Octave 3.8, jcobi/3

Percentage Accurate: 93.9% → 99.7%

Time: 12.7s

Alternatives: 18

Speedup: 2.9×

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: 93.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 2 \cdot 10^{+153}:\\ \;\;\;\;\frac{1 + \alpha}{t\_0 \cdot t\_0} \cdot \frac{1 + \beta}{\left(\beta + \alpha\right) + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(1 + \alpha\right) \cdot \left(1 + \frac{-1 - \alpha}{\beta}\right)}{\alpha + \left(\beta + 3\right)}}{2 + \left(\beta + \alpha\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 2e+153)
     (* (/ (+ 1.0 alpha) (* t_0 t_0)) (/ (+ 1.0 beta) (+ (+ beta alpha) 3.0)))
     (/
      (/
       (* (+ 1.0 alpha) (+ 1.0 (/ (- -1.0 alpha) beta)))
       (+ alpha (+ beta 3.0)))
      (+ 2.0 (+ beta alpha))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 2e+153) {
		tmp = ((1.0 + alpha) / (t_0 * t_0)) * ((1.0 + beta) / ((beta + alpha) + 3.0));
	} else {
		tmp = (((1.0 + alpha) * (1.0 + ((-1.0 - alpha) / beta))) / (alpha + (beta + 3.0))) / (2.0 + (beta + alpha));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    if (beta <= 2d+153) then
        tmp = ((1.0d0 + alpha) / (t_0 * t_0)) * ((1.0d0 + beta) / ((beta + alpha) + 3.0d0))
    else
        tmp = (((1.0d0 + alpha) * (1.0d0 + (((-1.0d0) - alpha) / beta))) / (alpha + (beta + 3.0d0))) / (2.0d0 + (beta + alpha))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 2e+153) {
		tmp = ((1.0 + alpha) / (t_0 * t_0)) * ((1.0 + beta) / ((beta + alpha) + 3.0));
	} else {
		tmp = (((1.0 + alpha) * (1.0 + ((-1.0 - alpha) / beta))) / (alpha + (beta + 3.0))) / (2.0 + (beta + alpha));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 2e+153:
		tmp = ((1.0 + alpha) / (t_0 * t_0)) * ((1.0 + beta) / ((beta + alpha) + 3.0))
	else:
		tmp = (((1.0 + alpha) * (1.0 + ((-1.0 - alpha) / beta))) / (alpha + (beta + 3.0))) / (2.0 + (beta + alpha))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 2e+153)
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(t_0 * t_0)) * Float64(Float64(1.0 + beta) / Float64(Float64(beta + alpha) + 3.0)));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + alpha) * Float64(1.0 + Float64(Float64(-1.0 - alpha) / beta))) / Float64(alpha + Float64(beta + 3.0))) / Float64(2.0 + Float64(beta + alpha)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 2e+153)
		tmp = ((1.0 + alpha) / (t_0 * t_0)) * ((1.0 + beta) / ((beta + alpha) + 3.0));
	else
		tmp = (((1.0 + alpha) * (1.0 + ((-1.0 - alpha) / beta))) / (alpha + (beta + 3.0))) / (2.0 + (beta + alpha));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2e+153], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 + alpha), $MachinePrecision] * N[(1.0 + N[(N[(-1.0 - alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2e153

   1. Initial program 97.0%

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

      1. associate-/l/ N/A

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

      2. associate-/l/ N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right) + 1\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      5. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\left(\alpha + \beta\right) + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      6. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\alpha + \left(\beta + 1\right)\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \alpha\right) + \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      8. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta + 1\right) \cdot \alpha + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\alpha \cdot \left(\beta + 1\right) + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\alpha + 1\right) \cdot \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \alpha\right) \cdot \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \alpha\right), \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\alpha + 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   3. Simplified 86.7%

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

      1. metadata-eval N/A

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

      2. associate-+l+ N/A

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

      3. metadata-eval N/A

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

      4. associate-*r* N/A

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

      5. times-frac N/A

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

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\alpha + 1}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)}\right), \color{blue}{\left(\frac{\beta + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\right)}\right) \]

   6. Applied egg-rr 97.9%

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

   if 2e153 < beta

   1. Initial program 78.7%

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

      1. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\alpha}, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 + \left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      4. *-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 + \left(\alpha + \left(\beta \cdot 1 + \beta \cdot \alpha\right)\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      5. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 + \left(\alpha + \beta \cdot \left(1 + \alpha\right)\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 + \left(\alpha + \beta \cdot \left(\alpha + 1\right)\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\left(1 + \alpha\right) + \beta \cdot \left(\alpha + 1\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\alpha}, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\left(\alpha + 1\right) + \beta \cdot \left(\alpha + 1\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      9. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 \cdot \left(\alpha + 1\right) + \beta \cdot \left(\alpha + 1\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      10. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\left(\alpha + 1\right) \cdot \left(1 + \beta\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\alpha}, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      12. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(\alpha, \beta\right)}, \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\alpha + 1\right), \left(\frac{\beta + 1}{\left(\alpha + \beta\right) + 2}\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(\alpha, \beta\right)}, \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\frac{\beta + 1}{\left(\alpha + \beta\right) + 2}\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\alpha}, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{/.f64}\left(\left(\beta + 1\right), \left(\left(\alpha + \beta\right) + 2\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \color{blue}{\beta}\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{/.f64}\left(\left(1 + \beta\right), \left(\left(\alpha + \beta\right) + 2\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      17. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \left(\left(\alpha + \beta\right) + 2\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      18. associate-+l+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \left(\alpha + \left(\beta + 2\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      19. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(\alpha, \left(\beta + 2\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      20. +-lowering-+.f64 100.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 2\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

   4. Applied egg-rr 100.0%

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

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(\alpha + 1\right) \cdot \frac{1 + \beta}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\right), \color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}\right) \]

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in beta around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(1 + -1 \cdot \frac{1 + \alpha}{\beta}\right)}, \mathsf{+.f64}\left(\alpha, 1\right)\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 3\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right) \]
   8. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \frac{-1 \cdot \left(1 + \alpha\right)}{\beta}\right), \mathsf{+.f64}\left(\alpha, 1\right)\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 3\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1 \cdot \left(1 + \alpha\right)}{\beta}\right)\right), \mathsf{+.f64}\left(\alpha, 1\right)\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 3\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(-1 \cdot \left(1 + \alpha\right)\right), \beta\right)\right), \mathsf{+.f64}\left(\alpha, 1\right)\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 3\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right) \]

      4. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(-1 \cdot 1 + -1 \cdot \alpha\right), \beta\right)\right), \mathsf{+.f64}\left(\alpha, 1\right)\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 3\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(-1 + -1 \cdot \alpha\right), \beta\right)\right), \mathsf{+.f64}\left(\alpha, 1\right)\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 3\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(-1 + \left(\mathsf{neg}\left(\alpha\right)\right)\right), \beta\right)\right), \mathsf{+.f64}\left(\alpha, 1\right)\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 3\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right) \]

      7. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(-1 - \alpha\right), \beta\right)\right), \mathsf{+.f64}\left(\alpha, 1\right)\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 3\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right) \]

      8. --lowering--.f64 94.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-1, \alpha\right), \beta\right)\right), \mathsf{+.f64}\left(\alpha, 1\right)\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 3\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right) \]

   9. Simplified 94.8%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2 \cdot 10^{+153}:\\ \;\;\;\;\frac{1 + \alpha}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \cdot \frac{1 + \beta}{\left(\beta + \alpha\right) + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(1 + \alpha\right) \cdot \left(1 + \frac{-1 - \alpha}{\beta}\right)}{\alpha + \left(\beta + 3\right)}}{2 + \left(\beta + \alpha\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 20000000000:\\ \;\;\;\;\frac{\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\beta + 2\right)}}{1 + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(1 + \alpha\right) \cdot \left(1 + \frac{-1 - \alpha}{\beta}\right)}{\alpha + \left(\beta + 3\right)}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (+ beta alpha))))
   (if (<= beta 20000000000.0)
     (/ (/ (+ 1.0 beta) (* (+ beta 2.0) (+ beta 2.0))) (+ 1.0 t_0))
     (/
      (/
       (* (+ 1.0 alpha) (+ 1.0 (/ (- -1.0 alpha) beta)))
       (+ alpha (+ beta 3.0)))
      t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta + alpha);
	double tmp;
	if (beta <= 20000000000.0) {
		tmp = ((1.0 + beta) / ((beta + 2.0) * (beta + 2.0))) / (1.0 + t_0);
	} else {
		tmp = (((1.0 + alpha) * (1.0 + ((-1.0 - alpha) / beta))) / (alpha + (beta + 3.0))) / t_0;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 + (beta + alpha)
    if (beta <= 20000000000.0d0) then
        tmp = ((1.0d0 + beta) / ((beta + 2.0d0) * (beta + 2.0d0))) / (1.0d0 + t_0)
    else
        tmp = (((1.0d0 + alpha) * (1.0d0 + (((-1.0d0) - alpha) / beta))) / (alpha + (beta + 3.0d0))) / t_0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta + alpha);
	double tmp;
	if (beta <= 20000000000.0) {
		tmp = ((1.0 + beta) / ((beta + 2.0) * (beta + 2.0))) / (1.0 + t_0);
	} else {
		tmp = (((1.0 + alpha) * (1.0 + ((-1.0 - alpha) / beta))) / (alpha + (beta + 3.0))) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = 2.0 + (beta + alpha)
	tmp = 0
	if beta <= 20000000000.0:
		tmp = ((1.0 + beta) / ((beta + 2.0) * (beta + 2.0))) / (1.0 + t_0)
	else:
		tmp = (((1.0 + alpha) * (1.0 + ((-1.0 - alpha) / beta))) / (alpha + (beta + 3.0))) / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(beta + alpha))
	tmp = 0.0
	if (beta <= 20000000000.0)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(Float64(beta + 2.0) * Float64(beta + 2.0))) / Float64(1.0 + t_0));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + alpha) * Float64(1.0 + Float64(Float64(-1.0 - alpha) / beta))) / Float64(alpha + Float64(beta + 3.0))) / t_0);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = 2.0 + (beta + alpha);
	tmp = 0.0;
	if (beta <= 20000000000.0)
		tmp = ((1.0 + beta) / ((beta + 2.0) * (beta + 2.0))) / (1.0 + t_0);
	else
		tmp = (((1.0 + alpha) * (1.0 + ((-1.0 - alpha) / beta))) / (alpha + (beta + 3.0))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 20000000000.0], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 + alpha), $MachinePrecision] * N[(1.0 + N[(N[(-1.0 - alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2e10

   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. Add Preprocessing

   3. Taylor expanded in alpha around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1 + \beta}{{\left(2 + \beta\right)}^{2}}\right)}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + \beta\right), \left({\left(2 + \beta\right)}^{2}\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)}, 1\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \left({\left(2 + \beta\right)}^{2}\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(\alpha, \beta\right)}, \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \left(\left(2 + \beta\right) \cdot \left(2 + \beta\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \color{blue}{\mathsf{*.f64}\left(2, 1\right)}\right), 1\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left(2 + \beta\right), \left(2 + \beta\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \color{blue}{\mathsf{*.f64}\left(2, 1\right)}\right), 1\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left(\beta + 2\right), \left(2 + \beta\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(\color{blue}{2}, 1\right)\right), 1\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(2 + \beta\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(\color{blue}{2}, 1\right)\right), 1\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(\beta + 2\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, \color{blue}{1}\right)\right), 1\right)\right) \]

      8. +-lowering-+.f64 69.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, \color{blue}{1}\right)\right), 1\right)\right) \]

   5. Simplified 69.3%

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

   if 2e10 < beta

   1. Initial program 82.5%

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

      1. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\alpha}, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 + \left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      4. *-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 + \left(\alpha + \left(\beta \cdot 1 + \beta \cdot \alpha\right)\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      5. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 + \left(\alpha + \beta \cdot \left(1 + \alpha\right)\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 + \left(\alpha + \beta \cdot \left(\alpha + 1\right)\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\left(1 + \alpha\right) + \beta \cdot \left(\alpha + 1\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\alpha}, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\left(\alpha + 1\right) + \beta \cdot \left(\alpha + 1\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      9. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 \cdot \left(\alpha + 1\right) + \beta \cdot \left(\alpha + 1\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      10. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\left(\alpha + 1\right) \cdot \left(1 + \beta\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\alpha}, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      12. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(\alpha, \beta\right)}, \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\alpha + 1\right), \left(\frac{\beta + 1}{\left(\alpha + \beta\right) + 2}\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(\alpha, \beta\right)}, \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\frac{\beta + 1}{\left(\alpha + \beta\right) + 2}\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\alpha}, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{/.f64}\left(\left(\beta + 1\right), \left(\left(\alpha + \beta\right) + 2\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \color{blue}{\beta}\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{/.f64}\left(\left(1 + \beta\right), \left(\left(\alpha + \beta\right) + 2\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      17. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \left(\left(\alpha + \beta\right) + 2\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      18. associate-+l+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \left(\alpha + \left(\beta + 2\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      19. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(\alpha, \left(\beta + 2\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      20. +-lowering-+.f64 99.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 2\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

   4. Applied egg-rr 99.7%

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

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(\alpha + 1\right) \cdot \frac{1 + \beta}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\right), \color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}\right) \]

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in beta around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(1 + -1 \cdot \frac{1 + \alpha}{\beta}\right)}, \mathsf{+.f64}\left(\alpha, 1\right)\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 3\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right) \]
   8. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \frac{-1 \cdot \left(1 + \alpha\right)}{\beta}\right), \mathsf{+.f64}\left(\alpha, 1\right)\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 3\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1 \cdot \left(1 + \alpha\right)}{\beta}\right)\right), \mathsf{+.f64}\left(\alpha, 1\right)\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 3\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(-1 \cdot \left(1 + \alpha\right)\right), \beta\right)\right), \mathsf{+.f64}\left(\alpha, 1\right)\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 3\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right) \]

      4. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(-1 \cdot 1 + -1 \cdot \alpha\right), \beta\right)\right), \mathsf{+.f64}\left(\alpha, 1\right)\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 3\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(-1 + -1 \cdot \alpha\right), \beta\right)\right), \mathsf{+.f64}\left(\alpha, 1\right)\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 3\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(-1 + \left(\mathsf{neg}\left(\alpha\right)\right)\right), \beta\right)\right), \mathsf{+.f64}\left(\alpha, 1\right)\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 3\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right) \]

      7. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(-1 - \alpha\right), \beta\right)\right), \mathsf{+.f64}\left(\alpha, 1\right)\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 3\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right) \]

      8. --lowering--.f64 76.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-1, \alpha\right), \beta\right)\right), \mathsf{+.f64}\left(\alpha, 1\right)\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 3\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right) \]

   9. Simplified 76.3%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 20000000000:\\ \;\;\;\;\frac{\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\beta + 2\right)}}{1 + \left(2 + \left(\beta + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(1 + \alpha\right) \cdot \left(1 + \frac{-1 - \alpha}{\beta}\right)}{\alpha + \left(\beta + 3\right)}}{2 + \left(\beta + \alpha\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\beta + \alpha\right)\\ \frac{\frac{\frac{1 + \beta}{t\_0} \cdot \left(1 + \alpha\right)}{\alpha + \left(\beta + 3\right)}}{t\_0} \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.4%

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

   1. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\alpha}, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

   3. associate-+l+ N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 + \left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

   4. *-rgt-identity N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 + \left(\alpha + \left(\beta \cdot 1 + \beta \cdot \alpha\right)\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

   5. distribute-lft-in N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 + \left(\alpha + \beta \cdot \left(1 + \alpha\right)\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

   6. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 + \left(\alpha + \beta \cdot \left(\alpha + 1\right)\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

   7. associate-+r+ N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\left(1 + \alpha\right) + \beta \cdot \left(\alpha + 1\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\alpha}, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

   8. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\left(\alpha + 1\right) + \beta \cdot \left(\alpha + 1\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

   9. *-lft-identity N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 \cdot \left(\alpha + 1\right) + \beta \cdot \left(\alpha + 1\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

   10. distribute-rgt-in N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\left(\alpha + 1\right) \cdot \left(1 + \beta\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\alpha}, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

   11. +-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

   12. associate-/l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(\alpha, \beta\right)}, \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\alpha + 1\right), \left(\frac{\beta + 1}{\left(\alpha + \beta\right) + 2}\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(\alpha, \beta\right)}, \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

   14. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\frac{\beta + 1}{\left(\alpha + \beta\right) + 2}\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\alpha}, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

   15. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{/.f64}\left(\left(\beta + 1\right), \left(\left(\alpha + \beta\right) + 2\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \color{blue}{\beta}\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

   16. +-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{/.f64}\left(\left(1 + \beta\right), \left(\left(\alpha + \beta\right) + 2\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

   17. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \left(\left(\alpha + \beta\right) + 2\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

   18. associate-+l+ N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \left(\alpha + \left(\beta + 2\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

   19. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(\alpha, \left(\beta + 2\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

   20. +-lowering-+.f64 99.8%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 2\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

4. Applied egg-rr 99.8%

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

   1. associate-/l/ N/A

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

   2. associate-/r* N/A

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

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(\alpha + 1\right) \cdot \frac{1 + \beta}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\right), \color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}\right) \]

6. Applied egg-rr 99.8%

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

7. Final simplification 99.8%

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

Alternative 4: 98.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 64000000000000:\\ \;\;\;\;\frac{\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\beta + 2\right)}}{1 + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 3\right)}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (+ beta alpha))))
   (if (<= beta 64000000000000.0)
     (/ (/ (+ 1.0 beta) (* (+ beta 2.0) (+ beta 2.0))) (+ 1.0 t_0))
     (/ (/ (+ 1.0 alpha) (+ alpha (+ beta 3.0))) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta + alpha);
	double tmp;
	if (beta <= 64000000000000.0) {
		tmp = ((1.0 + beta) / ((beta + 2.0) * (beta + 2.0))) / (1.0 + t_0);
	} else {
		tmp = ((1.0 + alpha) / (alpha + (beta + 3.0))) / t_0;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 + (beta + alpha)
    if (beta <= 64000000000000.0d0) then
        tmp = ((1.0d0 + beta) / ((beta + 2.0d0) * (beta + 2.0d0))) / (1.0d0 + t_0)
    else
        tmp = ((1.0d0 + alpha) / (alpha + (beta + 3.0d0))) / t_0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta + alpha);
	double tmp;
	if (beta <= 64000000000000.0) {
		tmp = ((1.0 + beta) / ((beta + 2.0) * (beta + 2.0))) / (1.0 + t_0);
	} else {
		tmp = ((1.0 + alpha) / (alpha + (beta + 3.0))) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = 2.0 + (beta + alpha)
	tmp = 0
	if beta <= 64000000000000.0:
		tmp = ((1.0 + beta) / ((beta + 2.0) * (beta + 2.0))) / (1.0 + t_0)
	else:
		tmp = ((1.0 + alpha) / (alpha + (beta + 3.0))) / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(beta + alpha))
	tmp = 0.0
	if (beta <= 64000000000000.0)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(Float64(beta + 2.0) * Float64(beta + 2.0))) / Float64(1.0 + t_0));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(beta + 3.0))) / t_0);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = 2.0 + (beta + alpha);
	tmp = 0.0;
	if (beta <= 64000000000000.0)
		tmp = ((1.0 + beta) / ((beta + 2.0) * (beta + 2.0))) / (1.0 + t_0);
	else
		tmp = ((1.0 + alpha) / (alpha + (beta + 3.0))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 64000000000000.0], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.4e13

   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. Add Preprocessing

   3. Taylor expanded in alpha around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1 + \beta}{{\left(2 + \beta\right)}^{2}}\right)}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + \beta\right), \left({\left(2 + \beta\right)}^{2}\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)}, 1\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \left({\left(2 + \beta\right)}^{2}\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(\alpha, \beta\right)}, \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \left(\left(2 + \beta\right) \cdot \left(2 + \beta\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \color{blue}{\mathsf{*.f64}\left(2, 1\right)}\right), 1\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left(2 + \beta\right), \left(2 + \beta\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \color{blue}{\mathsf{*.f64}\left(2, 1\right)}\right), 1\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left(\beta + 2\right), \left(2 + \beta\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(\color{blue}{2}, 1\right)\right), 1\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(2 + \beta\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(\color{blue}{2}, 1\right)\right), 1\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(\beta + 2\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, \color{blue}{1}\right)\right), 1\right)\right) \]

      8. +-lowering-+.f64 69.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, \color{blue}{1}\right)\right), 1\right)\right) \]

   5. Simplified 69.3%

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

   if 6.4e13 < beta

   1. Initial program 81.8%

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

      1. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\alpha}, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 + \left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      4. *-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 + \left(\alpha + \left(\beta \cdot 1 + \beta \cdot \alpha\right)\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      5. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 + \left(\alpha + \beta \cdot \left(1 + \alpha\right)\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 + \left(\alpha + \beta \cdot \left(\alpha + 1\right)\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\left(1 + \alpha\right) + \beta \cdot \left(\alpha + 1\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\alpha}, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\left(\alpha + 1\right) + \beta \cdot \left(\alpha + 1\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      9. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 \cdot \left(\alpha + 1\right) + \beta \cdot \left(\alpha + 1\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      10. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\left(\alpha + 1\right) \cdot \left(1 + \beta\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\alpha}, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      12. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(\alpha, \beta\right)}, \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\alpha + 1\right), \left(\frac{\beta + 1}{\left(\alpha + \beta\right) + 2}\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(\alpha, \beta\right)}, \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\frac{\beta + 1}{\left(\alpha + \beta\right) + 2}\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\alpha}, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{/.f64}\left(\left(\beta + 1\right), \left(\left(\alpha + \beta\right) + 2\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \color{blue}{\beta}\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{/.f64}\left(\left(1 + \beta\right), \left(\left(\alpha + \beta\right) + 2\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      17. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \left(\left(\alpha + \beta\right) + 2\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      18. associate-+l+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \left(\alpha + \left(\beta + 2\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      19. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(\alpha, \left(\beta + 2\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      20. +-lowering-+.f64 99.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 2\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

   4. Applied egg-rr 99.7%

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

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(\alpha + 1\right) \cdot \frac{1 + \beta}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\right), \color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}\right) \]

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in beta around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(1 + \alpha\right)}, \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 3\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right) \]
   8. Step-by-step derivation

      1. +-lowering-+.f64 76.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 3\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right) \]

   9. Simplified 76.8%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 64000000000000:\\ \;\;\;\;\frac{\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\beta + 2\right)}}{1 + \left(2 + \left(\beta + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 3\right)}}{2 + \left(\beta + \alpha\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 95000000000000:\\ \;\;\;\;\frac{1 + \beta}{12 + \beta \cdot \left(16 + \beta \cdot \left(\beta + 7\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 3\right)}}{2 + \left(\beta + \alpha\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 95000000000000.0)
   (/ (+ 1.0 beta) (+ 12.0 (* beta (+ 16.0 (* beta (+ beta 7.0))))))
   (/ (/ (+ 1.0 alpha) (+ alpha (+ beta 3.0))) (+ 2.0 (+ beta alpha)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 95000000000000.0) {
		tmp = (1.0 + beta) / (12.0 + (beta * (16.0 + (beta * (beta + 7.0)))));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (beta + 3.0))) / (2.0 + (beta + alpha));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 95000000000000.0d0) then
        tmp = (1.0d0 + beta) / (12.0d0 + (beta * (16.0d0 + (beta * (beta + 7.0d0)))))
    else
        tmp = ((1.0d0 + alpha) / (alpha + (beta + 3.0d0))) / (2.0d0 + (beta + alpha))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 95000000000000.0) {
		tmp = (1.0 + beta) / (12.0 + (beta * (16.0 + (beta * (beta + 7.0)))));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (beta + 3.0))) / (2.0 + (beta + alpha));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 95000000000000.0:
		tmp = (1.0 + beta) / (12.0 + (beta * (16.0 + (beta * (beta + 7.0)))))
	else:
		tmp = ((1.0 + alpha) / (alpha + (beta + 3.0))) / (2.0 + (beta + alpha))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 95000000000000.0)
		tmp = Float64(Float64(1.0 + beta) / Float64(12.0 + Float64(beta * Float64(16.0 + Float64(beta * Float64(beta + 7.0))))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(beta + 3.0))) / Float64(2.0 + Float64(beta + alpha)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 95000000000000.0)
		tmp = (1.0 + beta) / (12.0 + (beta * (16.0 + (beta * (beta + 7.0)))));
	else
		tmp = ((1.0 + alpha) / (alpha + (beta + 3.0))) / (2.0 + (beta + alpha));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 95000000000000.0], N[(N[(1.0 + beta), $MachinePrecision] / N[(12.0 + N[(beta * N[(16.0 + N[(beta * N[(beta + 7.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 9.5e13

   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/ N/A

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

      2. associate-/l/ N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right) + 1\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      5. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\left(\alpha + \beta\right) + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      6. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\alpha + \left(\beta + 1\right)\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \alpha\right) + \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      8. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta + 1\right) \cdot \alpha + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\alpha \cdot \left(\beta + 1\right) + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\alpha + 1\right) \cdot \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \alpha\right) \cdot \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \alpha\right), \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\alpha + 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   3. Simplified 93.9%

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

   5. Taylor expanded in alpha around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \beta\right), \color{blue}{\left({\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \left(\color{blue}{{\left(2 + \beta\right)}^{2}} \cdot \left(3 + \beta\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left({\left(2 + \beta\right)}^{2}\right), \color{blue}{\left(3 + \beta\right)}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left(\left(2 + \beta\right) \cdot \left(2 + \beta\right)\right), \left(\color{blue}{3} + \beta\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 + \beta\right), \left(2 + \beta\right)\right), \left(\color{blue}{3} + \beta\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\beta + 2\right), \left(2 + \beta\right)\right), \left(3 + \beta\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(2 + \beta\right)\right), \left(3 + \beta\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(\beta + 2\right)\right), \left(3 + \beta\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(3 + \beta\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(\beta + \color{blue}{3}\right)\right)\right) \]

      11. +-lowering-+.f64 67.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\beta, \color{blue}{3}\right)\right)\right) \]

   7. Simplified 67.8%

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

   8. Taylor expanded in beta around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \color{blue}{\left(12 + \beta \cdot \left(16 + \beta \cdot \left(7 + \beta\right)\right)\right)}\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(12, \color{blue}{\left(\beta \cdot \left(16 + \beta \cdot \left(7 + \beta\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(12, \mathsf{*.f64}\left(\beta, \color{blue}{\left(16 + \beta \cdot \left(7 + \beta\right)\right)}\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(12, \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(16, \color{blue}{\left(\beta \cdot \left(7 + \beta\right)\right)}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(12, \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(16, \mathsf{*.f64}\left(\beta, \color{blue}{\left(7 + \beta\right)}\right)\right)\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(12, \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(16, \mathsf{*.f64}\left(\beta, \left(\beta + \color{blue}{7}\right)\right)\right)\right)\right)\right) \]

      6. +-lowering-+.f64 67.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(12, \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(16, \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(\beta, \color{blue}{7}\right)\right)\right)\right)\right)\right) \]

   10. Simplified 67.9%

       \[\leadsto \frac{1 + \beta}{\color{blue}{12 + \beta \cdot \left(16 + \beta \cdot \left(\beta + 7\right)\right)}} \]

   if 9.5e13 < beta

   1. Initial program 81.8%

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

      1. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\alpha}, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 + \left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      4. *-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 + \left(\alpha + \left(\beta \cdot 1 + \beta \cdot \alpha\right)\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      5. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 + \left(\alpha + \beta \cdot \left(1 + \alpha\right)\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 + \left(\alpha + \beta \cdot \left(\alpha + 1\right)\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\left(1 + \alpha\right) + \beta \cdot \left(\alpha + 1\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\alpha}, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\left(\alpha + 1\right) + \beta \cdot \left(\alpha + 1\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      9. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 \cdot \left(\alpha + 1\right) + \beta \cdot \left(\alpha + 1\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      10. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\left(\alpha + 1\right) \cdot \left(1 + \beta\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\alpha}, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      12. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(\alpha, \beta\right)}, \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\alpha + 1\right), \left(\frac{\beta + 1}{\left(\alpha + \beta\right) + 2}\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(\alpha, \beta\right)}, \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\frac{\beta + 1}{\left(\alpha + \beta\right) + 2}\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\alpha}, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{/.f64}\left(\left(\beta + 1\right), \left(\left(\alpha + \beta\right) + 2\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \color{blue}{\beta}\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{/.f64}\left(\left(1 + \beta\right), \left(\left(\alpha + \beta\right) + 2\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      17. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \left(\left(\alpha + \beta\right) + 2\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      18. associate-+l+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \left(\alpha + \left(\beta + 2\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      19. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(\alpha, \left(\beta + 2\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      20. +-lowering-+.f64 99.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 2\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

   4. Applied egg-rr 99.7%

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

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(\alpha + 1\right) \cdot \frac{1 + \beta}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\right), \color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}\right) \]

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in beta around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(1 + \alpha\right)}, \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 3\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right) \]
   8. Step-by-step derivation

      1. +-lowering-+.f64 76.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 3\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right) \]

   9. Simplified 76.8%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 95000000000000:\\ \;\;\;\;\frac{1 + \beta}{12 + \beta \cdot \left(16 + \beta \cdot \left(\beta + 7\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 3\right)}}{2 + \left(\beta + \alpha\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 95000000000000:\\ \;\;\;\;\frac{1 + \beta}{12 + \beta \cdot \left(16 + \beta \cdot \left(\beta + 7\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{1 + \left(2 + \left(\beta + \alpha\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 95000000000000.0)
   (/ (+ 1.0 beta) (+ 12.0 (* beta (+ 16.0 (* beta (+ beta 7.0))))))
   (/ (/ (+ 1.0 alpha) beta) (+ 1.0 (+ 2.0 (+ beta alpha))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 95000000000000.0) {
		tmp = (1.0 + beta) / (12.0 + (beta * (16.0 + (beta * (beta + 7.0)))));
	} else {
		tmp = ((1.0 + alpha) / beta) / (1.0 + (2.0 + (beta + alpha)));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 95000000000000.0d0) then
        tmp = (1.0d0 + beta) / (12.0d0 + (beta * (16.0d0 + (beta * (beta + 7.0d0)))))
    else
        tmp = ((1.0d0 + alpha) / beta) / (1.0d0 + (2.0d0 + (beta + alpha)))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 95000000000000.0) {
		tmp = (1.0 + beta) / (12.0 + (beta * (16.0 + (beta * (beta + 7.0)))));
	} else {
		tmp = ((1.0 + alpha) / beta) / (1.0 + (2.0 + (beta + alpha)));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 95000000000000.0:
		tmp = (1.0 + beta) / (12.0 + (beta * (16.0 + (beta * (beta + 7.0)))))
	else:
		tmp = ((1.0 + alpha) / beta) / (1.0 + (2.0 + (beta + alpha)))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 95000000000000.0)
		tmp = Float64(Float64(1.0 + beta) / Float64(12.0 + Float64(beta * Float64(16.0 + Float64(beta * Float64(beta + 7.0))))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(1.0 + Float64(2.0 + Float64(beta + alpha))));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 95000000000000.0)
		tmp = (1.0 + beta) / (12.0 + (beta * (16.0 + (beta * (beta + 7.0)))));
	else
		tmp = ((1.0 + alpha) / beta) / (1.0 + (2.0 + (beta + alpha)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 95000000000000.0], N[(N[(1.0 + beta), $MachinePrecision] / N[(12.0 + N[(beta * N[(16.0 + N[(beta * N[(beta + 7.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(1.0 + N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 9.5e13

   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/ N/A

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

      2. associate-/l/ N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right) + 1\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      5. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\left(\alpha + \beta\right) + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      6. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\alpha + \left(\beta + 1\right)\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \alpha\right) + \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      8. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta + 1\right) \cdot \alpha + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\alpha \cdot \left(\beta + 1\right) + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\alpha + 1\right) \cdot \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \alpha\right) \cdot \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \alpha\right), \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\alpha + 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   3. Simplified 93.9%

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

   5. Taylor expanded in alpha around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \beta\right), \color{blue}{\left({\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \left(\color{blue}{{\left(2 + \beta\right)}^{2}} \cdot \left(3 + \beta\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left({\left(2 + \beta\right)}^{2}\right), \color{blue}{\left(3 + \beta\right)}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left(\left(2 + \beta\right) \cdot \left(2 + \beta\right)\right), \left(\color{blue}{3} + \beta\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 + \beta\right), \left(2 + \beta\right)\right), \left(\color{blue}{3} + \beta\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\beta + 2\right), \left(2 + \beta\right)\right), \left(3 + \beta\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(2 + \beta\right)\right), \left(3 + \beta\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(\beta + 2\right)\right), \left(3 + \beta\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(3 + \beta\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(\beta + \color{blue}{3}\right)\right)\right) \]

      11. +-lowering-+.f64 67.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\beta, \color{blue}{3}\right)\right)\right) \]

   7. Simplified 67.8%

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

   8. Taylor expanded in beta around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \color{blue}{\left(12 + \beta \cdot \left(16 + \beta \cdot \left(7 + \beta\right)\right)\right)}\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(12, \color{blue}{\left(\beta \cdot \left(16 + \beta \cdot \left(7 + \beta\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(12, \mathsf{*.f64}\left(\beta, \color{blue}{\left(16 + \beta \cdot \left(7 + \beta\right)\right)}\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(12, \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(16, \color{blue}{\left(\beta \cdot \left(7 + \beta\right)\right)}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(12, \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(16, \mathsf{*.f64}\left(\beta, \color{blue}{\left(7 + \beta\right)}\right)\right)\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(12, \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(16, \mathsf{*.f64}\left(\beta, \left(\beta + \color{blue}{7}\right)\right)\right)\right)\right)\right) \]

      6. +-lowering-+.f64 67.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(12, \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(16, \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(\beta, \color{blue}{7}\right)\right)\right)\right)\right)\right) \]

   10. Simplified 67.9%

       \[\leadsto \frac{1 + \beta}{\color{blue}{12 + \beta \cdot \left(16 + \beta \cdot \left(\beta + 7\right)\right)}} \]

   if 9.5e13 < beta

   1. Initial program 81.8%

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

   3. Taylor expanded in beta around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1 + \alpha}{\beta}\right)}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + \alpha\right), \beta\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)}, 1\right)\right) \]

      2. +-lowering-+.f64 76.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \beta\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(\alpha, \beta\right)}, \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

   5. Simplified 76.2%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 95000000000000:\\ \;\;\;\;\frac{1 + \beta}{12 + \beta \cdot \left(16 + \beta \cdot \left(\beta + 7\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{1 + \left(2 + \left(\beta + \alpha\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 95000000000000:\\ \;\;\;\;\frac{1 + \beta}{12 + \beta \cdot \left(16 + \beta \cdot \left(\beta + 7\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{2 + \left(\beta + \alpha\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 95000000000000.0)
   (/ (+ 1.0 beta) (+ 12.0 (* beta (+ 16.0 (* beta (+ beta 7.0))))))
   (/ (/ (+ 1.0 alpha) beta) (+ 2.0 (+ beta alpha)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 95000000000000.0) {
		tmp = (1.0 + beta) / (12.0 + (beta * (16.0 + (beta * (beta + 7.0)))));
	} else {
		tmp = ((1.0 + alpha) / beta) / (2.0 + (beta + alpha));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 95000000000000.0d0) then
        tmp = (1.0d0 + beta) / (12.0d0 + (beta * (16.0d0 + (beta * (beta + 7.0d0)))))
    else
        tmp = ((1.0d0 + alpha) / beta) / (2.0d0 + (beta + alpha))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 95000000000000.0) {
		tmp = (1.0 + beta) / (12.0 + (beta * (16.0 + (beta * (beta + 7.0)))));
	} else {
		tmp = ((1.0 + alpha) / beta) / (2.0 + (beta + alpha));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 95000000000000.0:
		tmp = (1.0 + beta) / (12.0 + (beta * (16.0 + (beta * (beta + 7.0)))))
	else:
		tmp = ((1.0 + alpha) / beta) / (2.0 + (beta + alpha))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 95000000000000.0)
		tmp = Float64(Float64(1.0 + beta) / Float64(12.0 + Float64(beta * Float64(16.0 + Float64(beta * Float64(beta + 7.0))))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(2.0 + Float64(beta + alpha)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 95000000000000.0)
		tmp = (1.0 + beta) / (12.0 + (beta * (16.0 + (beta * (beta + 7.0)))));
	else
		tmp = ((1.0 + alpha) / beta) / (2.0 + (beta + alpha));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 95000000000000.0], N[(N[(1.0 + beta), $MachinePrecision] / N[(12.0 + N[(beta * N[(16.0 + N[(beta * N[(beta + 7.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 9.5e13

   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/ N/A

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

      2. associate-/l/ N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right) + 1\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      5. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\left(\alpha + \beta\right) + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      6. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\alpha + \left(\beta + 1\right)\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \alpha\right) + \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      8. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta + 1\right) \cdot \alpha + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\alpha \cdot \left(\beta + 1\right) + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\alpha + 1\right) \cdot \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \alpha\right) \cdot \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \alpha\right), \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\alpha + 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   3. Simplified 93.9%

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

   5. Taylor expanded in alpha around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \beta\right), \color{blue}{\left({\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \left(\color{blue}{{\left(2 + \beta\right)}^{2}} \cdot \left(3 + \beta\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left({\left(2 + \beta\right)}^{2}\right), \color{blue}{\left(3 + \beta\right)}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left(\left(2 + \beta\right) \cdot \left(2 + \beta\right)\right), \left(\color{blue}{3} + \beta\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 + \beta\right), \left(2 + \beta\right)\right), \left(\color{blue}{3} + \beta\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\beta + 2\right), \left(2 + \beta\right)\right), \left(3 + \beta\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(2 + \beta\right)\right), \left(3 + \beta\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(\beta + 2\right)\right), \left(3 + \beta\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(3 + \beta\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(\beta + \color{blue}{3}\right)\right)\right) \]

      11. +-lowering-+.f64 67.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\beta, \color{blue}{3}\right)\right)\right) \]

   7. Simplified 67.8%

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

   8. Taylor expanded in beta around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \color{blue}{\left(12 + \beta \cdot \left(16 + \beta \cdot \left(7 + \beta\right)\right)\right)}\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(12, \color{blue}{\left(\beta \cdot \left(16 + \beta \cdot \left(7 + \beta\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(12, \mathsf{*.f64}\left(\beta, \color{blue}{\left(16 + \beta \cdot \left(7 + \beta\right)\right)}\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(12, \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(16, \color{blue}{\left(\beta \cdot \left(7 + \beta\right)\right)}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(12, \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(16, \mathsf{*.f64}\left(\beta, \color{blue}{\left(7 + \beta\right)}\right)\right)\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(12, \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(16, \mathsf{*.f64}\left(\beta, \left(\beta + \color{blue}{7}\right)\right)\right)\right)\right)\right) \]

      6. +-lowering-+.f64 67.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(12, \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(16, \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(\beta, \color{blue}{7}\right)\right)\right)\right)\right)\right) \]

   10. Simplified 67.9%

       \[\leadsto \frac{1 + \beta}{\color{blue}{12 + \beta \cdot \left(16 + \beta \cdot \left(\beta + 7\right)\right)}} \]

   if 9.5e13 < beta

   1. Initial program 81.8%

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

      1. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\alpha}, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 + \left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      4. *-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 + \left(\alpha + \left(\beta \cdot 1 + \beta \cdot \alpha\right)\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      5. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 + \left(\alpha + \beta \cdot \left(1 + \alpha\right)\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 + \left(\alpha + \beta \cdot \left(\alpha + 1\right)\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\left(1 + \alpha\right) + \beta \cdot \left(\alpha + 1\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\alpha}, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\left(\alpha + 1\right) + \beta \cdot \left(\alpha + 1\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      9. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 \cdot \left(\alpha + 1\right) + \beta \cdot \left(\alpha + 1\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      10. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\left(\alpha + 1\right) \cdot \left(1 + \beta\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\alpha}, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      12. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(\alpha, \beta\right)}, \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\alpha + 1\right), \left(\frac{\beta + 1}{\left(\alpha + \beta\right) + 2}\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(\alpha, \beta\right)}, \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\frac{\beta + 1}{\left(\alpha + \beta\right) + 2}\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\alpha}, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{/.f64}\left(\left(\beta + 1\right), \left(\left(\alpha + \beta\right) + 2\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \color{blue}{\beta}\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{/.f64}\left(\left(1 + \beta\right), \left(\left(\alpha + \beta\right) + 2\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      17. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \left(\left(\alpha + \beta\right) + 2\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      18. associate-+l+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \left(\alpha + \left(\beta + 2\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      19. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(\alpha, \left(\beta + 2\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      20. +-lowering-+.f64 99.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 2\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

   4. Applied egg-rr 99.7%

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

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(\alpha + 1\right) \cdot \frac{1 + \beta}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\right), \color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}\right) \]

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in beta around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1 + \alpha}{\beta}\right)}, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + \alpha\right), \beta\right), \mathsf{+.f64}\left(\color{blue}{2}, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right) \]

      2. +-lowering-+.f64 76.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \beta\right), \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right) \]

   9. Simplified 76.2%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 95000000000000:\\ \;\;\;\;\frac{1 + \beta}{12 + \beta \cdot \left(16 + \beta \cdot \left(\beta + 7\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{2 + \left(\beta + \alpha\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 90000000000000:\\ \;\;\;\;\frac{1 + \beta}{\left(\beta + 3\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{2 + \left(\beta + \alpha\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 90000000000000.0)
   (/ (+ 1.0 beta) (* (+ beta 3.0) (* (+ beta 2.0) (+ beta 2.0))))
   (/ (/ (+ 1.0 alpha) beta) (+ 2.0 (+ beta alpha)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 90000000000000.0) {
		tmp = (1.0 + beta) / ((beta + 3.0) * ((beta + 2.0) * (beta + 2.0)));
	} else {
		tmp = ((1.0 + alpha) / beta) / (2.0 + (beta + alpha));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 90000000000000.0d0) then
        tmp = (1.0d0 + beta) / ((beta + 3.0d0) * ((beta + 2.0d0) * (beta + 2.0d0)))
    else
        tmp = ((1.0d0 + alpha) / beta) / (2.0d0 + (beta + alpha))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 90000000000000.0) {
		tmp = (1.0 + beta) / ((beta + 3.0) * ((beta + 2.0) * (beta + 2.0)));
	} else {
		tmp = ((1.0 + alpha) / beta) / (2.0 + (beta + alpha));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 90000000000000.0:
		tmp = (1.0 + beta) / ((beta + 3.0) * ((beta + 2.0) * (beta + 2.0)))
	else:
		tmp = ((1.0 + alpha) / beta) / (2.0 + (beta + alpha))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 90000000000000.0)
		tmp = Float64(Float64(1.0 + beta) / Float64(Float64(beta + 3.0) * Float64(Float64(beta + 2.0) * Float64(beta + 2.0))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(2.0 + Float64(beta + alpha)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 90000000000000.0)
		tmp = (1.0 + beta) / ((beta + 3.0) * ((beta + 2.0) * (beta + 2.0)));
	else
		tmp = ((1.0 + alpha) / beta) / (2.0 + (beta + alpha));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 90000000000000.0], N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(beta + 3.0), $MachinePrecision] * N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 9e13

   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/ N/A

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

      2. associate-/l/ N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right) + 1\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      5. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\left(\alpha + \beta\right) + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      6. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\alpha + \left(\beta + 1\right)\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \alpha\right) + \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      8. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta + 1\right) \cdot \alpha + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\alpha \cdot \left(\beta + 1\right) + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\alpha + 1\right) \cdot \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \alpha\right) \cdot \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \alpha\right), \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\alpha + 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   3. Simplified 93.9%

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

   5. Taylor expanded in alpha around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \beta\right), \color{blue}{\left({\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \left(\color{blue}{{\left(2 + \beta\right)}^{2}} \cdot \left(3 + \beta\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left({\left(2 + \beta\right)}^{2}\right), \color{blue}{\left(3 + \beta\right)}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left(\left(2 + \beta\right) \cdot \left(2 + \beta\right)\right), \left(\color{blue}{3} + \beta\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 + \beta\right), \left(2 + \beta\right)\right), \left(\color{blue}{3} + \beta\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\beta + 2\right), \left(2 + \beta\right)\right), \left(3 + \beta\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(2 + \beta\right)\right), \left(3 + \beta\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(\beta + 2\right)\right), \left(3 + \beta\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(3 + \beta\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(\beta + \color{blue}{3}\right)\right)\right) \]

      11. +-lowering-+.f64 67.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\beta, \color{blue}{3}\right)\right)\right) \]

   7. Simplified 67.8%

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

   if 9e13 < beta

   1. Initial program 81.8%

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

      1. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\alpha}, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 + \left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      4. *-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 + \left(\alpha + \left(\beta \cdot 1 + \beta \cdot \alpha\right)\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      5. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 + \left(\alpha + \beta \cdot \left(1 + \alpha\right)\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 + \left(\alpha + \beta \cdot \left(\alpha + 1\right)\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\left(1 + \alpha\right) + \beta \cdot \left(\alpha + 1\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\alpha}, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\left(\alpha + 1\right) + \beta \cdot \left(\alpha + 1\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      9. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 \cdot \left(\alpha + 1\right) + \beta \cdot \left(\alpha + 1\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      10. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\left(\alpha + 1\right) \cdot \left(1 + \beta\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\alpha}, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      12. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(\alpha, \beta\right)}, \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\alpha + 1\right), \left(\frac{\beta + 1}{\left(\alpha + \beta\right) + 2}\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(\alpha, \beta\right)}, \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\frac{\beta + 1}{\left(\alpha + \beta\right) + 2}\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\alpha}, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{/.f64}\left(\left(\beta + 1\right), \left(\left(\alpha + \beta\right) + 2\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \color{blue}{\beta}\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{/.f64}\left(\left(1 + \beta\right), \left(\left(\alpha + \beta\right) + 2\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      17. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \left(\left(\alpha + \beta\right) + 2\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      18. associate-+l+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \left(\alpha + \left(\beta + 2\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      19. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(\alpha, \left(\beta + 2\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      20. +-lowering-+.f64 99.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 2\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

   4. Applied egg-rr 99.7%

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

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(\alpha + 1\right) \cdot \frac{1 + \beta}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\right), \color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}\right) \]

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in beta around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1 + \alpha}{\beta}\right)}, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + \alpha\right), \beta\right), \mathsf{+.f64}\left(\color{blue}{2}, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right) \]

      2. +-lowering-+.f64 76.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \beta\right), \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right) \]

   9. Simplified 76.2%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 90000000000000:\\ \;\;\;\;\frac{1 + \beta}{\left(\beta + 3\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{2 + \left(\beta + \alpha\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.4:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot \left(\alpha \cdot \left(\alpha \cdot 0.024691358024691357 + -0.011574074074074073\right) + -0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{2 + \left(\beta + \alpha\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.4)
   (+
    0.08333333333333333
    (*
     alpha
     (+
      (* alpha (+ (* alpha 0.024691358024691357) -0.011574074074074073))
      -0.027777777777777776)))
   (/ (/ (+ 1.0 alpha) beta) (+ 2.0 (+ beta alpha)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.4) {
		tmp = 0.08333333333333333 + (alpha * ((alpha * ((alpha * 0.024691358024691357) + -0.011574074074074073)) + -0.027777777777777776));
	} else {
		tmp = ((1.0 + alpha) / beta) / (2.0 + (beta + alpha));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.4d0) then
        tmp = 0.08333333333333333d0 + (alpha * ((alpha * ((alpha * 0.024691358024691357d0) + (-0.011574074074074073d0))) + (-0.027777777777777776d0)))
    else
        tmp = ((1.0d0 + alpha) / beta) / (2.0d0 + (beta + alpha))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.4) {
		tmp = 0.08333333333333333 + (alpha * ((alpha * ((alpha * 0.024691358024691357) + -0.011574074074074073)) + -0.027777777777777776));
	} else {
		tmp = ((1.0 + alpha) / beta) / (2.0 + (beta + alpha));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.4:
		tmp = 0.08333333333333333 + (alpha * ((alpha * ((alpha * 0.024691358024691357) + -0.011574074074074073)) + -0.027777777777777776))
	else:
		tmp = ((1.0 + alpha) / beta) / (2.0 + (beta + alpha))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.4)
		tmp = Float64(0.08333333333333333 + Float64(alpha * Float64(Float64(alpha * Float64(Float64(alpha * 0.024691358024691357) + -0.011574074074074073)) + -0.027777777777777776)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(2.0 + Float64(beta + alpha)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.4)
		tmp = 0.08333333333333333 + (alpha * ((alpha * ((alpha * 0.024691358024691357) + -0.011574074074074073)) + -0.027777777777777776));
	else
		tmp = ((1.0 + alpha) / beta) / (2.0 + (beta + alpha));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.4], N[(0.08333333333333333 + N[(alpha * N[(N[(alpha * N[(N[(alpha * 0.024691358024691357), $MachinePrecision] + -0.011574074074074073), $MachinePrecision]), $MachinePrecision] + -0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.39999999999999991

   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/ N/A

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

      2. associate-/l/ N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right) + 1\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      5. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\left(\alpha + \beta\right) + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      6. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\alpha + \left(\beta + 1\right)\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \alpha\right) + \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      8. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta + 1\right) \cdot \alpha + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\alpha \cdot \left(\beta + 1\right) + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\alpha + 1\right) \cdot \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \alpha\right) \cdot \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \alpha\right), \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\alpha + 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   3. Simplified 94.8%

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

   5. Taylor expanded in beta around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \alpha\right), \color{blue}{\left({\left(2 + \alpha\right)}^{2} \cdot \left(3 + \alpha\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left(\color{blue}{{\left(2 + \alpha\right)}^{2}} \cdot \left(3 + \alpha\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\left({\left(2 + \alpha\right)}^{2}\right), \color{blue}{\left(3 + \alpha\right)}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\left(\left(2 + \alpha\right) \cdot \left(2 + \alpha\right)\right), \left(\color{blue}{3} + \alpha\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 + \alpha\right), \left(2 + \alpha\right)\right), \left(\color{blue}{3} + \alpha\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \alpha\right), \left(2 + \alpha\right)\right), \left(3 + \alpha\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \alpha\right), \mathsf{+.f64}\left(2, \alpha\right)\right), \left(3 + \alpha\right)\right)\right) \]

      8. +-lowering-+.f64 92.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \alpha\right), \mathsf{+.f64}\left(2, \alpha\right)\right), \mathsf{+.f64}\left(3, \color{blue}{\alpha}\right)\right)\right) \]

   7. Simplified 92.4%

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

   8. Taylor expanded in alpha around 0

      \[\leadsto \color{blue}{\frac{1}{12} + \alpha \cdot \left(\alpha \cdot \left(\frac{2}{81} \cdot \alpha - \frac{5}{432}\right) - \frac{1}{36}\right)} \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \color{blue}{\left(\alpha \cdot \left(\alpha \cdot \left(\frac{2}{81} \cdot \alpha - \frac{5}{432}\right) - \frac{1}{36}\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\alpha, \color{blue}{\left(\alpha \cdot \left(\frac{2}{81} \cdot \alpha - \frac{5}{432}\right) - \frac{1}{36}\right)}\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\alpha, \left(\alpha \cdot \left(\frac{2}{81} \cdot \alpha - \frac{5}{432}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{36}\right)\right)}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\alpha, \left(\alpha \cdot \left(\frac{2}{81} \cdot \alpha - \frac{5}{432}\right) + \frac{-1}{36}\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\alpha, \mathsf{+.f64}\left(\left(\alpha \cdot \left(\frac{2}{81} \cdot \alpha - \frac{5}{432}\right)\right), \color{blue}{\frac{-1}{36}}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\alpha, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\alpha, \left(\frac{2}{81} \cdot \alpha - \frac{5}{432}\right)\right), \frac{-1}{36}\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\alpha, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\alpha, \left(\frac{2}{81} \cdot \alpha + \left(\mathsf{neg}\left(\frac{5}{432}\right)\right)\right)\right), \frac{-1}{36}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\alpha, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\alpha, \left(\frac{2}{81} \cdot \alpha + \frac{-5}{432}\right)\right), \frac{-1}{36}\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\alpha, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\alpha, \mathsf{+.f64}\left(\left(\frac{2}{81} \cdot \alpha\right), \frac{-5}{432}\right)\right), \frac{-1}{36}\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\alpha, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\alpha, \mathsf{+.f64}\left(\left(\alpha \cdot \frac{2}{81}\right), \frac{-5}{432}\right)\right), \frac{-1}{36}\right)\right)\right) \]

      11. *-lowering-*.f64 66.2%

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\alpha, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\alpha, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\alpha, \frac{2}{81}\right), \frac{-5}{432}\right)\right), \frac{-1}{36}\right)\right)\right) \]

   10. Simplified 66.2%

       \[\leadsto \color{blue}{0.08333333333333333 + \alpha \cdot \left(\alpha \cdot \left(\alpha \cdot 0.024691358024691357 + -0.011574074074074073\right) + -0.027777777777777776\right)} \]

   if 2.39999999999999991 < beta

   1. Initial program 83.2%

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

      1. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\alpha}, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 + \left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      4. *-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 + \left(\alpha + \left(\beta \cdot 1 + \beta \cdot \alpha\right)\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      5. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 + \left(\alpha + \beta \cdot \left(1 + \alpha\right)\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 + \left(\alpha + \beta \cdot \left(\alpha + 1\right)\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\left(1 + \alpha\right) + \beta \cdot \left(\alpha + 1\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\alpha}, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\left(\alpha + 1\right) + \beta \cdot \left(\alpha + 1\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      9. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 \cdot \left(\alpha + 1\right) + \beta \cdot \left(\alpha + 1\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      10. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\left(\alpha + 1\right) \cdot \left(1 + \beta\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\alpha}, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      12. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(\alpha, \beta\right)}, \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\alpha + 1\right), \left(\frac{\beta + 1}{\left(\alpha + \beta\right) + 2}\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(\alpha, \beta\right)}, \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\frac{\beta + 1}{\left(\alpha + \beta\right) + 2}\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\alpha}, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{/.f64}\left(\left(\beta + 1\right), \left(\left(\alpha + \beta\right) + 2\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \color{blue}{\beta}\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{/.f64}\left(\left(1 + \beta\right), \left(\left(\alpha + \beta\right) + 2\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      17. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \left(\left(\alpha + \beta\right) + 2\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      18. associate-+l+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \left(\alpha + \left(\beta + 2\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      19. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(\alpha, \left(\beta + 2\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      20. +-lowering-+.f64 99.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 2\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

   4. Applied egg-rr 99.6%

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

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(\alpha + 1\right) \cdot \frac{1 + \beta}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\right), \color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}\right) \]

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in beta around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1 + \alpha}{\beta}\right)}, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + \alpha\right), \beta\right), \mathsf{+.f64}\left(\color{blue}{2}, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right) \]

      2. +-lowering-+.f64 73.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \beta\right), \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right) \]

   9. Simplified 73.2%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.4:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot \left(\alpha \cdot \left(\alpha \cdot 0.024691358024691357 + -0.011574074074074073\right) + -0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{2 + \left(\beta + \alpha\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 96.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.95:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot -0.027777777777777776\\ \mathbf{elif}\;\beta \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.95)
   (+ 0.08333333333333333 (* alpha -0.027777777777777776))
   (if (<= beta 1.35e+154)
     (/ (+ 1.0 alpha) (* beta beta))
     (/ (/ alpha beta) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.95) {
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776);
	} else if (beta <= 1.35e+154) {
		tmp = (1.0 + alpha) / (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 <= 2.95d0) then
        tmp = 0.08333333333333333d0 + (alpha * (-0.027777777777777776d0))
    else if (beta <= 1.35d+154) then
        tmp = (1.0d0 + alpha) / (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 <= 2.95) {
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776);
	} else if (beta <= 1.35e+154) {
		tmp = (1.0 + alpha) / (beta * beta);
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.95:
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776)
	elif beta <= 1.35e+154:
		tmp = (1.0 + alpha) / (beta * beta)
	else:
		tmp = (alpha / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.95)
		tmp = Float64(0.08333333333333333 + Float64(alpha * -0.027777777777777776));
	elseif (beta <= 1.35e+154)
		tmp = Float64(Float64(1.0 + alpha) / 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 <= 2.95)
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776);
	elseif (beta <= 1.35e+154)
		tmp = (1.0 + alpha) / (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, 2.95], N[(0.08333333333333333 + N[(alpha * -0.027777777777777776), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 1.35e+154], N[(N[(1.0 + alpha), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / beta), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if beta < 2.9500000000000002

   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/ N/A

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

      2. associate-/l/ N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right) + 1\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      5. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\left(\alpha + \beta\right) + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      6. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\alpha + \left(\beta + 1\right)\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \alpha\right) + \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      8. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta + 1\right) \cdot \alpha + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\alpha \cdot \left(\beta + 1\right) + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\alpha + 1\right) \cdot \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \alpha\right) \cdot \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \alpha\right), \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\alpha + 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   3. Simplified 94.8%

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

   5. Taylor expanded in beta around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \alpha\right), \color{blue}{\left({\left(2 + \alpha\right)}^{2} \cdot \left(3 + \alpha\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left(\color{blue}{{\left(2 + \alpha\right)}^{2}} \cdot \left(3 + \alpha\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\left({\left(2 + \alpha\right)}^{2}\right), \color{blue}{\left(3 + \alpha\right)}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\left(\left(2 + \alpha\right) \cdot \left(2 + \alpha\right)\right), \left(\color{blue}{3} + \alpha\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 + \alpha\right), \left(2 + \alpha\right)\right), \left(\color{blue}{3} + \alpha\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \alpha\right), \left(2 + \alpha\right)\right), \left(3 + \alpha\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \alpha\right), \mathsf{+.f64}\left(2, \alpha\right)\right), \left(3 + \alpha\right)\right)\right) \]

      8. +-lowering-+.f64 92.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \alpha\right), \mathsf{+.f64}\left(2, \alpha\right)\right), \mathsf{+.f64}\left(3, \color{blue}{\alpha}\right)\right)\right) \]

   7. Simplified 92.4%

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

   8. Taylor expanded in alpha around 0

      \[\leadsto \color{blue}{\frac{1}{12} + \frac{-1}{36} \cdot \alpha} \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \color{blue}{\left(\frac{-1}{36} \cdot \alpha\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \left(\alpha \cdot \color{blue}{\frac{-1}{36}}\right)\right) \]

      3. *-lowering-*.f64 65.8%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\alpha, \color{blue}{\frac{-1}{36}}\right)\right) \]

   10. Simplified 65.8%

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

   if 2.9500000000000002 < beta < 1.35000000000000003e154

   1. Initial program 86.8%

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

      1. associate-/l/ N/A

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

      2. associate-/l/ N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right) + 1\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      5. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\left(\alpha + \beta\right) + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      6. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\alpha + \left(\beta + 1\right)\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \alpha\right) + \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      8. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta + 1\right) \cdot \alpha + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\alpha \cdot \left(\beta + 1\right) + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\alpha + 1\right) \cdot \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \alpha\right) \cdot \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \alpha\right), \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\alpha + 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   3. Simplified 57.1%

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

   5. Taylor expanded in beta around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \alpha\right), \color{blue}{\left({\beta}^{2}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left({\color{blue}{\beta}}^{2}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left(\beta \cdot \color{blue}{\beta}\right)\right) \]

      4. *-lowering-*.f64 55.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\beta, \color{blue}{\beta}\right)\right) \]

   7. Simplified 55.7%

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

   if 1.35000000000000003e154 < beta

   1. Initial program 78.7%

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

      1. associate-/l/ N/A

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

      2. associate-/l/ N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right) + 1\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      5. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\left(\alpha + \beta\right) + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      6. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\alpha + \left(\beta + 1\right)\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \alpha\right) + \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      8. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta + 1\right) \cdot \alpha + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\alpha \cdot \left(\beta + 1\right) + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\alpha + 1\right) \cdot \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \alpha\right) \cdot \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \alpha\right), \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\alpha + 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   3. Simplified 73.9%

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

   5. Taylor expanded in beta around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \alpha\right), \color{blue}{\left({\beta}^{2}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left({\color{blue}{\beta}}^{2}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left(\beta \cdot \color{blue}{\beta}\right)\right) \]

      4. *-lowering-*.f64 85.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\beta, \color{blue}{\beta}\right)\right) \]

   7. Simplified 85.6%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 + \alpha}{\beta}\right), \color{blue}{\beta}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + \alpha\right), \beta\right), \beta\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\alpha + 1\right), \beta\right), \beta\right) \]

      5. +-lowering-+.f64 95.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \beta\right), \beta\right) \]

   9. Applied egg-rr 95.0%

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

   10. Taylor expanded in alpha around inf

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{\alpha}{\beta}\right)}, \beta\right) \]
   11. Step-by-step derivation

       1. /-lowering-/.f64 90.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\alpha, \beta\right), \beta\right) \]

   12. Simplified 90.5%

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

Alternative 11: 97.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.45:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot \left(-0.027777777777777776 + \alpha \cdot -0.011574074074074073\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{2 + \left(\beta + \alpha\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.45)
   (+
    0.08333333333333333
    (* alpha (+ -0.027777777777777776 (* alpha -0.011574074074074073))))
   (/ (/ (+ 1.0 alpha) beta) (+ 2.0 (+ beta alpha)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.45) {
		tmp = 0.08333333333333333 + (alpha * (-0.027777777777777776 + (alpha * -0.011574074074074073)));
	} else {
		tmp = ((1.0 + alpha) / beta) / (2.0 + (beta + alpha));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.45d0) then
        tmp = 0.08333333333333333d0 + (alpha * ((-0.027777777777777776d0) + (alpha * (-0.011574074074074073d0))))
    else
        tmp = ((1.0d0 + alpha) / beta) / (2.0d0 + (beta + alpha))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.45) {
		tmp = 0.08333333333333333 + (alpha * (-0.027777777777777776 + (alpha * -0.011574074074074073)));
	} else {
		tmp = ((1.0 + alpha) / beta) / (2.0 + (beta + alpha));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.45:
		tmp = 0.08333333333333333 + (alpha * (-0.027777777777777776 + (alpha * -0.011574074074074073)))
	else:
		tmp = ((1.0 + alpha) / beta) / (2.0 + (beta + alpha))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.45)
		tmp = Float64(0.08333333333333333 + Float64(alpha * Float64(-0.027777777777777776 + Float64(alpha * -0.011574074074074073))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(2.0 + Float64(beta + alpha)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.45)
		tmp = 0.08333333333333333 + (alpha * (-0.027777777777777776 + (alpha * -0.011574074074074073)));
	else
		tmp = ((1.0 + alpha) / beta) / (2.0 + (beta + alpha));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.45], N[(0.08333333333333333 + N[(alpha * N[(-0.027777777777777776 + N[(alpha * -0.011574074074074073), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.4500000000000002

   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/ N/A

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

      2. associate-/l/ N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right) + 1\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      5. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\left(\alpha + \beta\right) + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      6. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\alpha + \left(\beta + 1\right)\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \alpha\right) + \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      8. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta + 1\right) \cdot \alpha + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\alpha \cdot \left(\beta + 1\right) + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\alpha + 1\right) \cdot \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \alpha\right) \cdot \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \alpha\right), \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\alpha + 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   3. Simplified 94.8%

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

   5. Taylor expanded in beta around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \alpha\right), \color{blue}{\left({\left(2 + \alpha\right)}^{2} \cdot \left(3 + \alpha\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left(\color{blue}{{\left(2 + \alpha\right)}^{2}} \cdot \left(3 + \alpha\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\left({\left(2 + \alpha\right)}^{2}\right), \color{blue}{\left(3 + \alpha\right)}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\left(\left(2 + \alpha\right) \cdot \left(2 + \alpha\right)\right), \left(\color{blue}{3} + \alpha\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 + \alpha\right), \left(2 + \alpha\right)\right), \left(\color{blue}{3} + \alpha\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \alpha\right), \left(2 + \alpha\right)\right), \left(3 + \alpha\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \alpha\right), \mathsf{+.f64}\left(2, \alpha\right)\right), \left(3 + \alpha\right)\right)\right) \]

      8. +-lowering-+.f64 92.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \alpha\right), \mathsf{+.f64}\left(2, \alpha\right)\right), \mathsf{+.f64}\left(3, \color{blue}{\alpha}\right)\right)\right) \]

   7. Simplified 92.4%

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

   8. Taylor expanded in alpha around 0

      \[\leadsto \color{blue}{\frac{1}{12} + \alpha \cdot \left(\frac{-5}{432} \cdot \alpha - \frac{1}{36}\right)} \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \color{blue}{\left(\alpha \cdot \left(\frac{-5}{432} \cdot \alpha - \frac{1}{36}\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\alpha, \color{blue}{\left(\frac{-5}{432} \cdot \alpha - \frac{1}{36}\right)}\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\alpha, \left(\frac{-5}{432} \cdot \alpha + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{36}\right)\right)}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\alpha, \left(\frac{-5}{432} \cdot \alpha + \frac{-1}{36}\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\alpha, \mathsf{+.f64}\left(\left(\frac{-5}{432} \cdot \alpha\right), \color{blue}{\frac{-1}{36}}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\alpha, \mathsf{+.f64}\left(\left(\alpha \cdot \frac{-5}{432}\right), \frac{-1}{36}\right)\right)\right) \]

      7. *-lowering-*.f64 65.9%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\alpha, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\alpha, \frac{-5}{432}\right), \frac{-1}{36}\right)\right)\right) \]

   10. Simplified 65.9%

       \[\leadsto \color{blue}{0.08333333333333333 + \alpha \cdot \left(\alpha \cdot -0.011574074074074073 + -0.027777777777777776\right)} \]

   if 2.4500000000000002 < beta

   1. Initial program 83.2%

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

      1. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\alpha}, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 + \left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      4. *-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 + \left(\alpha + \left(\beta \cdot 1 + \beta \cdot \alpha\right)\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      5. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 + \left(\alpha + \beta \cdot \left(1 + \alpha\right)\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 + \left(\alpha + \beta \cdot \left(\alpha + 1\right)\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\left(1 + \alpha\right) + \beta \cdot \left(\alpha + 1\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\alpha}, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\left(\alpha + 1\right) + \beta \cdot \left(\alpha + 1\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      9. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 \cdot \left(\alpha + 1\right) + \beta \cdot \left(\alpha + 1\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      10. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\left(\alpha + 1\right) \cdot \left(1 + \beta\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\alpha}, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      12. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\left(\alpha + \beta\right) + 2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(\alpha, \beta\right)}, \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\alpha + 1\right), \left(\frac{\beta + 1}{\left(\alpha + \beta\right) + 2}\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(\alpha, \beta\right)}, \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\frac{\beta + 1}{\left(\alpha + \beta\right) + 2}\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\alpha}, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{/.f64}\left(\left(\beta + 1\right), \left(\left(\alpha + \beta\right) + 2\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \color{blue}{\beta}\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{/.f64}\left(\left(1 + \beta\right), \left(\left(\alpha + \beta\right) + 2\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      17. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \left(\left(\alpha + \beta\right) + 2\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      18. associate-+l+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \left(\alpha + \left(\beta + 2\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      19. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(\alpha, \left(\beta + 2\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      20. +-lowering-+.f64 99.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 2\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

   4. Applied egg-rr 99.6%

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

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(\alpha + 1\right) \cdot \frac{1 + \beta}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\right), \color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}\right) \]

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in beta around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1 + \alpha}{\beta}\right)}, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + \alpha\right), \beta\right), \mathsf{+.f64}\left(\color{blue}{2}, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right) \]

      2. +-lowering-+.f64 73.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \beta\right), \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right) \]

   9. Simplified 73.2%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.45:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot \left(-0.027777777777777776 + \alpha \cdot -0.011574074074074073\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{2 + \left(\beta + \alpha\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 94.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.75:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot -0.027777777777777776\\ \mathbf{elif}\;\beta \leq 1.15 \cdot 10^{+159}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.75)
   (+ 0.08333333333333333 (* alpha -0.027777777777777776))
   (if (<= beta 1.15e+159) (/ (/ 1.0 beta) beta) (/ (/ alpha beta) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.75) {
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776);
	} else if (beta <= 1.15e+159) {
		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 <= 2.75d0) then
        tmp = 0.08333333333333333d0 + (alpha * (-0.027777777777777776d0))
    else if (beta <= 1.15d+159) 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 <= 2.75) {
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776);
	} else if (beta <= 1.15e+159) {
		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 <= 2.75:
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776)
	elif beta <= 1.15e+159:
		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 <= 2.75)
		tmp = Float64(0.08333333333333333 + Float64(alpha * -0.027777777777777776));
	elseif (beta <= 1.15e+159)
		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 <= 2.75)
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776);
	elseif (beta <= 1.15e+159)
		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, 2.75], N[(0.08333333333333333 + N[(alpha * -0.027777777777777776), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 1.15e+159], 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 < 2.75

   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/ N/A

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

      2. associate-/l/ N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right) + 1\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      5. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\left(\alpha + \beta\right) + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      6. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\alpha + \left(\beta + 1\right)\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \alpha\right) + \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      8. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta + 1\right) \cdot \alpha + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\alpha \cdot \left(\beta + 1\right) + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\alpha + 1\right) \cdot \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \alpha\right) \cdot \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \alpha\right), \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\alpha + 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   3. Simplified 94.8%

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

   5. Taylor expanded in beta around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \alpha\right), \color{blue}{\left({\left(2 + \alpha\right)}^{2} \cdot \left(3 + \alpha\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left(\color{blue}{{\left(2 + \alpha\right)}^{2}} \cdot \left(3 + \alpha\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\left({\left(2 + \alpha\right)}^{2}\right), \color{blue}{\left(3 + \alpha\right)}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\left(\left(2 + \alpha\right) \cdot \left(2 + \alpha\right)\right), \left(\color{blue}{3} + \alpha\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 + \alpha\right), \left(2 + \alpha\right)\right), \left(\color{blue}{3} + \alpha\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \alpha\right), \left(2 + \alpha\right)\right), \left(3 + \alpha\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \alpha\right), \mathsf{+.f64}\left(2, \alpha\right)\right), \left(3 + \alpha\right)\right)\right) \]

      8. +-lowering-+.f64 92.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \alpha\right), \mathsf{+.f64}\left(2, \alpha\right)\right), \mathsf{+.f64}\left(3, \color{blue}{\alpha}\right)\right)\right) \]

   7. Simplified 92.4%

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

   8. Taylor expanded in alpha around 0

      \[\leadsto \color{blue}{\frac{1}{12} + \frac{-1}{36} \cdot \alpha} \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \color{blue}{\left(\frac{-1}{36} \cdot \alpha\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \left(\alpha \cdot \color{blue}{\frac{-1}{36}}\right)\right) \]

      3. *-lowering-*.f64 65.8%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\alpha, \color{blue}{\frac{-1}{36}}\right)\right) \]

   10. Simplified 65.8%

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

   if 2.75 < beta < 1.14999999999999998e159

   1. Initial program 87.6%

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

      1. associate-/l/ N/A

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

      2. associate-/l/ N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right) + 1\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      5. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\left(\alpha + \beta\right) + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      6. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\alpha + \left(\beta + 1\right)\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \alpha\right) + \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      8. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta + 1\right) \cdot \alpha + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\alpha \cdot \left(\beta + 1\right) + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\alpha + 1\right) \cdot \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \alpha\right) \cdot \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \alpha\right), \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\alpha + 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   3. Simplified 56.3%

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

   5. Taylor expanded in beta around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \alpha\right), \color{blue}{\left({\beta}^{2}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left({\color{blue}{\beta}}^{2}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left(\beta \cdot \color{blue}{\beta}\right)\right) \]

      4. *-lowering-*.f64 55.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\beta, \color{blue}{\beta}\right)\right) \]

   7. Simplified 55.0%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 + \alpha}{\beta}\right), \color{blue}{\beta}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + \alpha\right), \beta\right), \beta\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\alpha + 1\right), \beta\right), \beta\right) \]

      5. +-lowering-+.f64 58.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \beta\right), \beta\right) \]

   9. Applied egg-rr 58.1%

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

   10. Taylor expanded in alpha around 0

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1}{\beta}\right)}, \beta\right) \]
   11. Step-by-step derivation

       1. /-lowering-/.f64 48.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \beta\right), \beta\right) \]

   12. Simplified 48.2%

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

   if 1.14999999999999998e159 < beta

   1. Initial program 76.9%

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

      1. associate-/l/ N/A

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

      2. associate-/l/ N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right) + 1\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      5. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\left(\alpha + \beta\right) + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      6. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\alpha + \left(\beta + 1\right)\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \alpha\right) + \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      8. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta + 1\right) \cdot \alpha + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\alpha \cdot \left(\beta + 1\right) + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\alpha + 1\right) \cdot \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \alpha\right) \cdot \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \alpha\right), \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\alpha + 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   3. Simplified 76.5%

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

   5. Taylor expanded in beta around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \alpha\right), \color{blue}{\left({\beta}^{2}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left({\color{blue}{\beta}}^{2}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left(\beta \cdot \color{blue}{\beta}\right)\right) \]

      4. *-lowering-*.f64 89.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\beta, \color{blue}{\beta}\right)\right) \]

   7. Simplified 89.2%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 + \alpha}{\beta}\right), \color{blue}{\beta}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + \alpha\right), \beta\right), \beta\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\alpha + 1\right), \beta\right), \beta\right) \]

      5. +-lowering-+.f64 94.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \beta\right), \beta\right) \]

   9. Applied egg-rr 94.5%

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

   10. Taylor expanded in alpha around inf

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{\alpha}{\beta}\right)}, \beta\right) \]
   11. Step-by-step derivation

       1. /-lowering-/.f64 94.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\alpha, \beta\right), \beta\right) \]

   12. Simplified 94.5%

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

Alternative 13: 97.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot \left(-0.027777777777777776 + \alpha \cdot -0.011574074074074073\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.0)
   (+
    0.08333333333333333
    (* alpha (+ -0.027777777777777776 (* alpha -0.011574074074074073))))
   (/ (/ (+ 1.0 alpha) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.0) {
		tmp = 0.08333333333333333 + (alpha * (-0.027777777777777776 + (alpha * -0.011574074074074073)));
	} else {
		tmp = ((1.0 + 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 <= 3.0d0) then
        tmp = 0.08333333333333333d0 + (alpha * ((-0.027777777777777776d0) + (alpha * (-0.011574074074074073d0))))
    else
        tmp = ((1.0d0 + 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 <= 3.0) {
		tmp = 0.08333333333333333 + (alpha * (-0.027777777777777776 + (alpha * -0.011574074074074073)));
	} else {
		tmp = ((1.0 + alpha) / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 3.0:
		tmp = 0.08333333333333333 + (alpha * (-0.027777777777777776 + (alpha * -0.011574074074074073)))
	else:
		tmp = ((1.0 + alpha) / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.0)
		tmp = Float64(0.08333333333333333 + Float64(alpha * Float64(-0.027777777777777776 + Float64(alpha * -0.011574074074074073))));
	else
		tmp = Float64(Float64(Float64(1.0 + 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 <= 3.0)
		tmp = 0.08333333333333333 + (alpha * (-0.027777777777777776 + (alpha * -0.011574074074074073)));
	else
		tmp = ((1.0 + 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, 3.0], N[(0.08333333333333333 + N[(alpha * N[(-0.027777777777777776 + N[(alpha * -0.011574074074074073), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3

   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/ N/A

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

      2. associate-/l/ N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right) + 1\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      5. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\left(\alpha + \beta\right) + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      6. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\alpha + \left(\beta + 1\right)\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \alpha\right) + \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      8. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta + 1\right) \cdot \alpha + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\alpha \cdot \left(\beta + 1\right) + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\alpha + 1\right) \cdot \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \alpha\right) \cdot \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \alpha\right), \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\alpha + 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   3. Simplified 94.8%

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

   5. Taylor expanded in beta around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \alpha\right), \color{blue}{\left({\left(2 + \alpha\right)}^{2} \cdot \left(3 + \alpha\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left(\color{blue}{{\left(2 + \alpha\right)}^{2}} \cdot \left(3 + \alpha\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\left({\left(2 + \alpha\right)}^{2}\right), \color{blue}{\left(3 + \alpha\right)}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\left(\left(2 + \alpha\right) \cdot \left(2 + \alpha\right)\right), \left(\color{blue}{3} + \alpha\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 + \alpha\right), \left(2 + \alpha\right)\right), \left(\color{blue}{3} + \alpha\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \alpha\right), \left(2 + \alpha\right)\right), \left(3 + \alpha\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \alpha\right), \mathsf{+.f64}\left(2, \alpha\right)\right), \left(3 + \alpha\right)\right)\right) \]

      8. +-lowering-+.f64 92.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \alpha\right), \mathsf{+.f64}\left(2, \alpha\right)\right), \mathsf{+.f64}\left(3, \color{blue}{\alpha}\right)\right)\right) \]

   7. Simplified 92.4%

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

   8. Taylor expanded in alpha around 0

      \[\leadsto \color{blue}{\frac{1}{12} + \alpha \cdot \left(\frac{-5}{432} \cdot \alpha - \frac{1}{36}\right)} \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \color{blue}{\left(\alpha \cdot \left(\frac{-5}{432} \cdot \alpha - \frac{1}{36}\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\alpha, \color{blue}{\left(\frac{-5}{432} \cdot \alpha - \frac{1}{36}\right)}\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\alpha, \left(\frac{-5}{432} \cdot \alpha + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{36}\right)\right)}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\alpha, \left(\frac{-5}{432} \cdot \alpha + \frac{-1}{36}\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\alpha, \mathsf{+.f64}\left(\left(\frac{-5}{432} \cdot \alpha\right), \color{blue}{\frac{-1}{36}}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\alpha, \mathsf{+.f64}\left(\left(\alpha \cdot \frac{-5}{432}\right), \frac{-1}{36}\right)\right)\right) \]

      7. *-lowering-*.f64 65.9%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\alpha, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\alpha, \frac{-5}{432}\right), \frac{-1}{36}\right)\right)\right) \]

   10. Simplified 65.9%

       \[\leadsto \color{blue}{0.08333333333333333 + \alpha \cdot \left(\alpha \cdot -0.011574074074074073 + -0.027777777777777776\right)} \]

   if 3 < beta

   1. Initial program 83.2%

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

      1. associate-/l/ N/A

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

      2. associate-/l/ N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right) + 1\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      5. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\left(\alpha + \beta\right) + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      6. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\alpha + \left(\beta + 1\right)\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \alpha\right) + \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      8. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta + 1\right) \cdot \alpha + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\alpha \cdot \left(\beta + 1\right) + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\alpha + 1\right) \cdot \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \alpha\right) \cdot \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \alpha\right), \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\alpha + 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   3. Simplified 64.5%

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

   5. Taylor expanded in beta around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \alpha\right), \color{blue}{\left({\beta}^{2}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left({\color{blue}{\beta}}^{2}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left(\beta \cdot \color{blue}{\beta}\right)\right) \]

      4. *-lowering-*.f64 68.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\beta, \color{blue}{\beta}\right)\right) \]

   7. Simplified 68.8%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 + \alpha}{\beta}\right), \color{blue}{\beta}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + \alpha\right), \beta\right), \beta\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\alpha + 1\right), \beta\right), \beta\right) \]

      5. +-lowering-+.f64 72.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \beta\right), \beta\right) \]

   9. Applied egg-rr 72.8%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot \left(-0.027777777777777776 + \alpha \cdot -0.011574074074074073\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 97.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.2:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.2)
   (+ 0.08333333333333333 (* alpha -0.027777777777777776))
   (/ (/ (+ 1.0 alpha) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.2) {
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776);
	} else {
		tmp = ((1.0 + 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 <= 3.2d0) then
        tmp = 0.08333333333333333d0 + (alpha * (-0.027777777777777776d0))
    else
        tmp = ((1.0d0 + 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 <= 3.2) {
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776);
	} else {
		tmp = ((1.0 + alpha) / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 3.2:
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776)
	else:
		tmp = ((1.0 + alpha) / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.2)
		tmp = Float64(0.08333333333333333 + Float64(alpha * -0.027777777777777776));
	else
		tmp = Float64(Float64(Float64(1.0 + 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 <= 3.2)
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776);
	else
		tmp = ((1.0 + 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, 3.2], N[(0.08333333333333333 + N[(alpha * -0.027777777777777776), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / 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/ N/A

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

      2. associate-/l/ N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right) + 1\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      5. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\left(\alpha + \beta\right) + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      6. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\alpha + \left(\beta + 1\right)\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \alpha\right) + \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      8. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta + 1\right) \cdot \alpha + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\alpha \cdot \left(\beta + 1\right) + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\alpha + 1\right) \cdot \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \alpha\right) \cdot \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \alpha\right), \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\alpha + 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   3. Simplified 94.8%

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

   5. Taylor expanded in beta around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \alpha\right), \color{blue}{\left({\left(2 + \alpha\right)}^{2} \cdot \left(3 + \alpha\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left(\color{blue}{{\left(2 + \alpha\right)}^{2}} \cdot \left(3 + \alpha\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\left({\left(2 + \alpha\right)}^{2}\right), \color{blue}{\left(3 + \alpha\right)}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\left(\left(2 + \alpha\right) \cdot \left(2 + \alpha\right)\right), \left(\color{blue}{3} + \alpha\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 + \alpha\right), \left(2 + \alpha\right)\right), \left(\color{blue}{3} + \alpha\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \alpha\right), \left(2 + \alpha\right)\right), \left(3 + \alpha\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \alpha\right), \mathsf{+.f64}\left(2, \alpha\right)\right), \left(3 + \alpha\right)\right)\right) \]

      8. +-lowering-+.f64 92.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \alpha\right), \mathsf{+.f64}\left(2, \alpha\right)\right), \mathsf{+.f64}\left(3, \color{blue}{\alpha}\right)\right)\right) \]

   7. Simplified 92.4%

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

   8. Taylor expanded in alpha around 0

      \[\leadsto \color{blue}{\frac{1}{12} + \frac{-1}{36} \cdot \alpha} \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \color{blue}{\left(\frac{-1}{36} \cdot \alpha\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \left(\alpha \cdot \color{blue}{\frac{-1}{36}}\right)\right) \]

      3. *-lowering-*.f64 65.8%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\alpha, \color{blue}{\frac{-1}{36}}\right)\right) \]

   10. Simplified 65.8%

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

   if 3.2000000000000002 < beta

   1. Initial program 83.2%

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

      1. associate-/l/ N/A

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

      2. associate-/l/ N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right) + 1\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      5. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\left(\alpha + \beta\right) + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      6. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\alpha + \left(\beta + 1\right)\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \alpha\right) + \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      8. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta + 1\right) \cdot \alpha + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\alpha \cdot \left(\beta + 1\right) + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\alpha + 1\right) \cdot \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \alpha\right) \cdot \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \alpha\right), \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\alpha + 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   3. Simplified 64.5%

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

   5. Taylor expanded in beta around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \alpha\right), \color{blue}{\left({\beta}^{2}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left({\color{blue}{\beta}}^{2}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left(\beta \cdot \color{blue}{\beta}\right)\right) \]

      4. *-lowering-*.f64 68.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\beta, \color{blue}{\beta}\right)\right) \]

   7. Simplified 68.8%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 + \alpha}{\beta}\right), \color{blue}{\beta}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + \alpha\right), \beta\right), \beta\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\alpha + 1\right), \beta\right), \beta\right) \]

      5. +-lowering-+.f64 72.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \beta\right), \beta\right) \]

   9. Applied egg-rr 72.8%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.2:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 91.6% accurate, 3.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.1:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.1)
   (+ 0.08333333333333333 (* alpha -0.027777777777777776))
   (/ (/ 1.0 beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.1) {
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776);
	} 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 <= 3.1d0) then
        tmp = 0.08333333333333333d0 + (alpha * (-0.027777777777777776d0))
    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 <= 3.1) {
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776);
	} else {
		tmp = (1.0 / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 3.1:
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776)
	else:
		tmp = (1.0 / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.1)
		tmp = Float64(0.08333333333333333 + Float64(alpha * -0.027777777777777776));
	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 <= 3.1)
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776);
	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, 3.1], N[(0.08333333333333333 + N[(alpha * -0.027777777777777776), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.10000000000000009

   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/ N/A

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

      2. associate-/l/ N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right) + 1\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      5. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\left(\alpha + \beta\right) + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      6. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\alpha + \left(\beta + 1\right)\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \alpha\right) + \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      8. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta + 1\right) \cdot \alpha + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\alpha \cdot \left(\beta + 1\right) + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\alpha + 1\right) \cdot \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \alpha\right) \cdot \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \alpha\right), \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\alpha + 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   3. Simplified 94.8%

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

   5. Taylor expanded in beta around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \alpha\right), \color{blue}{\left({\left(2 + \alpha\right)}^{2} \cdot \left(3 + \alpha\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left(\color{blue}{{\left(2 + \alpha\right)}^{2}} \cdot \left(3 + \alpha\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\left({\left(2 + \alpha\right)}^{2}\right), \color{blue}{\left(3 + \alpha\right)}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\left(\left(2 + \alpha\right) \cdot \left(2 + \alpha\right)\right), \left(\color{blue}{3} + \alpha\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 + \alpha\right), \left(2 + \alpha\right)\right), \left(\color{blue}{3} + \alpha\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \alpha\right), \left(2 + \alpha\right)\right), \left(3 + \alpha\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \alpha\right), \mathsf{+.f64}\left(2, \alpha\right)\right), \left(3 + \alpha\right)\right)\right) \]

      8. +-lowering-+.f64 92.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \alpha\right), \mathsf{+.f64}\left(2, \alpha\right)\right), \mathsf{+.f64}\left(3, \color{blue}{\alpha}\right)\right)\right) \]

   7. Simplified 92.4%

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

   8. Taylor expanded in alpha around 0

      \[\leadsto \color{blue}{\frac{1}{12} + \frac{-1}{36} \cdot \alpha} \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \color{blue}{\left(\frac{-1}{36} \cdot \alpha\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \left(\alpha \cdot \color{blue}{\frac{-1}{36}}\right)\right) \]

      3. *-lowering-*.f64 65.8%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\alpha, \color{blue}{\frac{-1}{36}}\right)\right) \]

   10. Simplified 65.8%

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

   if 3.10000000000000009 < beta

   1. Initial program 83.2%

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

      1. associate-/l/ N/A

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

      2. associate-/l/ N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right) + 1\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      5. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\left(\alpha + \beta\right) + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      6. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\alpha + \left(\beta + 1\right)\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \alpha\right) + \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      8. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta + 1\right) \cdot \alpha + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\alpha \cdot \left(\beta + 1\right) + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\alpha + 1\right) \cdot \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \alpha\right) \cdot \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \alpha\right), \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\alpha + 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   3. Simplified 64.5%

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

   5. Taylor expanded in beta around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \alpha\right), \color{blue}{\left({\beta}^{2}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left({\color{blue}{\beta}}^{2}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left(\beta \cdot \color{blue}{\beta}\right)\right) \]

      4. *-lowering-*.f64 68.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\beta, \color{blue}{\beta}\right)\right) \]

   7. Simplified 68.8%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 + \alpha}{\beta}\right), \color{blue}{\beta}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + \alpha\right), \beta\right), \beta\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\alpha + 1\right), \beta\right), \beta\right) \]

      5. +-lowering-+.f64 72.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \beta\right), \beta\right) \]

   9. Applied egg-rr 72.8%

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

   10. Taylor expanded in alpha around 0

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1}{\beta}\right)}, \beta\right) \]
   11. Step-by-step derivation

       1. /-lowering-/.f64 64.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \beta\right), \beta\right) \]

   12. Simplified 64.8%

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

Alternative 16: 91.2% accurate, 3.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.4:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.4)
   (+ 0.08333333333333333 (* alpha -0.027777777777777776))
   (/ 1.0 (* beta beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.4) {
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776);
	} 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 <= 2.4d0) then
        tmp = 0.08333333333333333d0 + (alpha * (-0.027777777777777776d0))
    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 <= 2.4) {
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776);
	} else {
		tmp = 1.0 / (beta * beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.4:
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776)
	else:
		tmp = 1.0 / (beta * beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.4)
		tmp = Float64(0.08333333333333333 + Float64(alpha * -0.027777777777777776));
	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 <= 2.4)
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776);
	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, 2.4], N[(0.08333333333333333 + N[(alpha * -0.027777777777777776), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.39999999999999991

   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/ N/A

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

      2. associate-/l/ N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right) + 1\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      5. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\left(\alpha + \beta\right) + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      6. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\alpha + \left(\beta + 1\right)\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \alpha\right) + \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      8. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta + 1\right) \cdot \alpha + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\alpha \cdot \left(\beta + 1\right) + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\alpha + 1\right) \cdot \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \alpha\right) \cdot \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \alpha\right), \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\alpha + 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   3. Simplified 94.8%

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

   5. Taylor expanded in beta around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \alpha\right), \color{blue}{\left({\left(2 + \alpha\right)}^{2} \cdot \left(3 + \alpha\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left(\color{blue}{{\left(2 + \alpha\right)}^{2}} \cdot \left(3 + \alpha\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\left({\left(2 + \alpha\right)}^{2}\right), \color{blue}{\left(3 + \alpha\right)}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\left(\left(2 + \alpha\right) \cdot \left(2 + \alpha\right)\right), \left(\color{blue}{3} + \alpha\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 + \alpha\right), \left(2 + \alpha\right)\right), \left(\color{blue}{3} + \alpha\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \alpha\right), \left(2 + \alpha\right)\right), \left(3 + \alpha\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \alpha\right), \mathsf{+.f64}\left(2, \alpha\right)\right), \left(3 + \alpha\right)\right)\right) \]

      8. +-lowering-+.f64 92.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \alpha\right), \mathsf{+.f64}\left(2, \alpha\right)\right), \mathsf{+.f64}\left(3, \color{blue}{\alpha}\right)\right)\right) \]

   7. Simplified 92.4%

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

   8. Taylor expanded in alpha around 0

      \[\leadsto \color{blue}{\frac{1}{12} + \frac{-1}{36} \cdot \alpha} \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \color{blue}{\left(\frac{-1}{36} \cdot \alpha\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \left(\alpha \cdot \color{blue}{\frac{-1}{36}}\right)\right) \]

      3. *-lowering-*.f64 65.8%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\alpha, \color{blue}{\frac{-1}{36}}\right)\right) \]

   10. Simplified 65.8%

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

   if 2.39999999999999991 < beta

   1. Initial program 83.2%

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

      1. associate-/l/ N/A

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

      2. associate-/l/ N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right) + 1\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      5. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\left(\alpha + \beta\right) + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      6. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\alpha + \left(\beta + 1\right)\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \alpha\right) + \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      8. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta + 1\right) \cdot \alpha + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\alpha \cdot \left(\beta + 1\right) + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\alpha + 1\right) \cdot \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \alpha\right) \cdot \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \alpha\right), \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\alpha + 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   3. Simplified 64.5%

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

   5. Taylor expanded in beta around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \alpha\right), \color{blue}{\left({\beta}^{2}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left({\color{blue}{\beta}}^{2}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left(\beta \cdot \color{blue}{\beta}\right)\right) \]

      4. *-lowering-*.f64 68.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\beta, \color{blue}{\beta}\right)\right) \]

   7. Simplified 68.8%

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

   8. Taylor expanded in alpha around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left({\beta}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\beta \cdot \color{blue}{\beta}\right)\right) \]

      3. *-lowering-*.f64 62.9%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\beta, \color{blue}{\beta}\right)\right) \]

   10. Simplified 62.9%

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

Alternative 17: 45.3% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.4%

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

   1. associate-/l/ N/A

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

   2. associate-/l/ N/A

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

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)}\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right) + 1\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   5. associate-+l+ N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\left(\alpha + \beta\right) + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   6. associate-+l+ N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\alpha + \left(\beta + 1\right)\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   7. associate-+r+ N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \alpha\right) + \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   8. distribute-lft1-in N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta + 1\right) \cdot \alpha + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\alpha \cdot \left(\beta + 1\right) + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   10. distribute-lft1-in N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\left(\alpha + 1\right) \cdot \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   11. +-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \alpha\right) \cdot \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \alpha\right), \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   13. +-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\alpha + 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   14. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   15. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

3. Simplified 84.8%

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

5. Taylor expanded in beta around 0

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(1 + \alpha\right), \color{blue}{\left({\left(2 + \alpha\right)}^{2} \cdot \left(3 + \alpha\right)\right)}\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left(\color{blue}{{\left(2 + \alpha\right)}^{2}} \cdot \left(3 + \alpha\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\left({\left(2 + \alpha\right)}^{2}\right), \color{blue}{\left(3 + \alpha\right)}\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\left(\left(2 + \alpha\right) \cdot \left(2 + \alpha\right)\right), \left(\color{blue}{3} + \alpha\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 + \alpha\right), \left(2 + \alpha\right)\right), \left(\color{blue}{3} + \alpha\right)\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \alpha\right), \left(2 + \alpha\right)\right), \left(3 + \alpha\right)\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \alpha\right), \mathsf{+.f64}\left(2, \alpha\right)\right), \left(3 + \alpha\right)\right)\right) \]

   8. +-lowering-+.f64 67.9%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \alpha\right), \mathsf{+.f64}\left(2, \alpha\right)\right), \mathsf{+.f64}\left(3, \color{blue}{\alpha}\right)\right)\right) \]

7. Simplified 67.9%

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

8. Taylor expanded in alpha around 0

   \[\leadsto \color{blue}{\frac{1}{12} + \frac{-1}{36} \cdot \alpha} \]
9. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \color{blue}{\left(\frac{-1}{36} \cdot \alpha\right)}\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \left(\alpha \cdot \color{blue}{\frac{-1}{36}}\right)\right) \]

   3. *-lowering-*.f64 45.3%

      \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\alpha, \color{blue}{\frac{-1}{36}}\right)\right) \]

10. Simplified 45.3%

    \[\leadsto \color{blue}{0.08333333333333333 + \alpha \cdot -0.027777777777777776} \]
11. Add Preprocessing

Alternative 18: 45.0% accurate, 35.0× speedup

Math

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

FPCore

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

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return 0.08333333333333333;
}

Fortran

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

Java

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

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return 0.08333333333333333

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return 0.08333333333333333
end

MATLAB

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

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := 0.08333333333333333

Derivation

1. Initial program 94.4%

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

   1. associate-/l/ N/A

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

   2. associate-/l/ N/A

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

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)}\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right) + 1\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   5. associate-+l+ N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\left(\alpha + \beta\right) + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   6. associate-+l+ N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\alpha + \left(\beta + 1\right)\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   7. associate-+r+ N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \alpha\right) + \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   8. distribute-lft1-in N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta + 1\right) \cdot \alpha + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\alpha \cdot \left(\beta + 1\right) + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   10. distribute-lft1-in N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\left(\alpha + 1\right) \cdot \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   11. +-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \alpha\right) \cdot \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \alpha\right), \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   13. +-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\alpha + 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   14. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   15. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

3. Simplified 84.8%

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

5. Taylor expanded in alpha around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \color{blue}{\left({\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)\right)}\right) \]
6. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\left({\left(2 + \beta\right)}^{2}\right), \color{blue}{\left(3 + \beta\right)}\right)\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\left(\left(2 + \beta\right) \cdot \left(2 + \beta\right)\right), \left(\color{blue}{3} + \beta\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 + \beta\right), \left(2 + \beta\right)\right), \left(\color{blue}{3} + \beta\right)\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\beta + 2\right), \left(2 + \beta\right)\right), \left(3 + \beta\right)\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(2 + \beta\right)\right), \left(3 + \beta\right)\right)\right) \]

   6. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(\beta + 2\right)\right), \left(3 + \beta\right)\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(3 + \beta\right)\right)\right) \]

   8. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(\beta + \color{blue}{3}\right)\right)\right) \]

   9. +-lowering-+.f64 64.1%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\beta, \color{blue}{3}\right)\right)\right) \]

7. Simplified 64.1%

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

8. Taylor expanded in beta around 0

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

   1. *-commutative N/A

      \[\leadsto \left(1 + \alpha\right) \cdot \color{blue}{\frac{1}{12}} \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(1 + \alpha\right), \color{blue}{\frac{1}{12}}\right) \]

   3. +-lowering-+.f64 45.2%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \frac{1}{12}\right) \]

10. Simplified 45.2%

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

11. Taylor expanded in alpha around 0

    \[\leadsto \color{blue}{\frac{1}{12}} \]
12. Step-by-step derivation

13. Simplified 45.6%

    \[\leadsto \color{blue}{0.08333333333333333} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024154 
(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)))