta + 3\right) + \alpha\right)} \cdot \left(\alpha + \left(\beta + 2\right)\right)}\right)} - 1 \]

      6. +-commutative 88.3%

         \[\leadsto e^{\mathsf{log1p}\left(\left(1 + \alpha\right) \cdot \frac{1 - \frac{1 + \alpha}{\beta}}{\left(\left(\beta + 3\right) + \alpha\right) \cdot \color{blue}{\left(\left(\beta + 2\right) + \alpha\right)}}\right)} - 1 \]

   8. Applied egg-rr 88.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(1 + \alpha\right) \cdot \frac{1 - \frac{1 + \alpha}{\beta}}{\left(\left(\beta + 3\right) + \alpha\right) \cdot \left(\left(\beta + 2\right) + \alpha\right)}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 90.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(1 + \alpha\right) \cdot \frac{1 - \frac{1 + \alpha}{\beta}}{\left(\left(\beta + 3\right) + \alpha\right) \cdot \left(\left(\beta + 2\right) + \alpha\right)}\right)\right)} \]

      2. expm1-log1p 90.8%

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

      3. associate-*r/ 76.7%

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

      4. times-frac 81.0%

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

      5. associate-+l+ 81.0%

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

      6. sub-neg 81.0%

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

      7. distribute-neg-frac 81.0%

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

      8. distribute-neg-in 81.0%

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

      9. metadata-eval 81.0%

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

      10. sub-neg 81.0%

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

      11. associate-+l+ 81.0%

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

   10. Simplified 81.0%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5 \cdot 10^{+139}:\\ \;\;\;\;\left(\alpha + 1\right) \cdot \frac{\frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\beta + \left(\alpha + 3\right)} \cdot \frac{1 + \frac{-1 - \alpha}{\beta}}{\beta + \left(\alpha + 2\right)}\\ \end{array} \]

Alternative 4: 98.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.8 \cdot 10^{+140}:\\ \;\;\;\;\left(\alpha + 1\right) \cdot \frac{\frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\beta + \left(\alpha + 3\right)} \cdot \frac{1 + \frac{-1 - \alpha}{\beta}}{\beta + \left(\alpha + 2\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.8e+140)
   (*
    (+ alpha 1.0)
    (/ (/ (+ beta 1.0) (+ alpha (+ beta 2.0))) (* (+ beta 3.0) (+ beta 2.0))))
   (*
    (/ (+ alpha 1.0) (+ beta (+ alpha 3.0)))
    (/ (+ 1.0 (/ (- -1.0 alpha) beta)) (+ beta (+ alpha 2.0))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.8e+140) {
		tmp = (alpha + 1.0) * (((beta + 1.0) / (alpha + (beta + 2.0))) / ((beta + 3.0) * (beta + 2.0)));
	} else {
		tmp = ((alpha + 1.0) / (beta + (alpha + 3.0))) * ((1.0 + ((-1.0 - alpha) / beta)) / (beta + (alpha + 2.0)));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 3.8d+140) then
        tmp = (alpha + 1.0d0) * (((beta + 1.0d0) / (alpha + (beta + 2.0d0))) / ((beta + 3.0d0) * (beta + 2.0d0)))
    else
        tmp = ((alpha + 1.0d0) / (beta + (alpha + 3.0d0))) * ((1.0d0 + (((-1.0d0) - alpha) / beta)) / (beta + (alpha + 2.0d0)))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.8e+140) {
		tmp = (alpha + 1.0) * (((beta + 1.0) / (alpha + (beta + 2.0))) / ((beta + 3.0) * (beta + 2.0)));
	} else {
		tmp = ((alpha + 1.0) / (beta + (alpha + 3.0))) * ((1.0 + ((-1.0 - alpha) / beta)) / (beta + (alpha + 2.0)));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 3.8e+140:
		tmp = (alpha + 1.0) * (((beta + 1.0) / (alpha + (beta + 2.0))) / ((beta + 3.0) * (beta + 2.0)))
	else:
		tmp = ((alpha + 1.0) / (beta + (alpha + 3.0))) * ((1.0 + ((-1.0 - alpha) / beta)) / (beta + (alpha + 2.0)))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.8e+140)
		tmp = Float64(Float64(alpha + 1.0) * Float64(Float64(Float64(beta + 1.0) / Float64(alpha + Float64(beta + 2.0))) / Float64(Float64(beta + 3.0) * Float64(beta + 2.0))));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(beta + Float64(alpha + 3.0))) * Float64(Float64(1.0 + Float64(Float64(-1.0 - alpha) / beta)) / Float64(beta + Float64(alpha + 2.0))));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.8e+140)
		tmp = (alpha + 1.0) * (((beta + 1.0) / (alpha + (beta + 2.0))) / ((beta + 3.0) * (beta + 2.0)));
	else
		tmp = ((alpha + 1.0) / (beta + (alpha + 3.0))) * ((1.0 + ((-1.0 - alpha) / beta)) / (beta + (alpha + 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 3.8e+140], N[(N[(alpha + 1.0), $MachinePrecision] * N[(N[(N[(beta + 1.0), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + 3.0), $MachinePrecision] * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(N[(-1.0 - alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.8000000000000001e140

   1. Initial program 98.5%

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

      1. associate-/l/ 96.5%

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

      2. associate-+l+ 96.5%

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

      3. +-commutative 96.5%

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

      4. associate-+r+ 96.5%

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

      5. associate-+l+ 96.5%

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

      6. distribute-rgt1-in 96.5%

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

      7. *-rgt-identity 96.5%

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

      8. distribute-lft-out 96.5%

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

      9. +-commutative 96.5%

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

      10. associate-*l/ 97.9%

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

      11. *-commutative 97.9%

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

      12. associate-*r/ 94.9%

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

   3. Simplified 94.9%

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

   4. Taylor expanded in alpha around 0 66.6%

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

   if 3.8000000000000001e140 < beta

   1. Initial program 63.4%

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

      1. associate-/l/ 61.6%

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

      2. associate-+l+ 61.6%

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

      3. +-commutative 61.6%

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

      4. associate-+r+ 61.6%

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

      5. associate-+l+ 61.6%

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

      6. distribute-rgt1-in 61.6%

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

      7. *-rgt-identity 61.6%

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

      8. distribute-lft-out 61.6%

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

      9. +-commutative 61.6%

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

      10. associate-*l/ 90.8%

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

      11. *-commutative 90.8%

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

      12. associate-*r/ 90.8%

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

   3. Simplified 90.8%

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

   4. Taylor expanded in beta around inf 90.8%

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

      1. mul-1-neg 90.8%

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

      2. unsub-neg 90.8%

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

   6. Simplified 90.8%

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

      1. expm1-log1p-u 90.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\alpha + 1\right) \cdot \frac{1 - \frac{1 + \alpha}{\beta}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}\right)\right)} \]

      2. expm1-udef 88.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\alpha + 1\right) \cdot \frac{1 - \frac{1 + \alpha}{\beta}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}\right)} - 1} \]

      3. +-commutative 88.3%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(1 + \alpha\right)} \cdot \frac{1 - \frac{1 + \alpha}{\beta}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}\right)} - 1 \]

      4. *-commutative 88.3%

         \[\leadsto e^{\mathsf{log1p}\left(\left(1 + \alpha\right) \cdot \frac{1 - \frac{1 + \alpha}{\beta}}{\color{blue}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}}\right)} - 1 \]

      5. +-commutative 88.3%

         \[\leadsto e^{\mathsf{log1p}\left(\left(1 + \alpha\right) \cdot \frac{1 - \frac{1 + \alpha}{\beta}}{\color{blue}{\left(\left(\beta + 3\right) + \alpha\right)} \cdot \left(\alpha + \left(\beta + 2\right)\right)}\right)} - 1 \]

      6. +-commutative 88.3%

         \[\leadsto e^{\mathsf{log1p}\left(\left(1 + \alpha\right) \cdot \frac{1 - \frac{1 + \alpha}{\beta}}{\left(\left(\beta + 3\right) + \alpha\right) \cdot \color{blue}{\left(\left(\beta + 2\right) + \alpha\right)}}\right)} - 1 \]

   8. Applied egg-rr 88.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(1 + \alpha\right) \cdot \frac{1 - \frac{1 + \alpha}{\beta}}{\left(\left(\beta + 3\right) + \alpha\right) \cdot \left(\left(\beta + 2\right) + \alpha\right)}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 90.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(1 + \alpha\right) \cdot \frac{1 - \frac{1 + \alpha}{\beta}}{\left(\left(\beta + 3\right) + \alpha\right) \cdot \left(\left(\beta + 2\right) + \alpha\right)}\right)\right)} \]

      2. expm1-log1p 90.8%

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

      3. associate-*r/ 76.7%

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

      4. times-frac 81.0%

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

      5. associate-+l+ 81.0%

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

      6. sub-neg 81.0%

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

      7. distribute-neg-frac 81.0%

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

      8. distribute-neg-in 81.0%

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

      9. metadata-eval 81.0%

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

      10. sub-neg 81.0%

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

      11. associate-+l+ 81.0%

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

   10. Simplified 81.0%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.8 \cdot 10^{+140}:\\ \;\;\;\;\left(\alpha + 1\right) \cdot \frac{\frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\beta + \left(\alpha + 3\right)} \cdot \frac{1 + \frac{-1 - \alpha}{\beta}}{\beta + \left(\alpha + 2\right)}\\ \end{array} \]

Alternative 5: 99.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 3.4 \cdot 10^{+36}:\\ \;\;\;\;\frac{\alpha + \left(\beta + 1\right)}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \left(t_0 \cdot t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\beta + \left(\alpha + 3\right)} \cdot \frac{1 + \frac{-1 - \alpha}{\beta}}{\beta + \left(\alpha + 2\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 3.4e+36)
     (/ (+ alpha (+ beta 1.0)) (* (+ alpha (+ beta 3.0)) (* t_0 t_0)))
     (*
      (/ (+ alpha 1.0) (+ beta (+ alpha 3.0)))
      (/ (+ 1.0 (/ (- -1.0 alpha) beta)) (+ beta (+ alpha 2.0)))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 3.4e+36) {
		tmp = (alpha + (beta + 1.0)) / ((alpha + (beta + 3.0)) * (t_0 * t_0));
	} else {
		tmp = ((alpha + 1.0) / (beta + (alpha + 3.0))) * ((1.0 + ((-1.0 - alpha) / beta)) / (beta + (alpha + 2.0)));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    if (beta <= 3.4d+36) then
        tmp = (alpha + (beta + 1.0d0)) / ((alpha + (beta + 3.0d0)) * (t_0 * t_0))
    else
        tmp = ((alpha + 1.0d0) / (beta + (alpha + 3.0d0))) * ((1.0d0 + (((-1.0d0) - alpha) / beta)) / (beta + (alpha + 2.0d0)))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 3.4e+36) {
		tmp = (alpha + (beta + 1.0)) / ((alpha + (beta + 3.0)) * (t_0 * t_0));
	} else {
		tmp = ((alpha + 1.0) / (beta + (alpha + 3.0))) * ((1.0 + ((-1.0 - alpha) / beta)) / (beta + (alpha + 2.0)));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 3.4e+36:
		tmp = (alpha + (beta + 1.0)) / ((alpha + (beta + 3.0)) * (t_0 * t_0))
	else:
		tmp = ((alpha + 1.0) / (beta + (alpha + 3.0))) * ((1.0 + ((-1.0 - alpha) / beta)) / (beta + (alpha + 2.0)))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 3.4e+36)
		tmp = Float64(Float64(alpha + Float64(beta + 1.0)) / Float64(Float64(alpha + Float64(beta + 3.0)) * Float64(t_0 * t_0)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(beta + Float64(alpha + 3.0))) * Float64(Float64(1.0 + Float64(Float64(-1.0 - alpha) / beta)) / Float64(beta + Float64(alpha + 2.0))));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 3.4e+36)
		tmp = (alpha + (beta + 1.0)) / ((alpha + (beta + 3.0)) * (t_0 * t_0));
	else
		tmp = ((alpha + 1.0) / (beta + (alpha + 3.0))) * ((1.0 + ((-1.0 - alpha) / beta)) / (beta + (alpha + 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 3.4e+36], N[(N[(alpha + N[(beta + 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(N[(-1.0 - alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.3999999999999998e36

   1. Initial program 99.8%

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

      1. associate-/l/ 99.1%

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

      2. associate-/l/ 95.9%

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

      3. associate-+l+ 95.9%

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

      4. associate-+l+ 95.9%

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

      5. *-commutative 95.9%

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

      6. fma-def 95.9%

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

      7. metadata-eval 95.9%

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

      8. associate-+l+ 95.9%

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

      9. associate-+l+ 95.9%

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

      10. metadata-eval 95.9%

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

      11. metadata-eval 95.9%

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

      12. associate-+l+ 95.9%

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

      13. metadata-eval 95.9%

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

      14. associate-+l+ 95.9%

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

   3. Simplified 95.9%

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

   4. Taylor expanded in beta around 0 95.9%

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

   5. Taylor expanded in alpha around 0 95.3%

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

   if 3.3999999999999998e36 < beta

   1. Initial program 71.7%

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

      1. associate-/l/ 65.6%

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

      2. associate-+l+ 65.6%

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

      3. +-commutative 65.6%

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

      4. associate-+r+ 65.6%

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

      5. associate-+l+ 65.6%

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

      6. distribute-rgt1-in 65.6%

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

      7. *-rgt-identity 65.6%

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

      8. distribute-lft-out 65.6%

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

      9. +-commutative 65.6%

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

      10. associate-*l/ 89.2%

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

      11. *-commutative 89.2%

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

      12. associate-*r/ 89.2%

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

   3. Simplified 89.2%

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

   4. Taylor expanded in beta around inf 89.2%

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

      1. mul-1-neg 89.2%

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

      2. unsub-neg 89.2%

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

   6. Simplified 89.2%

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

      1. expm1-log1p-u 89.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\alpha + 1\right) \cdot \frac{1 - \frac{1 + \alpha}{\beta}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}\right)\right)} \]

      2. expm1-udef 64.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\alpha + 1\right) \cdot \frac{1 - \frac{1 + \alpha}{\beta}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}\right)} - 1} \]

      3. +-commutative 64.1%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(1 + \alpha\right)} \cdot \frac{1 - \frac{1 + \alpha}{\beta}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}\right)} - 1 \]

      4. *-commutative 64.1%

         \[\leadsto e^{\mathsf{log1p}\left(\left(1 + \alpha\right) \cdot \frac{1 - \frac{1 + \alpha}{\beta}}{\color{blue}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}}\right)} - 1 \]

      5. +-commutative 64.1%

         \[\leadsto e^{\mathsf{log1p}\left(\left(1 + \alpha\right) \cdot \frac{1 - \frac{1 + \alpha}{\beta}}{\color{blue}{\left(\left(\beta + 3\right) + \alpha\right)} \cdot \left(\alpha + \left(\beta + 2\right)\right)}\right)} - 1 \]

      6. +-commutative 64.1%

         \[\leadsto e^{\mathsf{log1p}\left(\left(1 + \alpha\right) \cdot \frac{1 - \frac{1 + \alpha}{\beta}}{\left(\left(\beta + 3\right) + \alpha\right) \cdot \color{blue}{\left(\left(\beta + 2\right) + \alpha\right)}}\right)} - 1 \]

   8. Applied egg-rr 64.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(1 + \alpha\right) \cdot \frac{1 - \frac{1 + \alpha}{\beta}}{\left(\left(\beta + 3\right) + \alpha\right) \cdot \left(\left(\beta + 2\right) + \alpha\right)}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 89.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(1 + \alpha\right) \cdot \frac{1 - \frac{1 + \alpha}{\beta}}{\left(\left(\beta + 3\right) + \alpha\right) \cdot \left(\left(\beta + 2\right) + \alpha\right)}\right)\right)} \]

      2. expm1-log1p 89.2%

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

      3. associate-*r/ 73.2%

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

      4. times-frac 75.8%

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

      5. associate-+l+ 75.8%

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

      6. sub-neg 75.8%

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

      7. distribute-neg-frac 75.8%

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

      8. distribute-neg-in 75.8%

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

      9. metadata-eval 75.8%

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

      10. sub-neg 75.8%

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

      11. associate-+l+ 75.8%

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

   10. Simplified 75.8%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.4 \cdot 10^{+36}:\\ \;\;\;\;\frac{\alpha + \left(\beta + 1\right)}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\beta + \left(\alpha + 3\right)} \cdot \frac{1 + \frac{-1 - \alpha}{\beta}}{\beta + \left(\alpha + 2\right)}\\ \end{array} \]

Alternative 6: 98.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 10^{+49}:\\ \;\;\;\;\left(\alpha + 1\right) \cdot \frac{\frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1e+49)
   (*
    (+ alpha 1.0)
    (/ (/ (+ beta 1.0) (+ alpha (+ beta 2.0))) (* (+ beta 3.0) (+ beta 2.0))))
   (/ (/ (+ alpha 1.0) beta) (+ beta (+ alpha 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1e+49) {
		tmp = (alpha + 1.0) * (((beta + 1.0) / (alpha + (beta + 2.0))) / ((beta + 3.0) * (beta + 2.0)));
	} else {
		tmp = ((alpha + 1.0) / beta) / (beta + (alpha + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1d+49) then
        tmp = (alpha + 1.0d0) * (((beta + 1.0d0) / (alpha + (beta + 2.0d0))) / ((beta + 3.0d0) * (beta + 2.0d0)))
    else
        tmp = ((alpha + 1.0d0) / beta) / (beta + (alpha + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1e+49) {
		tmp = (alpha + 1.0) * (((beta + 1.0) / (alpha + (beta + 2.0))) / ((beta + 3.0) * (beta + 2.0)));
	} else {
		tmp = ((alpha + 1.0) / beta) / (beta + (alpha + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1e+49:
		tmp = (alpha + 1.0) * (((beta + 1.0) / (alpha + (beta + 2.0))) / ((beta + 3.0) * (beta + 2.0)))
	else:
		tmp = ((alpha + 1.0) / beta) / (beta + (alpha + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1e+49)
		tmp = Float64(Float64(alpha + 1.0) * Float64(Float64(Float64(beta + 1.0) / Float64(alpha + Float64(beta + 2.0))) / Float64(Float64(beta + 3.0) * Float64(beta + 2.0))));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / Float64(beta + Float64(alpha + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1e+49)
		tmp = (alpha + 1.0) * (((beta + 1.0) / (alpha + (beta + 2.0))) / ((beta + 3.0) * (beta + 2.0)));
	else
		tmp = ((alpha + 1.0) / beta) / (beta + (alpha + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1e+49], N[(N[(alpha + 1.0), $MachinePrecision] * N[(N[(N[(beta + 1.0), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + 3.0), $MachinePrecision] * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 9.99999999999999946e48

   1. Initial program 99.3%

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

      1. associate-/l/ 98.2%

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

      2. associate-+l+ 98.2%

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

      3. +-commutative 98.2%

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

      4. associate-+r+ 98.2%

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

      5. associate-+l+ 98.2%

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

      6. distribute-rgt1-in 98.2%

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

      7. *-rgt-identity 98.2%

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

      8. distribute-lft-out 98.2%

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

      9. +-commutative 98.2%

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

      10. associate-*l/ 98.6%

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

      11. *-commutative 98.6%

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

      12. associate-*r/ 95.4%

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

   3. Simplified 95.4%

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

   4. Taylor expanded in alpha around 0 65.1%

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

   if 9.99999999999999946e48 < beta

   1. Initial program 72.0%

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

   2. Taylor expanded in beta around -inf 79.0%

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

   3. Taylor expanded in beta around inf 78.4%

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

      1. associate-*r/ 78.4%

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

      2. neg-mul-1 78.4%

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

      3. sub-neg 78.4%

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

      4. metadata-eval 78.4%

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

      5. metadata-eval 78.4%

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

      6. distribute-lft-in 78.4%

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

      7. +-commutative 78.4%

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

      8. distribute-lft-in 78.4%

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

      9. metadata-eval 78.4%

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

      10. neg-mul-1 78.4%

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

      11. unsub-neg 78.4%

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

   5. Simplified 78.4%

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

      1. expm1-log1p-u 78.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{-\left(-1 - \alpha\right)}{\beta}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\right)\right)} \]

      2. expm1-udef 65.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{-\left(-1 - \alpha\right)}{\beta}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\right)} - 1} \]

      3. metadata-eval 65.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{-\left(-1 - \alpha\right)}{\beta}}{\left(\left(\alpha + \beta\right) + \color{blue}{2}\right) + 1}\right)} - 1 \]

      4. associate-+l+ 65.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{-\left(-1 - \alpha\right)}{\beta}}{\color{blue}{\left(\alpha + \beta\right) + \left(2 + 1\right)}}\right)} - 1 \]

      5. metadata-eval 65.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{-\left(-1 - \alpha\right)}{\beta}}{\left(\alpha + \beta\right) + \color{blue}{3}}\right)} - 1 \]

      6. associate-+r+ 65.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{-\left(-1 - \alpha\right)}{\beta}}{\color{blue}{\alpha + \left(\beta + 3\right)}}\right)} - 1 \]

      7. +-commutative 65.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{-\left(-1 - \alpha\right)}{\beta}}{\color{blue}{\left(\beta + 3\right) + \alpha}}\right)} - 1 \]

   7. Applied egg-rr 65.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{-\left(-1 - \alpha\right)}{\beta}}{\left(\beta + 3\right) + \alpha}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 78.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{-\left(-1 - \alpha\right)}{\beta}}{\left(\beta + 3\right) + \alpha}\right)\right)} \]

      2. expm1-log1p 78.4%

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

      3. distribute-frac-neg 78.4%

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

      4. distribute-frac-neg 78.4%

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

      5. associate-+l+ 78.4%

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

   9. Simplified 78.4%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 10^{+49}:\\ \;\;\;\;\left(\alpha + 1\right) \cdot \frac{\frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]

Alternative 7: 98.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2 \cdot 10^{+35}:\\ \;\;\;\;\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\left(\beta + 2\right) \cdot \left(\left(\beta + 3\right) \cdot \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2e+35)
   (*
    (+ alpha 1.0)
    (/ (+ beta 1.0) (* (+ beta 2.0) (* (+ beta 3.0) (+ beta 2.0)))))
   (/ (/ (+ alpha 1.0) beta) (+ beta (+ alpha 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2e+35) {
		tmp = (alpha + 1.0) * ((beta + 1.0) / ((beta + 2.0) * ((beta + 3.0) * (beta + 2.0))));
	} else {
		tmp = ((alpha + 1.0) / beta) / (beta + (alpha + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2d+35) then
        tmp = (alpha + 1.0d0) * ((beta + 1.0d0) / ((beta + 2.0d0) * ((beta + 3.0d0) * (beta + 2.0d0))))
    else
        tmp = ((alpha + 1.0d0) / beta) / (beta + (alpha + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2e+35) {
		tmp = (alpha + 1.0) * ((beta + 1.0) / ((beta + 2.0) * ((beta + 3.0) * (beta + 2.0))));
	} else {
		tmp = ((alpha + 1.0) / beta) / (beta + (alpha + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2e+35:
		tmp = (alpha + 1.0) * ((beta + 1.0) / ((beta + 2.0) * ((beta + 3.0) * (beta + 2.0))))
	else:
		tmp = ((alpha + 1.0) / beta) / (beta + (alpha + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2e+35)
		tmp = Float64(Float64(alpha + 1.0) * Float64(Float64(beta + 1.0) / Float64(Float64(beta + 2.0) * Float64(Float64(beta + 3.0) * Float64(beta + 2.0)))));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / Float64(beta + Float64(alpha + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2e+35)
		tmp = (alpha + 1.0) * ((beta + 1.0) / ((beta + 2.0) * ((beta + 3.0) * (beta + 2.0))));
	else
		tmp = ((alpha + 1.0) / beta) / (beta + (alpha + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2e+35], N[(N[(alpha + 1.0), $MachinePrecision] * N[(N[(beta + 1.0), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(N[(beta + 3.0), $MachinePrecision] * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.9999999999999999e35

   1. Initial program 99.8%

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

      1. associate-/l/ 99.1%

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

      2. associate-+l+ 99.1%

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

      3. +-commutative 99.1%

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

      4. associate-+r+ 99.1%

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

      5. associate-+l+ 99.1%

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

      6. distribute-rgt1-in 99.1%

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

      7. *-rgt-identity 99.1%

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

      8. distribute-lft-out 99.1%

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

      9. +-commutative 99.1%

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

      10. associate-*l/ 99.1%

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

      11. *-commutative 99.1%

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

      12. associate-*r/ 95.8%

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

   3. Simplified 95.8%

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

   4. Taylor expanded in alpha around 0 65.1%

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

   5. Taylor expanded in alpha around 0 64.7%

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

      1. *-un-lft-identity 64.7%

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

      2. associate-/l/ 64.7%

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

      3. +-commutative 64.7%

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

      4. *-commutative 64.7%

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

   7. Applied egg-rr 64.7%

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

      1. *-lft-identity 64.7%

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

      2. +-commutative 64.7%

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

      3. associate-*l* 64.7%

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

   9. Simplified 64.7%

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

   if 1.9999999999999999e35 < beta

   1. Initial program 71.7%

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

   2. Taylor expanded in beta around -inf 76.9%

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

   3. Taylor expanded in beta around inf 76.1%

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

      1. associate-*r/ 76.1%

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

      2. neg-mul-1 76.1%

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

      3. sub-neg 76.1%

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

      4. metadata-eval 76.1%

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

      5. metadata-eval 76.1%

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

      6. distribute-lft-in 76.1%

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

      7. +-commutative 76.1%

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

      8. distribute-lft-in 76.1%

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

      9. metadata-eval 76.1%

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

      10. neg-mul-1 76.1%

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

      11. unsub-neg 76.1%

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

   5. Simplified 76.1%

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

      1. expm1-log1p-u 76.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{-\left(-1 - \alpha\right)}{\beta}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\right)\right)} \]

      2. expm1-udef 64.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{-\left(-1 - \alpha\right)}{\beta}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\right)} - 1} \]

      3. metadata-eval 64.1%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{-\left(-1 - \alpha\right)}{\beta}}{\left(\left(\alpha + \beta\right) + \color{blue}{2}\right) + 1}\right)} - 1 \]

      4. associate-+l+ 64.1%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{-\left(-1 - \alpha\right)}{\beta}}{\color{blue}{\left(\alpha + \beta\right) + \left(2 + 1\right)}}\right)} - 1 \]

      5. metadata-eval 64.1%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{-\left(-1 - \alpha\right)}{\beta}}{\left(\alpha + \beta\right) + \color{blue}{3}}\right)} - 1 \]

      6. associate-+r+ 64.1%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{-\left(-1 - \alpha\right)}{\beta}}{\color{blue}{\alpha + \left(\beta + 3\right)}}\right)} - 1 \]

      7. +-commutative 64.1%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{-\left(-1 - \alpha\right)}{\beta}}{\color{blue}{\left(\beta + 3\right) + \alpha}}\right)} - 1 \]

   7. Applied egg-rr 64.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{-\left(-1 - \alpha\right)}{\beta}}{\left(\beta + 3\right) + \alpha}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 76.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{-\left(-1 - \alpha\right)}{\beta}}{\left(\beta + 3\right) + \alpha}\right)\right)} \]

      2. expm1-log1p 76.1%

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

      3. distribute-frac-neg 76.1%

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

      4. distribute-frac-neg 76.1%

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

      5. associate-+l+ 76.1%

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

   9. Simplified 76.1%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2 \cdot 10^{+35}:\\ \;\;\;\;\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\left(\beta + 2\right) \cdot \left(\left(\beta + 3\right) \cdot \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]

Alternative 8: 96.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.5:\\ \;\;\;\;\left(\alpha + 1\right) \cdot \left(0.08333333333333333 + \beta \cdot -0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.5)
   (* (+ alpha 1.0) (+ 0.08333333333333333 (* beta -0.027777777777777776)))
   (/ (/ (+ alpha 1.0) beta) (+ beta (+ alpha 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.5) {
		tmp = (alpha + 1.0) * (0.08333333333333333 + (beta * -0.027777777777777776));
	} else {
		tmp = ((alpha + 1.0) / beta) / (beta + (alpha + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.5d0) then
        tmp = (alpha + 1.0d0) * (0.08333333333333333d0 + (beta * (-0.027777777777777776d0)))
    else
        tmp = ((alpha + 1.0d0) / beta) / (beta + (alpha + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.5) {
		tmp = (alpha + 1.0) * (0.08333333333333333 + (beta * -0.027777777777777776));
	} else {
		tmp = ((alpha + 1.0) / beta) / (beta + (alpha + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.5:
		tmp = (alpha + 1.0) * (0.08333333333333333 + (beta * -0.027777777777777776))
	else:
		tmp = ((alpha + 1.0) / beta) / (beta + (alpha + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.5)
		tmp = Float64(Float64(alpha + 1.0) * Float64(0.08333333333333333 + Float64(beta * -0.027777777777777776)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / Float64(beta + Float64(alpha + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.5)
		tmp = (alpha + 1.0) * (0.08333333333333333 + (beta * -0.027777777777777776));
	else
		tmp = ((alpha + 1.0) / beta) / (beta + (alpha + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.5], N[(N[(alpha + 1.0), $MachinePrecision] * N[(0.08333333333333333 + N[(beta * -0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.5

   1. Initial program 99.8%

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

      1. associate-/l/ 99.1%

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

      2. associate-+l+ 99.1%

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

      3. +-commutative 99.1%

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

      4. associate-+r+ 99.1%

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

      5. associate-+l+ 99.1%

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

      6. distribute-rgt1-in 99.1%

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

      7. *-rgt-identity 99.1%

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

      8. distribute-lft-out 99.1%

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

      9. +-commutative 99.1%

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

      10. associate-*l/ 99.0%

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

      11. *-commutative 99.0%

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

      12. associate-*r/ 95.5%

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

   3. Simplified 95.5%

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

   4. Taylor expanded in alpha around 0 64.3%

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

   5. Taylor expanded in alpha around 0 63.8%

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

   6. Taylor expanded in beta around 0 63.2%

      \[\leadsto \left(\alpha + 1\right) \cdot \color{blue}{\left(0.08333333333333333 + -0.027777777777777776 \cdot \beta\right)} \]
   7. Step-by-step derivation

      1. *-commutative 63.2%

         \[\leadsto \left(\alpha + 1\right) \cdot \left(0.08333333333333333 + \color{blue}{\beta \cdot -0.027777777777777776}\right) \]

   8. Simplified 63.2%

      \[\leadsto \left(\alpha + 1\right) \cdot \color{blue}{\left(0.08333333333333333 + \beta \cdot -0.027777777777777776\right)} \]

   if 2.5 < beta

   1. Initial program 77.6%

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

   2. Taylor expanded in beta around -inf 74.8%

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

   3. Taylor expanded in beta around inf 74.0%

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

      1. associate-*r/ 74.0%

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

      2. neg-mul-1 74.0%

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

      3. sub-neg 74.0%

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

      4. metadata-eval 74.0%

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

      5. metadata-eval 74.0%

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

      6. distribute-lft-in 74.0%

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

      7. +-commutative 74.0%

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

      8. distribute-lft-in 74.0%

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

      9. metadata-eval 74.0%

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

      10. neg-mul-1 74.0%

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

      11. unsub-neg 74.0%

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

   5. Simplified 74.0%

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

      1. expm1-log1p-u 74.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{-\left(-1 - \alpha\right)}{\beta}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\right)\right)} \]

      2. expm1-udef 54.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{-\left(-1 - \alpha\right)}{\beta}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\right)} - 1} \]

      3. metadata-eval 54.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{-\left(-1 - \alpha\right)}{\beta}}{\left(\left(\alpha + \beta\right) + \color{blue}{2}\right) + 1}\right)} - 1 \]

      4. associate-+l+ 54.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{-\left(-1 - \alpha\right)}{\beta}}{\color{blue}{\left(\alpha + \beta\right) + \left(2 + 1\right)}}\right)} - 1 \]

      5. metadata-eval 54.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{-\left(-1 - \alpha\right)}{\beta}}{\left(\alpha + \beta\right) + \color{blue}{3}}\right)} - 1 \]

      6. associate-+r+ 54.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{-\left(-1 - \alpha\right)}{\beta}}{\color{blue}{\alpha + \left(\beta + 3\right)}}\right)} - 1 \]

      7. +-commutative 54.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{-\left(-1 - \alpha\right)}{\beta}}{\color{blue}{\left(\beta + 3\right) + \alpha}}\right)} - 1 \]

   7. Applied egg-rr 54.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{-\left(-1 - \alpha\right)}{\beta}}{\left(\beta + 3\right) + \alpha}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 74.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{-\left(-1 - \alpha\right)}{\beta}}{\left(\beta + 3\right) + \alpha}\right)\right)} \]

      2. expm1-log1p 74.0%

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

      3. distribute-frac-neg 74.0%

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

      4. distribute-frac-neg 74.0%

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

      5. associate-+l+ 74.0%

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

   9. Simplified 74.0%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.5:\\ \;\;\;\;\left(\alpha + 1\right) \cdot \left(0.08333333333333333 + \beta \cdot -0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]

Alternative 9: 95.8% accurate, 3.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.4:\\ \;\;\;\;\left(\alpha + 1\right) \cdot 0.08333333333333333\\ \mathbf{elif}\;\beta \leq 5 \cdot 10^{+154}:\\ \;\;\;\;\frac{\alpha + 1}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.4)
   (* (+ alpha 1.0) 0.08333333333333333)
   (if (<= beta 5e+154)
     (/ (+ alpha 1.0) (* beta beta))
     (/ (/ alpha beta) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.4) {
		tmp = (alpha + 1.0) * 0.08333333333333333;
	} else if (beta <= 5e+154) {
		tmp = (alpha + 1.0) / (beta * beta);
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 3.4d0) then
        tmp = (alpha + 1.0d0) * 0.08333333333333333d0
    else if (beta <= 5d+154) then
        tmp = (alpha + 1.0d0) / (beta * beta)
    else
        tmp = (alpha / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.4) {
		tmp = (alpha + 1.0) * 0.08333333333333333;
	} else if (beta <= 5e+154) {
		tmp = (alpha + 1.0) / (beta * beta);
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 3.4:
		tmp = (alpha + 1.0) * 0.08333333333333333
	elif beta <= 5e+154:
		tmp = (alpha + 1.0) / (beta * beta)
	else:
		tmp = (alpha / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.4)
		tmp = Float64(Float64(alpha + 1.0) * 0.08333333333333333);
	elseif (beta <= 5e+154)
		tmp = Float64(Float64(alpha + 1.0) / Float64(beta * beta));
	else
		tmp = Float64(Float64(alpha / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.4)
		tmp = (alpha + 1.0) * 0.08333333333333333;
	elseif (beta <= 5e+154)
		tmp = (alpha + 1.0) / (beta * beta);
	else
		tmp = (alpha / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 3.4], N[(N[(alpha + 1.0), $MachinePrecision] * 0.08333333333333333), $MachinePrecision], If[LessEqual[beta, 5e+154], N[(N[(alpha + 1.0), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / beta), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if beta < 3.39999999999999991

   1. Initial program 99.8%

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

      1. associate-/l/ 99.1%

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

      2. associate-+l+ 99.1%

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

      3. +-commutative 99.1%

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

      4. associate-+r+ 99.1%

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

      5. associate-+l+ 99.1%

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

      6. distribute-rgt1-in 99.1%

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

      7. *-rgt-identity 99.1%

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

      8. distribute-lft-out 99.1%

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

      9. +-commutative 99.1%

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

      10. associate-*l/ 99.0%

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

      11. *-commutative 99.0%

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

      12. associate-*r/ 95.5%

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

   3. Simplified 95.5%

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

   4. Taylor expanded in alpha around 0 64.3%

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

   5. Taylor expanded in alpha around 0 63.8%

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

   6. Taylor expanded in beta around 0 62.4%

      \[\leadsto \left(\alpha + 1\right) \cdot \color{blue}{0.08333333333333333} \]

   if 3.39999999999999991 < beta < 5.00000000000000004e154

   1. Initial program 91.6%

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

      1. associate-/l/ 83.9%

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

      2. associate-+l+ 83.9%

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

      3. +-commutative 83.9%

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

      4. associate-+r+ 83.9%

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

      5. associate-+l+ 83.9%

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

      6. distribute-rgt1-in 83.9%

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

      7. *-rgt-identity 83.9%

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

      8. distribute-lft-out 83.9%

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

      9. +-commutative 83.9%

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

      10. associate-*l/ 92.0%

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

      11. *-commutative 92.0%

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

      12. associate-*r/ 92.0%

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

   3. Simplified 92.0%

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

   4. Taylor expanded in beta around inf 67.2%

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

      1. unpow2 67.2%

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

   6. Simplified 67.2%

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

   if 5.00000000000000004e154 < beta

   1. Initial program 62.4%

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

      1. associate-/l/ 60.5%

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

      2. associate-+l+ 60.5%

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

      3. +-commutative 60.5%

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

      4. associate-+r+ 60.5%

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

      5. associate-+l+ 60.5%

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

      6. distribute-rgt1-in 60.5%

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

      7. *-rgt-identity 60.5%

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

      8. distribute-lft-out 60.5%

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

      9. +-commutative 60.5%

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

      10. associate-*l/ 90.6%

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

      11. *-commutative 90.6%

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

      12. associate-*r/ 90.6%

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

   3. Simplified 90.6%

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

   4. Taylor expanded in beta around inf 90.6%

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

      1. unpow2 90.6%

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

   6. Simplified 90.6%

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

      1. clear-num 90.6%

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

      2. inv-pow 90.6%

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

      3. +-commutative 90.6%

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

   8. Applied egg-rr 90.6%

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

      1. unpow-1 90.6%

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

      2. associate-/l* 79.3%

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

      3. +-commutative 79.3%

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

   10. Simplified 79.3%

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

       1. *-un-lft-identity 79.3%

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

       2. associate-/r/ 80.4%

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

       3. +-commutative 80.4%

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

   12. Applied egg-rr 80.4%

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

       1. *-lft-identity 80.4%

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

       2. associate-*l/ 80.5%

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

       3. *-lft-identity 80.5%

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

       4. +-commutative 80.5%

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

   14. Simplified 80.5%

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

   15. Taylor expanded in alpha around inf 79.3%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.4:\\ \;\;\;\;\left(\alpha + 1\right) \cdot 0.08333333333333333\\ \mathbf{elif}\;\beta \leq 5 \cdot 10^{+154}:\\ \;\;\;\;\frac{\alpha + 1}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]

Alternative 10: 96.8% accurate, 3.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.8:\\ \;\;\;\;\left(\alpha + 1\right) \cdot \left(0.08333333333333333 + \beta \cdot -0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.8)
   (* (+ alpha 1.0) (+ 0.08333333333333333 (* beta -0.027777777777777776)))
   (/ (/ (+ alpha 1.0) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.8) {
		tmp = (alpha + 1.0) * (0.08333333333333333 + (beta * -0.027777777777777776));
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.8d0) then
        tmp = (alpha + 1.0d0) * (0.08333333333333333d0 + (beta * (-0.027777777777777776d0)))
    else
        tmp = ((alpha + 1.0d0) / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.8) {
		tmp = (alpha + 1.0) * (0.08333333333333333 + (beta * -0.027777777777777776));
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.8:
		tmp = (alpha + 1.0) * (0.08333333333333333 + (beta * -0.027777777777777776))
	else:
		tmp = ((alpha + 1.0) / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.8)
		tmp = Float64(Float64(alpha + 1.0) * Float64(0.08333333333333333 + Float64(beta * -0.027777777777777776)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.8)
		tmp = (alpha + 1.0) * (0.08333333333333333 + (beta * -0.027777777777777776));
	else
		tmp = ((alpha + 1.0) / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.8], N[(N[(alpha + 1.0), $MachinePrecision] * N[(0.08333333333333333 + N[(beta * -0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.7999999999999998

   1. Initial program 99.8%

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

      1. associate-/l/ 99.1%

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

      2. associate-+l+ 99.1%

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

      3. +-commutative 99.1%

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

      4. associate-+r+ 99.1%

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

      5. associate-+l+ 99.1%

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

      6. distribute-rgt1-in 99.1%

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

      7. *-rgt-identity 99.1%

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

      8. distribute-lft-out 99.1%

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

      9. +-commutative 99.1%

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

      10. associate-*l/ 99.0%

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

      11. *-commutative 99.0%

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

      12. associate-*r/ 95.5%

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

   3. Simplified 95.5%

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

   4. Taylor expanded in alpha around 0 64.3%

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

   5. Taylor expanded in alpha around 0 63.8%

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

   6. Taylor expanded in beta around 0 63.2%

      \[\leadsto \left(\alpha + 1\right) \cdot \color{blue}{\left(0.08333333333333333 + -0.027777777777777776 \cdot \beta\right)} \]
   7. Step-by-step derivation

      1. *-commutative 63.2%

         \[\leadsto \left(\alpha + 1\right) \cdot \left(0.08333333333333333 + \color{blue}{\beta \cdot -0.027777777777777776}\right) \]

   8. Simplified 63.2%

      \[\leadsto \left(\alpha + 1\right) \cdot \color{blue}{\left(0.08333333333333333 + \beta \cdot -0.027777777777777776\right)} \]

   if 2.7999999999999998 < beta

   1. Initial program 77.6%

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

      1. associate-/l/ 72.7%

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

      2. associate-+l+ 72.7%

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

      3. +-commutative 72.7%

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

      4. associate-+r+ 72.7%

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

      5. associate-+l+ 72.7%

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

      6. distribute-rgt1-in 72.7%

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

      7. *-rgt-identity 72.7%

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

      8. distribute-lft-out 72.7%

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

      9. +-commutative 72.7%

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

      10. associate-*l/ 91.3%

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

      11. *-commutative 91.3%

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

      12. associate-*r/ 91.3%

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

   3. Simplified 91.3%

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

   4. Taylor expanded in beta around inf 78.4%

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

      1. unpow2 78.4%

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

   6. Simplified 78.4%

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

      1. clear-num 78.4%

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

      2. inv-pow 78.4%

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

      3. +-commutative 78.4%

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

   8. Applied egg-rr 78.4%

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

      1. unpow-1 78.4%

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

      2. associate-/l* 73.0%

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

      3. +-commutative 73.0%

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

   10. Simplified 73.0%

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

       1. *-un-lft-identity 73.0%

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

       2. associate-/r/ 73.4%

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

       3. +-commutative 73.4%

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

   12. Applied egg-rr 73.4%

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

       1. *-lft-identity 73.4%

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

       2. associate-*l/ 73.6%

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

       3. *-lft-identity 73.6%

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

       4. +-commutative 73.6%

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

   14. Simplified 73.6%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.8:\\ \;\;\;\;\left(\alpha + 1\right) \cdot \left(0.08333333333333333 + \beta \cdot -0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \]

Alternative 11: 93.1% accurate, 3.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.3:\\ \;\;\;\;\left(\alpha + 1\right) \cdot 0.08333333333333333\\ \mathbf{elif}\;\beta \leq 1.15 \cdot 10^{+160}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.3)
   (* (+ alpha 1.0) 0.08333333333333333)
   (if (<= beta 1.15e+160) (/ (/ 1.0 beta) beta) (/ (/ alpha beta) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.3) {
		tmp = (alpha + 1.0) * 0.08333333333333333;
	} else if (beta <= 1.15e+160) {
		tmp = (1.0 / beta) / beta;
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 3.3d0) then
        tmp = (alpha + 1.0d0) * 0.08333333333333333d0
    else if (beta <= 1.15d+160) then
        tmp = (1.0d0 / beta) / beta
    else
        tmp = (alpha / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.3) {
		tmp = (alpha + 1.0) * 0.08333333333333333;
	} else if (beta <= 1.15e+160) {
		tmp = (1.0 / beta) / beta;
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 3.3:
		tmp = (alpha + 1.0) * 0.08333333333333333
	elif beta <= 1.15e+160:
		tmp = (1.0 / beta) / beta
	else:
		tmp = (alpha / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.3)
		tmp = Float64(Float64(alpha + 1.0) * 0.08333333333333333);
	elseif (beta <= 1.15e+160)
		tmp = Float64(Float64(1.0 / beta) / beta);
	else
		tmp = Float64(Float64(alpha / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.3)
		tmp = (alpha + 1.0) * 0.08333333333333333;
	elseif (beta <= 1.15e+160)
		tmp = (1.0 / beta) / beta;
	else
		tmp = (alpha / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 3.3], N[(N[(alpha + 1.0), $MachinePrecision] * 0.08333333333333333), $MachinePrecision], If[LessEqual[beta, 1.15e+160], N[(N[(1.0 / beta), $MachinePrecision] / beta), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / beta), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if beta < 3.2999999999999998

   1. Initial program 99.8%

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

      1. associate-/l/ 99.1%

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

      2. associate-+l+ 99.1%

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

      3. +-commutative 99.1%

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

      4. associate-+r+ 99.1%

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

      5. associate-+l+ 99.1%

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

      6. distribute-rgt1-in 99.1%

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

      7. *-rgt-identity 99.1%

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

      8. distribute-lft-out 99.1%

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

      9. +-commutative 99.1%

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

      10. associate-*l/ 99.0%

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

      11. *-commutative 99.0%

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

      12. associate-*r/ 95.5%

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

   3. Simplified 95.5%

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

   4. Taylor expanded in alpha around 0 64.3%

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

   5. Taylor expanded in alpha around 0 63.8%

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

   6. Taylor expanded in beta around 0 62.4%

      \[\leadsto \left(\alpha + 1\right) \cdot \color{blue}{0.08333333333333333} \]

   if 3.2999999999999998 < beta < 1.14999999999999994e160

   1. Initial program 91.8%

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

      1. associate-/l/ 83.3%

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

      2. associate-+l+ 83.3%

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

      3. +-commutative 83.3%

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

      4. associate-+r+ 83.3%

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

      5. associate-+l+ 83.3%

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

      6. distribute-rgt1-in 83.3%

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

      7. *-rgt-identity 83.3%

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

      8. distribute-lft-out 83.3%

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

      9. +-commutative 83.3%

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

      10. associate-*l/ 91.2%

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

      11. *-commutative 91.2%

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

      12. associate-*r/ 91.2%

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

   3. Simplified 91.2%

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

   4. Taylor expanded in beta around inf 67.0%

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

      1. unpow2 67.0%

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

   6. Simplified 67.0%

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

      1. clear-num 67.0%

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

      2. inv-pow 67.0%

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

      3. +-commutative 67.0%

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

   8. Applied egg-rr 67.0%

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

      1. unpow-1 67.0%

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

      2. associate-/l* 67.0%

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

      3. +-commutative 67.0%

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

   10. Simplified 67.0%

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

       1. *-un-lft-identity 67.0%

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

       2. associate-/r/ 67.9%

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

       3. +-commutative 67.9%

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

   12. Applied egg-rr 67.9%

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

       1. *-lft-identity 67.9%

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

       2. associate-*l/ 68.1%

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

       3. *-lft-identity 68.1%

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

       4. +-commutative 68.1%

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

   14. Simplified 68.1%

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

   15. Taylor expanded in alpha around 0 65.1%

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

   if 1.14999999999999994e160 < beta

   1. Initial program 61.2%

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

      1. associate-/l/ 60.6%

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

      2. associate-+l+ 60.6%

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

      3. +-commutative 60.6%

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

      4. associate-+r+ 60.6%

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

      5. associate-+l+ 60.6%

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

      6. distribute-rgt1-in 60.6%

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

      7. *-rgt-identity 60.6%

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

      8. distribute-lft-out 60.6%

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

      9. +-commutative 60.6%

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

      10. associate-*l/ 91.6%

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

      11. *-commutative 91.6%

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

      12. associate-*r/ 91.6%

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

   3. Simplified 91.6%

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

   4. Taylor expanded in beta around inf 91.6%

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

      1. unpow2 91.6%

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

   6. Simplified 91.6%

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

      1. clear-num 91.6%

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

      2. inv-pow 91.6%

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

      3. +-commutative 91.6%

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

   8. Applied egg-rr 91.6%

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

      1. unpow-1 91.6%

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

      2. associate-/l* 79.9%

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

      3. +-commutative 79.9%

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

   10. Simplified 79.9%

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

       1. *-un-lft-identity 79.9%

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

       2. associate-/r/ 79.9%

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

       3. +-commutative 79.9%

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

   12. Applied egg-rr 79.9%

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

       1. *-lft-identity 79.9%

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

       2. associate-*l/ 79.9%

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

       3. *-lft-identity 79.9%

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

       4. +-commutative 79.9%

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

   14. Simplified 79.9%

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

   15. Taylor expanded in alpha around inf 79.9%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.3:\\ \;\;\;\;\left(\alpha + 1\right) \cdot 0.08333333333333333\\ \mathbf{elif}\;\beta \leq 1.15 \cdot 10^{+160}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]

Alternative 12: 96.3% accurate, 3.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.4:\\ \;\;\;\;\left(\alpha + 1\right) \cdot 0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.4)
   (* (+ alpha 1.0) 0.08333333333333333)
   (/ (/ (+ alpha 1.0) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.4) {
		tmp = (alpha + 1.0) * 0.08333333333333333;
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 3.4d0) then
        tmp = (alpha + 1.0d0) * 0.08333333333333333d0
    else
        tmp = ((alpha + 1.0d0) / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.4) {
		tmp = (alpha + 1.0) * 0.08333333333333333;
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 3.4:
		tmp = (alpha + 1.0) * 0.08333333333333333
	else:
		tmp = ((alpha + 1.0) / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.4)
		tmp = Float64(Float64(alpha + 1.0) * 0.08333333333333333);
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.4)
		tmp = (alpha + 1.0) * 0.08333333333333333;
	else
		tmp = ((alpha + 1.0) / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 3.4], N[(N[(alpha + 1.0), $MachinePrecision] * 0.08333333333333333), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.39999999999999991

   1. Initial program 99.8%

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

      1. associate-/l/ 99.1%

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

      2. associate-+l+ 99.1%

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

      3. +-commutative 99.1%

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

      4. associate-+r+ 99.1%

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

      5. associate-+l+ 99.1%

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

      6. distribute-rgt1-in 99.1%

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

      7. *-rgt-identity 99.1%

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

      8. distribute-lft-out 99.1%

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

      9. +-commutative 99.1%

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

      10. associate-*l/ 99.0%

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

      11. *-commutative 99.0%

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

      12. associate-*r/ 95.5%

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

   3. Simplified 95.5%

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

   4. Taylor expanded in alpha around 0 64.3%

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

   5. Taylor expanded in alpha around 0 63.8%

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

   6. Taylor expanded in beta around 0 62.4%

      \[\leadsto \left(\alpha + 1\right) \cdot \color{blue}{0.08333333333333333} \]

   if 3.39999999999999991 < beta

   1. Initial program 77.6%

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

      1. associate-/l/ 72.7%

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

      2. associate-+l+ 72.7%

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

      3. +-commutative 72.7%

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

      4. associate-+r+ 72.7%

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

      5. associate-+l+ 72.7%

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

      6. distribute-rgt1-in 72.7%

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

      7. *-rgt-identity 72.7%

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

      8. distribute-lft-out 72.7%

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

      9. +-commutative 72.7%

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

      10. associate-*l/ 91.3%

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

      11. *-commutative 91.3%

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

      12. associate-*r/ 91.3%

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

   3. Simplified 91.3%

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

   4. Taylor expanded in beta around inf 78.4%

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

      1. unpow2 78.4%

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

   6. Simplified 78.4%

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

      1. clear-num 78.4%

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

      2. inv-pow 78.4%

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

      3. +-commutative 78.4%

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

   8. Applied egg-rr 78.4%

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

      1. unpow-1 78.4%

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

      2. associate-/l* 73.0%

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

      3. +-commutative 73.0%

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

   10. Simplified 73.0%

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

       1. *-un-lft-identity 73.0%

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

       2. associate-/r/ 73.4%

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

       3. +-commutative 73.4%

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

   12. Applied egg-rr 73.4%

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

       1. *-lft-identity 73.4%

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

       2. associate-*l/ 73.6%

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

       3. *-lft-identity 73.6%

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

       4. +-commutative 73.6%

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

   14. Simplified 73.6%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.4:\\ \;\;\;\;\left(\alpha + 1\right) \cdot 0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \]

Alternative 13: 46.5% accurate, 5.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 9.2:\\ \;\;\;\;\left(\alpha + 1\right) \cdot 0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 9.2) (* (+ alpha 1.0) 0.08333333333333333) (/ 1.0 beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 9.2) {
		tmp = (alpha + 1.0) * 0.08333333333333333;
	} else {
		tmp = 1.0 / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 9.2d0) then
        tmp = (alpha + 1.0d0) * 0.08333333333333333d0
    else
        tmp = 1.0d0 / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 9.2) {
		tmp = (alpha + 1.0) * 0.08333333333333333;
	} else {
		tmp = 1.0 / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 9.2:
		tmp = (alpha + 1.0) * 0.08333333333333333
	else:
		tmp = 1.0 / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 9.2)
		tmp = Float64(Float64(alpha + 1.0) * 0.08333333333333333);
	else
		tmp = Float64(1.0 / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 9.2)
		tmp = (alpha + 1.0) * 0.08333333333333333;
	else
		tmp = 1.0 / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 9.2], N[(N[(alpha + 1.0), $MachinePrecision] * 0.08333333333333333), $MachinePrecision], N[(1.0 / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 9.1999999999999993

   1. Initial program 99.8%

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

      1. associate-/l/ 99.1%

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

      2. associate-+l+ 99.1%

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

      3. +-commutative 99.1%

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

      4. associate-+r+ 99.1%

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

      5. associate-+l+ 99.1%

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

      6. distribute-rgt1-in 99.1%

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

      7. *-rgt-identity 99.1%

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

      8. distribute-lft-out 99.1%

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

      9. +-commutative 99.1%

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

      10. associate-*l/ 99.0%

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

      11. *-commutative 99.0%

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

      12. associate-*r/ 95.5%

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

   3. Simplified 95.5%

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

   4. Taylor expanded in alpha around 0 64.3%

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

   5. Taylor expanded in alpha around 0 63.8%

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

   6. Taylor expanded in beta around 0 62.4%

      \[\leadsto \left(\alpha + 1\right) \cdot \color{blue}{0.08333333333333333} \]

   if 9.1999999999999993 < beta

   1. Initial program 77.6%

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

   2. Taylor expanded in beta around -inf 74.0%

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

   3. Taylor expanded in alpha around inf 6.8%

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

4. Final simplification 47.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 9.2:\\ \;\;\;\;\left(\alpha + 1\right) \cdot 0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta}\\ \end{array} \]

Alternative 14: 90.2% accurate, 5.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.3:\\ \;\;\;\;\left(\alpha + 1\right) \cdot 0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.3)
   (* (+ alpha 1.0) 0.08333333333333333)
   (/ 1.0 (* beta beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.3) {
		tmp = (alpha + 1.0) * 0.08333333333333333;
	} else {
		tmp = 1.0 / (beta * beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 3.3d0) then
        tmp = (alpha + 1.0d0) * 0.08333333333333333d0
    else
        tmp = 1.0d0 / (beta * beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.3) {
		tmp = (alpha + 1.0) * 0.08333333333333333;
	} else {
		tmp = 1.0 / (beta * beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 3.3:
		tmp = (alpha + 1.0) * 0.08333333333333333
	else:
		tmp = 1.0 / (beta * beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.3)
		tmp = Float64(Float64(alpha + 1.0) * 0.08333333333333333);
	else
		tmp = Float64(1.0 / Float64(beta * beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.3)
		tmp = (alpha + 1.0) * 0.08333333333333333;
	else
		tmp = 1.0 / (beta * beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 3.3], N[(N[(alpha + 1.0), $MachinePrecision] * 0.08333333333333333), $MachinePrecision], N[(1.0 / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.2999999999999998

   1. Initial program 99.8%

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

      1. associate-/l/ 99.1%

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

      2. associate-+l+ 99.1%

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

      3. +-commutative 99.1%

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

      4. associate-+r+ 99.1%

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

      5. associate-+l+ 99.1%

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

      6. distribute-rgt1-in 99.1%

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

      7. *-rgt-identity 99.1%

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

      8. distribute-lft-out 99.1%

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

      9. +-commutative 99.1%

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

      10. associate-*l/ 99.0%

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

      11. *-commutative 99.0%

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

      12. associate-*r/ 95.5%

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

   3. Simplified 95.5%

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

   4. Taylor expanded in alpha around 0 64.3%

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

   5. Taylor expanded in alpha around 0 63.8%

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

   6. Taylor expanded in beta around 0 62.4%

      \[\leadsto \left(\alpha + 1\right) \cdot \color{blue}{0.08333333333333333} \]

   if 3.2999999999999998 < beta

   1. Initial program 77.6%

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

      1. associate-/l/ 72.7%

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

      2. associate-+l+ 72.7%

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

      3. +-commutative 72.7%

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

      4. associate-+r+ 72.7%

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

      5. associate-+l+ 72.7%

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

      6. distribute-rgt1-in 72.7%

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

      7. *-rgt-identity 72.7%

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

      8. distribute-lft-out 72.7%

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

      9. +-commutative 72.7%

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

      10. associate-*l/ 91.3%

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

      11. *-commutative 91.3%

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

      12. associate-*r/ 91.3%

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

   3. Simplified 91.3%

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

   4. Taylor expanded in beta around inf 78.4%

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

      1. unpow2 78.4%

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

   6. Simplified 78.4%

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

      1. clear-num 78.4%

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

      2. inv-pow 78.4%

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

      3. +-commutative 78.4%

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

   8. Applied egg-rr 78.4%

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

      1. unpow-1 78.4%

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

      2. associate-/l* 73.0%

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

      3. +-commutative 73.0%

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

   10. Simplified 73.0%

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

   11. Taylor expanded in alpha around 0 76.8%

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

       1. unpow2 76.8%

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

   13. Simplified 76.8%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.3:\\ \;\;\;\;\left(\alpha + 1\right) \cdot 0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \]

Alternative 15: 90.7% accurate, 5.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.2:\\ \;\;\;\;\left(\alpha + 1\right) \cdot 0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.2)
   (* (+ alpha 1.0) 0.08333333333333333)
   (/ (/ 1.0 beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.2) {
		tmp = (alpha + 1.0) * 0.08333333333333333;
	} else {
		tmp = (1.0 / beta) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 3.2d0) then
        tmp = (alpha + 1.0d0) * 0.08333333333333333d0
    else
        tmp = (1.0d0 / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.2) {
		tmp = (alpha + 1.0) * 0.08333333333333333;
	} else {
		tmp = (1.0 / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 3.2:
		tmp = (alpha + 1.0) * 0.08333333333333333
	else:
		tmp = (1.0 / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.2)
		tmp = Float64(Float64(alpha + 1.0) * 0.08333333333333333);
	else
		tmp = Float64(Float64(1.0 / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.2)
		tmp = (alpha + 1.0) * 0.08333333333333333;
	else
		tmp = (1.0 / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 3.2], N[(N[(alpha + 1.0), $MachinePrecision] * 0.08333333333333333), $MachinePrecision], N[(N[(1.0 / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.2000000000000002

   1. Initial program 99.8%

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

      1. associate-/l/ 99.1%

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

      2. associate-+l+ 99.1%

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

      3. +-commutative 99.1%

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

      4. associate-+r+ 99.1%

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

      5. associate-+l+ 99.1%

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

      6. distribute-rgt1-in 99.1%

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

      7. *-rgt-identity 99.1%

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

      8. distribute-lft-out 99.1%

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

      9. +-commutative 99.1%

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

      10. associate-*l/ 99.0%

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

      11. *-commutative 99.0%

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

      12. associate-*r/ 95.5%

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

   3. Simplified 95.5%

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

   4. Taylor expanded in alpha around 0 64.3%

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

   5. Taylor expanded in alpha around 0 63.8%

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

   6. Taylor expanded in beta around 0 62.4%

      \[\leadsto \left(\alpha + 1\right) \cdot \color{blue}{0.08333333333333333} \]

   if 3.2000000000000002 < beta

   1. Initial program 77.6%

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

      1. associate-/l/ 72.7%

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

      2. associate-+l+ 72.7%

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

      3. +-commutative 72.7%

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

      4. associate-+r+ 72.7%

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

      5. associate-+l+ 72.7%

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

      6. distribute-rgt1-in 72.7%

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

      7. *-rgt-identity 72.7%

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

      8. distribute-lft-out 72.7%

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

      9. +-commutative 72.7%

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

      10. associate-*l/ 91.3%

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

      11. *-commutative 91.3%

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

      12. associate-*r/ 91.3%

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

   3. Simplified 91.3%

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

   4. Taylor expanded in beta around inf 78.4%

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

      1. unpow2 78.4%

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

   6. Simplified 78.4%

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

      1. clear-num 78.4%

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

      2. inv-pow 78.4%

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

      3. +-commutative 78.4%

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

   8. Applied egg-rr 78.4%

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

      1. unpow-1 78.4%

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

      2. associate-/l* 73.0%

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

      3. +-commutative 73.0%

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

   10. Simplified 73.0%

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

       1. *-un-lft-identity 73.0%

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

       2. associate-/r/ 73.4%

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

       3. +-commutative 73.4%

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

   12. Applied egg-rr 73.4%

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

       1. *-lft-identity 73.4%

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

       2. associate-*l/ 73.6%

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

       3. *-lft-identity 73.6%

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

       4. +-commutative 73.6%

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

   14. Simplified 73.6%

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

   15. Taylor expanded in alpha around 0 77.4%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.2:\\ \;\;\;\;\left(\alpha + 1\right) \cdot 0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta}\\ \end{array} \]

Alternative 16: 3.7% accurate, 11.7× speedup

Math

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

FPCore

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

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return -1.0 / 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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.7%

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

   1. associate-/l/ 91.8%

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

   2. associate-+l+ 91.8%

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

   3. +-commutative 91.8%

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

   4. associate-+r+ 91.8%

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

   5. associate-+l+ 91.8%

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

   6. distribute-rgt1-in 91.8%

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

   7. *-rgt-identity 91.8%

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

   8. distribute-lft-out 91.8%

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

   9. +-commutative 91.8%

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

   10. associate-*l/ 96.9%

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

   11. *-commutative 96.9%

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

   12. associate-*r/ 94.4%

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

3. Simplified 94.4%

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

4. Taylor expanded in beta around inf 28.7%

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

   1. mul-1-neg 28.7%

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

   2. unsub-neg 28.7%

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

6. Simplified 28.7%

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

7. Taylor expanded in alpha around inf 3.4%

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

8. Final simplification 3.4%

   \[\leadsto \frac{-1}{\beta} \]

Alternative 17: 6.0% accurate, 11.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.7%

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

2. Taylor expanded in beta around -inf 22.6%

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

3. Taylor expanded in alpha around inf 4.0%

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

4. Final simplification 4.0%

   \[\leadsto \frac{1}{\beta} \]

Reproduce

herbie shell --seed 2023193 
(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)))