Octave 3.8, jcobi/3

Percentage Accurate: 94.5% → 99.8%

Time: 13.8s

Alternatives: 20

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: 94.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.1%

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

   1. +-commutative N/A

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

   2. associate-+l+ N/A

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

   3. *-rgt-identity N/A

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

   4. distribute-lft-in N/A

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

   5. +-commutative N/A

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

   6. associate-+r+ N/A

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

   7. +-commutative N/A

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

   8. *-lft-identity N/A

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

   9. distribute-rgt-in N/A

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

   10. +-commutative N/A

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

   11. associate-/l* N/A

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

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

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

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

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

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

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

   15. +-commutative N/A

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

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

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

   17. metadata-eval 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. /-lowering-/.f64 N/A

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

6. Applied egg-rr 99.9%

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

   1. associate-+r+ N/A

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

   2. metadata-eval N/A

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

   3. associate-+l+ N/A

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

   4. metadata-eval N/A

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

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

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

8. Applied egg-rr 99.9%

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

Alternative 2: 99.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\alpha + \beta\right)\\ t_1 := \alpha + \left(\beta + 3\right)\\ \mathbf{if}\;\beta \leq 2.6 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\beta + 2\right)}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(1 + \alpha\right) \cdot \frac{\beta}{t\_0}}{t\_0}}{t\_1}\\ \end{array} \end{array} \]

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 (+ alpha beta))) (t_1 (+ alpha (+ beta 3.0))))
   (if (<= beta 2.6e+16)
     (/ (/ (+ 1.0 beta) (* (+ beta 2.0) (+ beta 2.0))) t_1)
     (/ (/ (* (+ 1.0 alpha) (/ beta t_0)) t_0) t_1))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (alpha + beta);
	double t_1 = alpha + (beta + 3.0);
	double tmp;
	if (beta <= 2.6e+16) {
		tmp = ((1.0 + beta) / ((beta + 2.0) * (beta + 2.0))) / t_1;
	} else {
		tmp = (((1.0 + alpha) * (beta / t_0)) / t_0) / t_1;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 2.0d0 + (alpha + beta)
    t_1 = alpha + (beta + 3.0d0)
    if (beta <= 2.6d+16) then
        tmp = ((1.0d0 + beta) / ((beta + 2.0d0) * (beta + 2.0d0))) / t_1
    else
        tmp = (((1.0d0 + alpha) * (beta / t_0)) / t_0) / t_1
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = 2.0 + (alpha + beta);
	double t_1 = alpha + (beta + 3.0);
	double tmp;
	if (beta <= 2.6e+16) {
		tmp = ((1.0 + beta) / ((beta + 2.0) * (beta + 2.0))) / t_1;
	} else {
		tmp = (((1.0 + alpha) * (beta / t_0)) / t_0) / t_1;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = 2.0 + (alpha + beta)
	t_1 = alpha + (beta + 3.0)
	tmp = 0
	if beta <= 2.6e+16:
		tmp = ((1.0 + beta) / ((beta + 2.0) * (beta + 2.0))) / t_1
	else:
		tmp = (((1.0 + alpha) * (beta / t_0)) / t_0) / t_1
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(alpha + beta))
	t_1 = Float64(alpha + Float64(beta + 3.0))
	tmp = 0.0
	if (beta <= 2.6e+16)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(Float64(beta + 2.0) * Float64(beta + 2.0))) / t_1);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + alpha) * Float64(beta / t_0)) / t_0) / t_1);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = 2.0 + (alpha + beta);
	t_1 = alpha + (beta + 3.0);
	tmp = 0.0;
	if (beta <= 2.6e+16)
		tmp = ((1.0 + beta) / ((beta + 2.0) * (beta + 2.0))) / t_1;
	else
		tmp = (((1.0 + alpha) * (beta / t_0)) / t_0) / t_1;
	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[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2.6e+16], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(N[(N[(1.0 + alpha), $MachinePrecision] * N[(beta / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.6e16

   1. Initial program 99.9%

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. *-rgt-identity N/A

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

      4. distribute-lft-in N/A

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

      5. +-commutative N/A

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

      6. associate-+r+ N/A

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

      7. +-commutative N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

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

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

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

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

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

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

      15. +-commutative N/A

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

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

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

      17. metadata-eval 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 99.9%

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

   4. Applied egg-rr 99.9%

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

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

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in alpha around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

      6. +-lowering-+.f64 71.2%

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

   9. Simplified 71.2%

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

   if 2.6e16 < beta

   1. Initial program 81.6%

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. *-rgt-identity N/A

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

      4. distribute-lft-in N/A

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

      5. +-commutative N/A

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

      6. associate-+r+ N/A

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

      7. +-commutative N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

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

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

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

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

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

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

      15. +-commutative N/A

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

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

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

      17. metadata-eval 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. /-lowering-/.f64 N/A

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in beta around inf

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

   9. Simplified 99.8%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.6 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\beta + 2\right)}}{\alpha + \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(1 + \alpha\right) \cdot \frac{\beta}{2 + \left(\alpha + \beta\right)}}{2 + \left(\alpha + \beta\right)}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	t_0 = 2.0 + (alpha + beta);
	tmp = (((1.0 + alpha) * ((1.0 + beta) / t_0)) / t_0) / (alpha + (beta + 3.0));
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[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(1.0 + alpha), $MachinePrecision] * N[(N[(1.0 + beta), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 94.1%

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

   1. +-commutative N/A

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

   2. associate-+l+ N/A

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

   3. *-rgt-identity N/A

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

   4. distribute-lft-in N/A

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

   5. +-commutative N/A

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

   6. associate-+r+ N/A

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

   7. +-commutative N/A

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

   8. *-lft-identity N/A

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

   9. distribute-rgt-in N/A

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

   10. +-commutative N/A

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

   11. associate-/l* N/A

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

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

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

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

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

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

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

   15. +-commutative N/A

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

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

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

   17. metadata-eval 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. /-lowering-/.f64 N/A

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

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

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

Alternative 4: 99.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.1%

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

   1. +-commutative N/A

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

   2. associate-+l+ N/A

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

   3. *-rgt-identity N/A

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

   4. distribute-lft-in N/A

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

   5. +-commutative N/A

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

   6. associate-+r+ N/A

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

   7. +-commutative N/A

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

   8. *-lft-identity N/A

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

   9. distribute-rgt-in N/A

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

   10. +-commutative N/A

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

   11. associate-/l* N/A

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

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

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

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

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

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

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

   15. +-commutative N/A

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

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

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

   17. metadata-eval 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. /-lowering-/.f64 N/A

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

6. Applied egg-rr 99.9%

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

   1. associate-/l* N/A

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

   2. associate-+r+ N/A

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

   3. metadata-eval N/A

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

   4. associate-+l+ N/A

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

   5. metadata-eval N/A

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

   6. associate-/l* N/A

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

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

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

   8. /-lowering-/.f64 N/A

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

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

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

   10. +-commutative N/A

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

   11. associate-+l+ N/A

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

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

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

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

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

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

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

8. Applied egg-rr 99.8%

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

Alternative 5: 98.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.1%

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

   1. +-commutative N/A

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

   2. associate-+l+ N/A

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

   3. *-rgt-identity N/A

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

   4. distribute-lft-in N/A

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

   5. +-commutative N/A

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

   6. associate-+r+ N/A

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

   7. +-commutative N/A

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

   8. *-lft-identity N/A

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

   9. distribute-rgt-in N/A

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

   10. +-commutative N/A

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

   11. associate-/l* N/A

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

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

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

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

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

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

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

   15. +-commutative N/A

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

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

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

   17. metadata-eval 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. /-lowering-/.f64 N/A

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

6. Applied egg-rr 99.9%

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

   1. associate-+r+ N/A

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

   2. metadata-eval N/A

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

   3. associate-+l+ N/A

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

   4. metadata-eval N/A

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

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

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

8. Applied egg-rr 99.9%

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

9. Taylor expanded in alpha around 0

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

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

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

    2. +-commutative N/A

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

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

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

    4. +-lowering-+.f64 73.0%

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

11. Simplified 73.0%

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

Alternative 6: 98.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 3\right)\\ \mathbf{if}\;\beta \leq 5 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\beta + 2\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{2 + \left(\alpha + \beta\right)}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 3.0))))
   (if (<= beta 5e+16)
     (/ (/ (+ 1.0 beta) (* (+ beta 2.0) (+ beta 2.0))) t_0)
     (/ (/ (+ 1.0 alpha) (+ 2.0 (+ alpha beta))) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 3.0);
	double tmp;
	if (beta <= 5e+16) {
		tmp = ((1.0 + beta) / ((beta + 2.0) * (beta + 2.0))) / t_0;
	} else {
		tmp = ((1.0 + alpha) / (2.0 + (alpha + beta))) / t_0;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 3.0d0)
    if (beta <= 5d+16) then
        tmp = ((1.0d0 + beta) / ((beta + 2.0d0) * (beta + 2.0d0))) / t_0
    else
        tmp = ((1.0d0 + alpha) / (2.0d0 + (alpha + beta))) / t_0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 3.0);
	double tmp;
	if (beta <= 5e+16) {
		tmp = ((1.0 + beta) / ((beta + 2.0) * (beta + 2.0))) / t_0;
	} else {
		tmp = ((1.0 + alpha) / (2.0 + (alpha + beta))) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 3.0)
	tmp = 0
	if beta <= 5e+16:
		tmp = ((1.0 + beta) / ((beta + 2.0) * (beta + 2.0))) / t_0
	else:
		tmp = ((1.0 + alpha) / (2.0 + (alpha + beta))) / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 3.0))
	tmp = 0.0
	if (beta <= 5e+16)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(Float64(beta + 2.0) * Float64(beta + 2.0))) / t_0);
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(2.0 + Float64(alpha + beta))) / t_0);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 3.0);
	tmp = 0.0;
	if (beta <= 5e+16)
		tmp = ((1.0 + beta) / ((beta + 2.0) * (beta + 2.0))) / t_0;
	else
		tmp = ((1.0 + alpha) / (2.0 + (alpha + beta))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 5e+16], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 5e16

   1. Initial program 99.9%

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. *-rgt-identity N/A

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

      4. distribute-lft-in N/A

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

      5. +-commutative N/A

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

      6. associate-+r+ N/A

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

      7. +-commutative N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

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

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

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

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

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

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

      15. +-commutative N/A

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

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

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

      17. metadata-eval 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 99.9%

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

   4. Applied egg-rr 99.9%

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

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

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in alpha around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

      6. +-lowering-+.f64 71.2%

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

   9. Simplified 71.2%

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

   if 5e16 < beta

   1. Initial program 81.6%

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. *-rgt-identity N/A

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

      4. distribute-lft-in N/A

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

      5. +-commutative N/A

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

      6. associate-+r+ N/A

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

      7. +-commutative N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

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

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

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

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

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

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

      15. +-commutative N/A

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

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

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

      17. metadata-eval 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. /-lowering-/.f64 N/A

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in beta around inf

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

      1. +-lowering-+.f64 78.8%

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

   9. Simplified 78.8%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\beta + 2\right)}}{\alpha + \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{2 + \left(\alpha + \beta\right)}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.6 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\beta + 2\right)}}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{2 + \left(\alpha + \beta\right)}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.6e+16)
   (/ (/ (+ 1.0 beta) (* (+ beta 2.0) (+ beta 2.0))) (+ beta 3.0))
   (/ (/ (+ 1.0 alpha) (+ 2.0 (+ alpha beta))) (+ alpha (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.6e+16) {
		tmp = ((1.0 + beta) / ((beta + 2.0) * (beta + 2.0))) / (beta + 3.0);
	} else {
		tmp = ((1.0 + alpha) / (2.0 + (alpha + beta))) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.6d+16) then
        tmp = ((1.0d0 + beta) / ((beta + 2.0d0) * (beta + 2.0d0))) / (beta + 3.0d0)
    else
        tmp = ((1.0d0 + alpha) / (2.0d0 + (alpha + beta))) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.6e+16) {
		tmp = ((1.0 + beta) / ((beta + 2.0) * (beta + 2.0))) / (beta + 3.0);
	} else {
		tmp = ((1.0 + alpha) / (2.0 + (alpha + beta))) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.6e+16:
		tmp = ((1.0 + beta) / ((beta + 2.0) * (beta + 2.0))) / (beta + 3.0)
	else:
		tmp = ((1.0 + alpha) / (2.0 + (alpha + beta))) / (alpha + (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.6e+16)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(Float64(beta + 2.0) * Float64(beta + 2.0))) / Float64(beta + 3.0));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(2.0 + Float64(alpha + beta))) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.6e+16)
		tmp = ((1.0 + beta) / ((beta + 2.0) * (beta + 2.0))) / (beta + 3.0);
	else
		tmp = ((1.0 + alpha) / (2.0 + (alpha + beta))) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.6e+16], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.6e16

   1. Initial program 99.9%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. 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 94.6%

      \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\beta + \left(\alpha + 2\right)\right) \cdot \left(\left(\beta + \left(\alpha + 2\right)\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)\right)}} \]
   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. associate-/r* N/A

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

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

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

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

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

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

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

      5. unpow2 N/A

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

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

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

      7. +-commutative N/A

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

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

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

      9. +-commutative N/A

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

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

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

      11. +-commutative N/A

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

      12. +-lowering-+.f64 69.6%

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

   7. Simplified 69.6%

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

   if 2.6e16 < beta

   1. Initial program 81.6%

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. *-rgt-identity N/A

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

      4. distribute-lft-in N/A

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

      5. +-commutative N/A

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

      6. associate-+r+ N/A

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

      7. +-commutative N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

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

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

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

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

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

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

      15. +-commutative N/A

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

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

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

      17. metadata-eval 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. /-lowering-/.f64 N/A

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in beta around inf

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

      1. +-lowering-+.f64 78.8%

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

   9. Simplified 78.8%

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

Alternative 8: 97.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\alpha + \beta\right)\\ \mathbf{if}\;\beta \leq 2:\\ \;\;\;\;\frac{0.25}{1 + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{t\_0}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

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

C

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

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 + (alpha + beta)
    if (beta <= 2.0d0) then
        tmp = 0.25d0 / (1.0d0 + t_0)
    else
        tmp = ((1.0d0 + alpha) / t_0) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = 2.0 + (alpha + beta);
	tmp = 0.0;
	if (beta <= 2.0)
		tmp = 0.25 / (1.0 + t_0);
	else
		tmp = ((1.0 + alpha) / t_0) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2.0], N[(0.25 / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2

   1. Initial program 99.9%

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

   3. Taylor expanded in beta around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

      6. +-lowering-+.f64 98.3%

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

   5. Simplified 98.3%

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

   6. Taylor expanded in alpha around 0

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

   8. Simplified 70.7%

      \[\leadsto \frac{\color{blue}{0.25}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   if 2 < beta

   1. Initial program 82.3%

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. *-rgt-identity N/A

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

      4. distribute-lft-in N/A

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

      5. +-commutative N/A

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

      6. associate-+r+ N/A

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

      7. +-commutative N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

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

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

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

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

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

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

      15. +-commutative N/A

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

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

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

      17. metadata-eval 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. /-lowering-/.f64 N/A

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in beta around inf

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

      1. +-lowering-+.f64 77.2%

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

   9. Simplified 77.2%

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

4. Final simplification 72.9%

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

Alternative 9: 96.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.75:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{elif}\;\beta \leq 2.65 \cdot 10^{+154}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.75)
   (+ 0.08333333333333333 (* beta -0.027777777777777776))
   (if (<= beta 2.65e+154)
     (/ (+ 1.0 alpha) (* beta beta))
     (/ (/ alpha beta) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.75) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else if (beta <= 2.65e+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 <= 2.75d0) then
        tmp = 0.08333333333333333d0 + (beta * (-0.027777777777777776d0))
    else if (beta <= 2.65d+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 <= 2.75) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else if (beta <= 2.65e+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 <= 2.75:
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776)
	elif beta <= 2.65e+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 <= 2.75)
		tmp = Float64(0.08333333333333333 + Float64(beta * -0.027777777777777776));
	elseif (beta <= 2.65e+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 <= 2.75)
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	elseif (beta <= 2.65e+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, 2.75], N[(0.08333333333333333 + N[(beta * -0.027777777777777776), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 2.65e+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 < 2.75

   1. Initial program 99.9%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. 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 95.1%

      \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\beta + \left(\alpha + 2\right)\right) \cdot \left(\left(\beta + \left(\alpha + 2\right)\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)\right)}} \]
   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. associate-/r* N/A

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

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

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

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

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

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

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

      5. unpow2 N/A

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

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

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

      7. +-commutative N/A

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

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

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

      9. +-commutative N/A

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

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

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

      11. +-commutative N/A

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

      12. +-lowering-+.f64 69.6%

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

   7. Simplified 69.6%

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

   8. Taylor expanded in beta around 0

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

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

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 69.1%

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

   10. Simplified 69.1%

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

   if 2.75 < beta < 2.65000000000000012e154

   1. Initial program 86.6%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 43.1%

      \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\beta + \left(\alpha + 2\right)\right) \cdot \left(\left(\beta + \left(\alpha + 2\right)\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)\right)}} \]
   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 67.3%

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

   7. Simplified 67.3%

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

   if 2.65000000000000012e154 < beta

   1. Initial program 77.3%

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

      1. associate-/l/ 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 69.6%

      \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\beta + \left(\alpha + 2\right)\right) \cdot \left(\left(\beta + \left(\alpha + 2\right)\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)\right)}} \]
   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 79.0%

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

   7. Simplified 79.0%

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

   8. Taylor expanded in alpha around inf

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 79.0%

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

   10. Simplified 79.0%

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

       1. associate-/r* N/A

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

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

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

       3. /-lowering-/.f64 86.2%

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

   12. Applied egg-rr 86.2%

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

Alternative 10: 97.5% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 4.2d0) then
        tmp = 0.25d0 / (1.0d0 + (2.0d0 + (alpha + beta)))
    else
        tmp = ((1.0d0 + alpha) / beta) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4.2)
		tmp = 0.25 / (1.0 + (2.0 + (alpha + beta)));
	else
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 4.2], N[(0.25 / N[(1.0 + N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.20000000000000018

   1. Initial program 99.9%

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

   3. Taylor expanded in beta around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

      6. +-lowering-+.f64 98.3%

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

   5. Simplified 98.3%

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

   6. Taylor expanded in alpha around 0

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

   8. Simplified 70.7%

      \[\leadsto \frac{\color{blue}{0.25}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   if 4.20000000000000018 < beta

   1. Initial program 82.3%

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. *-rgt-identity N/A

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

      4. distribute-lft-in N/A

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

      5. +-commutative N/A

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

      6. associate-+r+ N/A

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

      7. +-commutative N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

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

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

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

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

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

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

      15. +-commutative N/A

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

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

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

      17. metadata-eval 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. /-lowering-/.f64 N/A

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in beta around inf

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

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

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

      2. +-lowering-+.f64 76.5%

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

   9. Simplified 76.5%

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

4. Final simplification 72.6%

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

Alternative 11: 93.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.75:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{elif}\;\beta \leq 5.8 \cdot 10^{+150}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.75)
   (+ 0.08333333333333333 (* beta -0.027777777777777776))
   (if (<= beta 5.8e+150) (/ 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 + (beta * -0.027777777777777776);
	} else if (beta <= 5.8e+150) {
		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 + (beta * (-0.027777777777777776d0))
    else if (beta <= 5.8d+150) 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 + (beta * -0.027777777777777776);
	} else if (beta <= 5.8e+150) {
		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 + (beta * -0.027777777777777776)
	elif beta <= 5.8e+150:
		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(beta * -0.027777777777777776));
	elseif (beta <= 5.8e+150)
		tmp = Float64(1.0 / Float64(beta * beta));
	else
		tmp = Float64(Float64(alpha / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.75)
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	elseif (beta <= 5.8e+150)
		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[(beta * -0.027777777777777776), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 5.8e+150], N[(1.0 / N[(beta * beta), $MachinePrecision]), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / beta), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if beta < 2.75

   1. Initial program 99.9%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. 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 95.1%

      \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\beta + \left(\alpha + 2\right)\right) \cdot \left(\left(\beta + \left(\alpha + 2\right)\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)\right)}} \]
   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. associate-/r* N/A

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

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

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

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

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

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

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

      5. unpow2 N/A

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

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

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

      7. +-commutative N/A

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

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

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

      9. +-commutative N/A

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

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

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

      11. +-commutative N/A

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

      12. +-lowering-+.f64 69.6%

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

   7. Simplified 69.6%

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

   8. Taylor expanded in beta around 0

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

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

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 69.1%

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

   10. Simplified 69.1%

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

   if 2.75 < beta < 5.80000000000000022e150

   1. Initial program 88.2%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 44.9%

      \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\beta + \left(\alpha + 2\right)\right) \cdot \left(\left(\beta + \left(\alpha + 2\right)\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)\right)}} \]
   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 68.0%

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

   7. Simplified 68.0%

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

   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 63.2%

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

   10. Simplified 63.2%

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

   if 5.80000000000000022e150 < beta

   1. Initial program 76.0%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 66.5%

      \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\beta + \left(\alpha + 2\right)\right) \cdot \left(\left(\beta + \left(\alpha + 2\right)\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)\right)}} \]
   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 77.7%

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

   7. Simplified 77.7%

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

   8. Taylor expanded in alpha around inf

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 76.0%

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

   10. Simplified 76.0%

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

       1. associate-/r* N/A

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

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

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

       3. /-lowering-/.f64 82.8%

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

   12. Applied egg-rr 82.8%

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

Alternative 12: 97.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 7.2:\\ \;\;\;\;\frac{0.25}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 7.2)
   (/ 0.25 (+ 1.0 (+ 2.0 (+ alpha beta))))
   (/ (/ (+ 1.0 alpha) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 7.2) {
		tmp = 0.25 / (1.0 + (2.0 + (alpha + beta)));
	} 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 <= 7.2d0) then
        tmp = 0.25d0 / (1.0d0 + (2.0d0 + (alpha + beta)))
    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 <= 7.2) {
		tmp = 0.25 / (1.0 + (2.0 + (alpha + beta)));
	} else {
		tmp = ((1.0 + alpha) / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 7.2:
		tmp = 0.25 / (1.0 + (2.0 + (alpha + beta)))
	else:
		tmp = ((1.0 + alpha) / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 7.2)
		tmp = Float64(0.25 / Float64(1.0 + Float64(2.0 + Float64(alpha + beta))));
	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 <= 7.2)
		tmp = 0.25 / (1.0 + (2.0 + (alpha + beta)));
	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, 7.2], N[(0.25 / N[(1.0 + N[(2.0 + N[(alpha + beta), $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 < 7.20000000000000018

   1. Initial program 99.9%

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

   3. Taylor expanded in beta around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

      6. +-lowering-+.f64 98.3%

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

   5. Simplified 98.3%

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

   6. Taylor expanded in alpha around 0

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

   8. Simplified 70.7%

      \[\leadsto \frac{\color{blue}{0.25}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   if 7.20000000000000018 < beta

   1. Initial program 82.3%

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

      1. associate-/l/ 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 55.4%

      \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\beta + \left(\alpha + 2\right)\right) \cdot \left(\left(\beta + \left(\alpha + 2\right)\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)\right)}} \]
   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 72.8%

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

   7. Simplified 72.8%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]
   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 76.2%

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

   9. Applied egg-rr 76.2%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 7.2:\\ \;\;\;\;\frac{0.25}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 97.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.7:\\ \;\;\;\;0.08333333333333333 + \beta \cdot \left(\beta \cdot -0.011574074074074073 + -0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.7)
   (+
    0.08333333333333333
    (* beta (+ (* beta -0.011574074074074073) -0.027777777777777776)))
   (/ (/ (+ 1.0 alpha) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.7) {
		tmp = 0.08333333333333333 + (beta * ((beta * -0.011574074074074073) + -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 <= 1.7d0) then
        tmp = 0.08333333333333333d0 + (beta * ((beta * (-0.011574074074074073d0)) + (-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 <= 1.7) {
		tmp = 0.08333333333333333 + (beta * ((beta * -0.011574074074074073) + -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 <= 1.7:
		tmp = 0.08333333333333333 + (beta * ((beta * -0.011574074074074073) + -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 <= 1.7)
		tmp = Float64(0.08333333333333333 + Float64(beta * Float64(Float64(beta * -0.011574074074074073) + -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 <= 1.7)
		tmp = 0.08333333333333333 + (beta * ((beta * -0.011574074074074073) + -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, 1.7], N[(0.08333333333333333 + N[(beta * N[(N[(beta * -0.011574074074074073), $MachinePrecision] + -0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.69999999999999996

   1. Initial program 99.9%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 95.1%

      \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\beta + \left(\alpha + 2\right)\right) \cdot \left(\left(\beta + \left(\alpha + 2\right)\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)\right)}} \]
   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. associate-/r* N/A

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

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

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

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

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

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

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

      5. unpow2 N/A

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

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

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

      7. +-commutative N/A

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

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

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

      9. +-commutative N/A

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

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

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

      11. +-commutative N/A

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

      12. +-lowering-+.f64 69.6%

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

   7. Simplified 69.6%

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

   8. Taylor expanded in beta around 0

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 69.5%

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

   10. Simplified 69.5%

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

   if 1.69999999999999996 < beta

   1. Initial program 82.3%

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

      1. associate-/l/ 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 55.4%

      \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\beta + \left(\alpha + 2\right)\right) \cdot \left(\left(\beta + \left(\alpha + 2\right)\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)\right)}} \]
   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 72.8%

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

   7. Simplified 72.8%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]
   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 76.2%

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

   9. Applied egg-rr 76.2%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.7:\\ \;\;\;\;0.08333333333333333 + \beta \cdot \left(\beta \cdot -0.011574074074074073 + -0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 97.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.75:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.75)
   (+ 0.08333333333333333 (* beta -0.027777777777777776))
   (/ (/ (+ 1.0 alpha) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.75) {
		tmp = 0.08333333333333333 + (beta * -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 <= 2.75d0) then
        tmp = 0.08333333333333333d0 + (beta * (-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 <= 2.75) {
		tmp = 0.08333333333333333 + (beta * -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 <= 2.75:
		tmp = 0.08333333333333333 + (beta * -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 <= 2.75)
		tmp = Float64(0.08333333333333333 + Float64(beta * -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 <= 2.75)
		tmp = 0.08333333333333333 + (beta * -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, 2.75], N[(0.08333333333333333 + N[(beta * -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 < 2.75

   1. Initial program 99.9%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. 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 95.1%

      \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\beta + \left(\alpha + 2\right)\right) \cdot \left(\left(\beta + \left(\alpha + 2\right)\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)\right)}} \]
   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. associate-/r* N/A

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

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

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

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

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

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

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

      5. unpow2 N/A

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

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

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

      7. +-commutative N/A

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

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

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

      9. +-commutative N/A

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

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

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

      11. +-commutative N/A

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

      12. +-lowering-+.f64 69.6%

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

   7. Simplified 69.6%

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

   8. Taylor expanded in beta around 0

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

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

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 69.1%

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

   10. Simplified 69.1%

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

   if 2.75 < beta

   1. Initial program 82.3%

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

      1. associate-/l/ 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 55.4%

      \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\beta + \left(\alpha + 2\right)\right) \cdot \left(\left(\beta + \left(\alpha + 2\right)\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)\right)}} \]
   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 72.8%

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

   7. Simplified 72.8%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]
   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 76.2%

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

   9. Applied egg-rr 76.2%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.75:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 91.2% accurate, 3.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.75:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.75)
   (+ 0.08333333333333333 (* beta -0.027777777777777776))
   (/ 1.0 (* beta beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.75) {
		tmp = 0.08333333333333333 + (beta * -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 <= 2.75d0) then
        tmp = 0.08333333333333333d0 + (beta * (-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 <= 2.75) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = 1.0 / (beta * beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.75:
		tmp = 0.08333333333333333 + (beta * -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 <= 2.75)
		tmp = Float64(0.08333333333333333 + Float64(beta * -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 <= 2.75)
		tmp = 0.08333333333333333 + (beta * -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, 2.75], N[(0.08333333333333333 + N[(beta * -0.027777777777777776), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.75

   1. Initial program 99.9%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. 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 95.1%

      \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\beta + \left(\alpha + 2\right)\right) \cdot \left(\left(\beta + \left(\alpha + 2\right)\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)\right)}} \]
   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. associate-/r* N/A

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

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

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

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

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

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

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

      5. unpow2 N/A

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

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

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

      7. +-commutative N/A

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

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

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

      9. +-commutative N/A

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

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

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

      11. +-commutative N/A

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

      12. +-lowering-+.f64 69.6%

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

   7. Simplified 69.6%

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

   8. Taylor expanded in beta around 0

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

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

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 69.1%

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

   10. Simplified 69.1%

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

   if 2.75 < beta

   1. Initial program 82.3%

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

      1. associate-/l/ 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 55.4%

      \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\beta + \left(\alpha + 2\right)\right) \cdot \left(\left(\beta + \left(\alpha + 2\right)\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)\right)}} \]
   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 72.8%

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

   7. Simplified 72.8%

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

   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 69.3%

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

   10. Simplified 69.3%

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

Alternative 16: 46.9% accurate, 3.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.6:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.6)
   (+ 0.08333333333333333 (* beta -0.027777777777777776))
   (/ 0.25 beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.6) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = 0.25 / 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 <= 2.6d0) then
        tmp = 0.08333333333333333d0 + (beta * (-0.027777777777777776d0))
    else
        tmp = 0.25d0 / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.6) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = 0.25 / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.6:
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776)
	else:
		tmp = 0.25 / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.6)
		tmp = Float64(0.08333333333333333 + Float64(beta * -0.027777777777777776));
	else
		tmp = Float64(0.25 / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.6)
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	else
		tmp = 0.25 / 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, 2.6], N[(0.08333333333333333 + N[(beta * -0.027777777777777776), $MachinePrecision]), $MachinePrecision], N[(0.25 / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.60000000000000009

   1. Initial program 99.9%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 95.1%

      \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\beta + \left(\alpha + 2\right)\right) \cdot \left(\left(\beta + \left(\alpha + 2\right)\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)\right)}} \]
   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. associate-/r* N/A

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

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

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

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

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

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

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

      5. unpow2 N/A

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

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

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

      7. +-commutative N/A

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

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

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

      9. +-commutative N/A

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

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

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

      11. +-commutative N/A

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

      12. +-lowering-+.f64 69.6%

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

   7. Simplified 69.6%

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

   8. Taylor expanded in beta around 0

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

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

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 69.1%

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

   10. Simplified 69.1%

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

   if 2.60000000000000009 < beta

   1. Initial program 82.3%

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

   3. Taylor expanded in beta around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

      6. +-lowering-+.f64 19.3%

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

   5. Simplified 19.3%

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

   6. Taylor expanded in alpha around 0

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

   8. Simplified 7.2%

      \[\leadsto \frac{\color{blue}{0.25}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   9. Taylor expanded in beta around inf

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

       1. /-lowering-/.f64 6.5%

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{4}, \color{blue}{\beta}\right) \]

   11. Simplified 6.5%

       \[\leadsto \color{blue}{\frac{0.25}{\beta}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 46.9% accurate, 3.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.1:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.1)
   (+ 0.08333333333333333 (* alpha -0.027777777777777776))
   (/ 0.25 beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.1) {
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776);
	} else {
		tmp = 0.25 / 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 <= 2.1d0) then
        tmp = 0.08333333333333333d0 + (alpha * (-0.027777777777777776d0))
    else
        tmp = 0.25d0 / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.1) {
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776);
	} else {
		tmp = 0.25 / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.1:
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776)
	else:
		tmp = 0.25 / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.1)
		tmp = Float64(0.08333333333333333 + Float64(alpha * -0.027777777777777776));
	else
		tmp = Float64(0.25 / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.1)
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776);
	else
		tmp = 0.25 / 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, 2.1], N[(0.08333333333333333 + N[(alpha * -0.027777777777777776), $MachinePrecision]), $MachinePrecision], N[(0.25 / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.10000000000000009

   1. Initial program 99.9%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. 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 95.1%

      \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\beta + \left(\alpha + 2\right)\right) \cdot \left(\left(\beta + \left(\alpha + 2\right)\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)\right)}} \]
   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 92.9%

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

   7. Simplified 92.9%

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

   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 67.4%

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

   10. Simplified 67.4%

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

   if 2.10000000000000009 < beta

   1. Initial program 82.3%

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

   3. Taylor expanded in beta around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

      6. +-lowering-+.f64 19.3%

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

   5. Simplified 19.3%

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

   6. Taylor expanded in alpha around 0

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

   8. Simplified 7.2%

      \[\leadsto \frac{\color{blue}{0.25}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   9. Taylor expanded in beta around inf

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

       1. /-lowering-/.f64 6.5%

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{4}, \color{blue}{\beta}\right) \]

   11. Simplified 6.5%

       \[\leadsto \color{blue}{\frac{0.25}{\beta}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 46.5% accurate, 4.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{\beta}\\ \end{array} \end{array} \]

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 (/ 0.25 beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.0) {
		tmp = 0.08333333333333333;
	} else {
		tmp = 0.25 / 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
    else
        tmp = 0.25d0 / 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;
	} else {
		tmp = 0.25 / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 3.0:
		tmp = 0.08333333333333333
	else:
		tmp = 0.25 / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.0)
		tmp = 0.08333333333333333;
	else
		tmp = Float64(0.25 / 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;
	else
		tmp = 0.25 / 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], 0.08333333333333333, N[(0.25 / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3

   1. Initial program 99.9%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. 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 95.1%

      \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\beta + \left(\alpha + 2\right)\right) \cdot \left(\left(\beta + \left(\alpha + 2\right)\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)\right)}} \]
   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 92.9%

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

   7. Simplified 92.9%

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

   8. Taylor expanded in alpha around 0

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

   10. Simplified 67.4%

       \[\leadsto \color{blue}{0.08333333333333333} \]

   if 3 < beta

   1. Initial program 82.3%

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

   3. Taylor expanded in beta around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

      6. +-lowering-+.f64 19.3%

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

   5. Simplified 19.3%

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

   6. Taylor expanded in alpha around 0

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

   8. Simplified 7.2%

      \[\leadsto \frac{\color{blue}{0.25}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   9. Taylor expanded in beta around inf

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

       1. /-lowering-/.f64 6.5%

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{4}, \color{blue}{\beta}\right) \]

   11. Simplified 6.5%

       \[\leadsto \color{blue}{\frac{0.25}{\beta}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 46.9% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.1%

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

3. Taylor expanded in beta around 0

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

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

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

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

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

   3. unpow2 N/A

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

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

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

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

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

   6. +-lowering-+.f64 72.4%

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

5. Simplified 72.4%

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

6. Taylor expanded in alpha around 0

   \[\leadsto \color{blue}{\frac{\frac{1}{4}}{3 + \beta}} \]
7. Step-by-step derivation

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

      \[\leadsto \mathsf{/.f64}\left(\frac{1}{4}, \color{blue}{\left(3 + \beta\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{1}{4}, \left(\beta + \color{blue}{3}\right)\right) \]

   3. +-lowering-+.f64 48.6%

      \[\leadsto \mathsf{/.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\beta, \color{blue}{3}\right)\right) \]

8. Simplified 48.6%

   \[\leadsto \color{blue}{\frac{0.25}{\beta + 3}} \]
9. Add Preprocessing

Alternative 20: 44.8% 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 94.1%

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

   1. 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 82.1%

   \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\beta + \left(\alpha + 2\right)\right) \cdot \left(\left(\beta + \left(\alpha + 2\right)\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)\right)}} \]
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 67.7%

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

7. Simplified 67.7%

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

8. Taylor expanded in alpha around 0

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

10. Simplified 46.6%

    \[\leadsto \color{blue}{0.08333333333333333} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024144 
(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)))