Octave 3.8, jcobi/3

Percentage Accurate: 94.4% → 99.8%

Time: 15.3s

Alternatives: 25

Speedup: 1.4×

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: 94.4% 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

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

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ 2.0 beta))))
   (/ (/ (/ (+ 1.0 alpha) (/ t_0 (+ 1.0 beta))) t_0) (+ (+ alpha beta) 3.0))))

C

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

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    t_0 = alpha + (2.0d0 + beta)
    code = (((1.0d0 + alpha) / (t_0 / (1.0d0 + beta))) / t_0) / ((alpha + beta) + 3.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(alpha, beta)
	t_0 = alpha + (2.0 + beta);
	tmp = (((1.0 + alpha) / (t_0 / (1.0 + beta))) / t_0) / ((alpha + beta) + 3.0);
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(2.0 + beta), $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[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 93.7%

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

   1. metadata-eval N/A

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

   2. +-commutative N/A

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

   3. associate-+l+ N/A

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

   4. *-rgt-identity N/A

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

   5. distribute-lft-in N/A

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

   6. +-commutative N/A

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

   7. associate-+r+ N/A

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

   8. +-commutative N/A

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

   9. *-lft-identity N/A

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

   10. distribute-rgt-in N/A

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

   11. +-commutative N/A

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

   12. associate-/l* N/A

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

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

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

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

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

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

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

   16. +-commutative N/A

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

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

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

   18. associate-+l+ N/A

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

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

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

   20. +-lowering-+.f64 99.8%

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

4. Applied egg-rr 99.8%

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

   1. /-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%

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

Alternative 2: 98.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \alpha + \left(2 + \beta\right)\\ t_1 := 2 + \left(\alpha + \beta\right)\\ t_2 := \left(\alpha + \beta\right) + 3\\ \mathbf{if}\;\alpha \leq -2 \cdot 10^{-15}:\\ \;\;\;\;\frac{1 + \alpha}{t\_1 \cdot \left(t\_2 \cdot t\_1\right)}\\ \mathbf{elif}\;\alpha \leq 1.75:\\ \;\;\;\;\frac{\frac{\frac{1 + \beta}{2 + \beta}}{t\_0}}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\alpha}{\frac{t\_0}{1 + \beta}}}{t\_0}}{t\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ 2.0 beta)))
        (t_1 (+ 2.0 (+ alpha beta)))
        (t_2 (+ (+ alpha beta) 3.0)))
   (if (<= alpha -2e-15)
     (/ (+ 1.0 alpha) (* t_1 (* t_2 t_1)))
     (if (<= alpha 1.75)
       (/ (/ (/ (+ 1.0 beta) (+ 2.0 beta)) t_0) t_2)
       (/ (/ (/ alpha (/ t_0 (+ 1.0 beta))) t_0) t_2)))))

C

double code(double alpha, double beta) {
	double t_0 = alpha + (2.0 + beta);
	double t_1 = 2.0 + (alpha + beta);
	double t_2 = (alpha + beta) + 3.0;
	double tmp;
	if (alpha <= -2e-15) {
		tmp = (1.0 + alpha) / (t_1 * (t_2 * t_1));
	} else if (alpha <= 1.75) {
		tmp = (((1.0 + beta) / (2.0 + beta)) / t_0) / t_2;
	} else {
		tmp = ((alpha / (t_0 / (1.0 + beta))) / t_0) / t_2;
	}
	return tmp;
}

Fortran

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) :: t_2
    real(8) :: tmp
    t_0 = alpha + (2.0d0 + beta)
    t_1 = 2.0d0 + (alpha + beta)
    t_2 = (alpha + beta) + 3.0d0
    if (alpha <= (-2d-15)) then
        tmp = (1.0d0 + alpha) / (t_1 * (t_2 * t_1))
    else if (alpha <= 1.75d0) then
        tmp = (((1.0d0 + beta) / (2.0d0 + beta)) / t_0) / t_2
    else
        tmp = ((alpha / (t_0 / (1.0d0 + beta))) / t_0) / t_2
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = alpha + (2.0 + beta);
	double t_1 = 2.0 + (alpha + beta);
	double t_2 = (alpha + beta) + 3.0;
	double tmp;
	if (alpha <= -2e-15) {
		tmp = (1.0 + alpha) / (t_1 * (t_2 * t_1));
	} else if (alpha <= 1.75) {
		tmp = (((1.0 + beta) / (2.0 + beta)) / t_0) / t_2;
	} else {
		tmp = ((alpha / (t_0 / (1.0 + beta))) / t_0) / t_2;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = alpha + (2.0 + beta)
	t_1 = 2.0 + (alpha + beta)
	t_2 = (alpha + beta) + 3.0
	tmp = 0
	if alpha <= -2e-15:
		tmp = (1.0 + alpha) / (t_1 * (t_2 * t_1))
	elif alpha <= 1.75:
		tmp = (((1.0 + beta) / (2.0 + beta)) / t_0) / t_2
	else:
		tmp = ((alpha / (t_0 / (1.0 + beta))) / t_0) / t_2
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(alpha + Float64(2.0 + beta))
	t_1 = Float64(2.0 + Float64(alpha + beta))
	t_2 = Float64(Float64(alpha + beta) + 3.0)
	tmp = 0.0
	if (alpha <= -2e-15)
		tmp = Float64(Float64(1.0 + alpha) / Float64(t_1 * Float64(t_2 * t_1)));
	elseif (alpha <= 1.75)
		tmp = Float64(Float64(Float64(Float64(1.0 + beta) / Float64(2.0 + beta)) / t_0) / t_2);
	else
		tmp = Float64(Float64(Float64(alpha / Float64(t_0 / Float64(1.0 + beta))) / t_0) / t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = alpha + (2.0 + beta);
	t_1 = 2.0 + (alpha + beta);
	t_2 = (alpha + beta) + 3.0;
	tmp = 0.0;
	if (alpha <= -2e-15)
		tmp = (1.0 + alpha) / (t_1 * (t_2 * t_1));
	elseif (alpha <= 1.75)
		tmp = (((1.0 + beta) / (2.0 + beta)) / t_0) / t_2;
	else
		tmp = ((alpha / (t_0 / (1.0 + beta))) / t_0) / t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision]}, If[LessEqual[alpha, -2e-15], N[(N[(1.0 + alpha), $MachinePrecision] / N[(t$95$1 * N[(t$95$2 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[alpha, 1.75], N[(N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(N[(alpha / N[(t$95$0 / N[(1.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$2), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if alpha < -2.0000000000000002e-15

   1. Initial program 99.1%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 99.2%

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

   5. Taylor expanded in beta around 0

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

      1. +-lowering-+.f64 99.2%

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

   7. Simplified 99.2%

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

   if -2.0000000000000002e-15 < alpha < 1.75

   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. Add Preprocessing
   3. Step-by-step derivation

      1. metadata-eval N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. *-rgt-identity N/A

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

      5. distribute-lft-in N/A

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

      6. +-commutative N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. *-lft-identity N/A

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

      10. distribute-rgt-in N/A

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

      11. +-commutative N/A

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

      12. associate-/l* N/A

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

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

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

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

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

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

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

      16. +-commutative N/A

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

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

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

      18. associate-+l+ N/A

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

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

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

      20. +-lowering-+.f64 99.9%

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

   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%

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

   7. Taylor expanded in alpha around 0

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

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

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

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

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 99.8%

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

   9. Simplified 99.8%

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

   if 1.75 < alpha

   1. Initial program 81.7%

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

      1. metadata-eval N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. *-rgt-identity N/A

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

      5. distribute-lft-in N/A

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

      6. +-commutative N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. *-lft-identity N/A

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

      10. distribute-rgt-in N/A

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

      11. +-commutative N/A

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

      12. associate-/l* N/A

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

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

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

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

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

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

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

      16. +-commutative N/A

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

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

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

      18. associate-+l+ N/A

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

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

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

      20. +-lowering-+.f64 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%

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

   7. Taylor expanded in alpha around inf

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

   9. Simplified 98.4%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq -2 \cdot 10^{-15}:\\ \;\;\;\;\frac{1 + \alpha}{\left(2 + \left(\alpha + \beta\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(2 + \left(\alpha + \beta\right)\right)\right)}\\ \mathbf{elif}\;\alpha \leq 1.75:\\ \;\;\;\;\frac{\frac{\frac{1 + \beta}{2 + \beta}}{\alpha + \left(2 + \beta\right)}}{\left(\alpha + \beta\right) + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\alpha}{\frac{\alpha + \left(2 + \beta\right)}{1 + \beta}}}{\alpha + \left(2 + \beta\right)}}{\left(\alpha + \beta\right) + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 3\\ t_1 := \alpha + \left(2 + \beta\right)\\ \mathbf{if}\;\beta \leq 1.05 \cdot 10^{+16}:\\ \;\;\;\;\frac{1 + \beta}{t\_1 \cdot t\_1} \cdot \frac{1 + \alpha}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{t\_1}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) 3.0)) (t_1 (+ alpha (+ 2.0 beta))))
   (if (<= beta 1.05e+16)
     (* (/ (+ 1.0 beta) (* t_1 t_1)) (/ (+ 1.0 alpha) t_0))
     (/ (/ (+ 1.0 alpha) t_1) t_0))))

C

double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + 3.0;
	double t_1 = alpha + (2.0 + beta);
	double tmp;
	if (beta <= 1.05e+16) {
		tmp = ((1.0 + beta) / (t_1 * t_1)) * ((1.0 + alpha) / t_0);
	} else {
		tmp = ((1.0 + alpha) / t_1) / t_0;
	}
	return tmp;
}

Fortran

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 = (alpha + beta) + 3.0d0
    t_1 = alpha + (2.0d0 + beta)
    if (beta <= 1.05d+16) then
        tmp = ((1.0d0 + beta) / (t_1 * t_1)) * ((1.0d0 + alpha) / t_0)
    else
        tmp = ((1.0d0 + alpha) / t_1) / t_0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + 3.0;
	double t_1 = alpha + (2.0 + beta);
	double tmp;
	if (beta <= 1.05e+16) {
		tmp = ((1.0 + beta) / (t_1 * t_1)) * ((1.0 + alpha) / t_0);
	} else {
		tmp = ((1.0 + alpha) / t_1) / t_0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (alpha + beta) + 3.0
	t_1 = alpha + (2.0 + beta)
	tmp = 0
	if beta <= 1.05e+16:
		tmp = ((1.0 + beta) / (t_1 * t_1)) * ((1.0 + alpha) / t_0)
	else:
		tmp = ((1.0 + alpha) / t_1) / t_0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(alpha + beta) + 3.0)
	t_1 = Float64(alpha + Float64(2.0 + beta))
	tmp = 0.0
	if (beta <= 1.05e+16)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(t_1 * t_1)) * Float64(Float64(1.0 + alpha) / t_0));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / t_1) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (alpha + beta) + 3.0;
	t_1 = alpha + (2.0 + beta);
	tmp = 0.0;
	if (beta <= 1.05e+16)
		tmp = ((1.0 + beta) / (t_1 * t_1)) * ((1.0 + alpha) / t_0);
	else
		tmp = ((1.0 + alpha) / t_1) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision]}, Block[{t$95$1 = N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1.05e+16], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.05e16

   1. Initial program 99.8%

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

      1. associate-/l/ N/A

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

      2. associate-/l/ N/A

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

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

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. associate-+l+ N/A

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

      7. associate-+r+ N/A

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

      8. distribute-lft1-in N/A

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

      9. *-commutative N/A

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

      10. distribute-lft1-in N/A

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

      11. +-commutative N/A

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

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

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

      13. +-commutative N/A

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

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

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

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

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

   3. Simplified 96.8%

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

      1. *-commutative N/A

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

      2. metadata-eval N/A

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

      3. associate-+l+ N/A

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

      4. metadata-eval N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

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

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

   6. Applied egg-rr 99.2%

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

   if 1.05e16 < beta

   1. Initial program 75.6%

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

      1. metadata-eval N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. *-rgt-identity N/A

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

      5. distribute-lft-in N/A

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

      6. +-commutative N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. *-lft-identity N/A

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

      10. distribute-rgt-in N/A

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

      11. +-commutative N/A

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

      12. associate-/l* N/A

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

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

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

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

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

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

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

      16. +-commutative N/A

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

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

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

      18. associate-+l+ N/A

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

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

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

      20. +-lowering-+.f64 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. /-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%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

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

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

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

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

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

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

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

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

      10. +-lowering-+.f64 99.6%

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

   8. Applied egg-rr 99.6%

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

   9. Taylor expanded in beta around inf

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

       1. +-commutative N/A

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

       2. +-lowering-+.f64 81.9%

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

   11. Simplified 81.9%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.05 \cdot 10^{+16}:\\ \;\;\;\;\frac{1 + \beta}{\left(\alpha + \left(2 + \beta\right)\right) \cdot \left(\alpha + \left(2 + \beta\right)\right)} \cdot \frac{1 + \alpha}{\left(\alpha + \beta\right) + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\left(\alpha + \beta\right) + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.8 \cdot 10^{-8}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\left(\alpha + 2\right) \cdot \left(\alpha + 2\right)}}{\left(\alpha + \beta\right) + 3}\\ \mathbf{elif}\;\beta \leq 3 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{1 + \beta}{2 + \beta}}{\left(6 + \beta \cdot 3\right) + \beta \cdot \left(2 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{2 + \left(\alpha + \beta\right)}}{\beta \cdot \left(\frac{\alpha - -3}{\beta} - -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.8e-8)
   (/ (/ (+ 1.0 alpha) (* (+ alpha 2.0) (+ alpha 2.0))) (+ (+ alpha beta) 3.0))
   (if (<= beta 3e+16)
     (/
      (/ (+ 1.0 beta) (+ 2.0 beta))
      (+ (+ 6.0 (* beta 3.0)) (* beta (+ 2.0 beta))))
     (/
      (/ (+ 1.0 alpha) (+ 2.0 (+ alpha beta)))
      (* beta (- (/ (- alpha -3.0) beta) -1.0))))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.8e-8) {
		tmp = ((1.0 + alpha) / ((alpha + 2.0) * (alpha + 2.0))) / ((alpha + beta) + 3.0);
	} else if (beta <= 3e+16) {
		tmp = ((1.0 + beta) / (2.0 + beta)) / ((6.0 + (beta * 3.0)) + (beta * (2.0 + beta)));
	} else {
		tmp = ((1.0 + alpha) / (2.0 + (alpha + beta))) / (beta * (((alpha - -3.0) / beta) - -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.8d-8) then
        tmp = ((1.0d0 + alpha) / ((alpha + 2.0d0) * (alpha + 2.0d0))) / ((alpha + beta) + 3.0d0)
    else if (beta <= 3d+16) then
        tmp = ((1.0d0 + beta) / (2.0d0 + beta)) / ((6.0d0 + (beta * 3.0d0)) + (beta * (2.0d0 + beta)))
    else
        tmp = ((1.0d0 + alpha) / (2.0d0 + (alpha + beta))) / (beta * (((alpha - (-3.0d0)) / beta) - (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.8e-8) {
		tmp = ((1.0 + alpha) / ((alpha + 2.0) * (alpha + 2.0))) / ((alpha + beta) + 3.0);
	} else if (beta <= 3e+16) {
		tmp = ((1.0 + beta) / (2.0 + beta)) / ((6.0 + (beta * 3.0)) + (beta * (2.0 + beta)));
	} else {
		tmp = ((1.0 + alpha) / (2.0 + (alpha + beta))) / (beta * (((alpha - -3.0) / beta) - -1.0));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.8e-8:
		tmp = ((1.0 + alpha) / ((alpha + 2.0) * (alpha + 2.0))) / ((alpha + beta) + 3.0)
	elif beta <= 3e+16:
		tmp = ((1.0 + beta) / (2.0 + beta)) / ((6.0 + (beta * 3.0)) + (beta * (2.0 + beta)))
	else:
		tmp = ((1.0 + alpha) / (2.0 + (alpha + beta))) / (beta * (((alpha - -3.0) / beta) - -1.0))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.8e-8)
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(Float64(alpha + 2.0) * Float64(alpha + 2.0))) / Float64(Float64(alpha + beta) + 3.0));
	elseif (beta <= 3e+16)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(2.0 + beta)) / Float64(Float64(6.0 + Float64(beta * 3.0)) + Float64(beta * Float64(2.0 + beta))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(2.0 + Float64(alpha + beta))) / Float64(beta * Float64(Float64(Float64(alpha - -3.0) / beta) - -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.8e-8)
		tmp = ((1.0 + alpha) / ((alpha + 2.0) * (alpha + 2.0))) / ((alpha + beta) + 3.0);
	elseif (beta <= 3e+16)
		tmp = ((1.0 + beta) / (2.0 + beta)) / ((6.0 + (beta * 3.0)) + (beta * (2.0 + beta)));
	else
		tmp = ((1.0 + alpha) / (2.0 + (alpha + beta))) / (beta * (((alpha - -3.0) / beta) - -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.8e-8], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(N[(alpha + 2.0), $MachinePrecision] * N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 3e+16], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] / N[(N[(6.0 + N[(beta * 3.0), $MachinePrecision]), $MachinePrecision] + N[(beta * N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(beta * N[(N[(N[(alpha - -3.0), $MachinePrecision] / beta), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if beta < 2.7999999999999999e-8

   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. metadata-eval N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. *-rgt-identity N/A

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

      5. distribute-lft-in N/A

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

      6. +-commutative N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. *-lft-identity N/A

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

      10. distribute-rgt-in N/A

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

      11. +-commutative N/A

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

      12. associate-/l* N/A

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

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

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

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

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

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

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

      16. +-commutative N/A

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

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

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

      18. associate-+l+ N/A

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

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

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

      20. +-lowering-+.f64 99.9%

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

   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%

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

   7. Taylor expanded in beta around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

      6. +-lowering-+.f64 98.7%

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

   9. Simplified 98.7%

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

   if 2.7999999999999999e-8 < beta < 3e16

   1. Initial program 99.2%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 98.7%

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

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

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

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

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

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

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

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

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

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), 6\right), \left(\left(\alpha + \color{blue}{\beta}\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), 6\right), \mathsf{*.f64}\left(\left(\alpha + \beta\right), \color{blue}{\left(\left(\alpha + \beta\right) + 2\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), 6\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \left(\color{blue}{\left(\alpha + \beta\right)} + 2\right)\right)\right)\right)\right) \]

      12. associate-+l+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), 6\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \left(\alpha + \color{blue}{\left(\beta + 2\right)}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), 6\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{+.f64}\left(\alpha, \color{blue}{\left(\beta + 2\right)}\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 99.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), 6\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, \color{blue}{2}\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 99.2%

      \[\leadsto \frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \color{blue}{\left(\left(\left(\alpha + \beta\right) \cdot 3 + 6\right) + \left(\alpha + \beta\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)}} \]

   7. Taylor expanded in alpha around 0

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

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{1 + \beta}{2 + \beta}}{\color{blue}{6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 + \beta}{2 + \beta}\right), \color{blue}{\left(6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + \beta\right), \left(2 + \beta\right)\right), \left(\color{blue}{6} + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\beta + 1\right), \left(2 + \beta\right)\right), \left(6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \left(2 + \beta\right)\right), \left(6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \left(\beta + 2\right)\right), \left(6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)\right) \]

      8. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(\left(6 + 3 \cdot \beta\right) + \color{blue}{\beta \cdot \left(2 + \beta\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\left(6 + 3 \cdot \beta\right), \color{blue}{\left(\beta \cdot \left(2 + \beta\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(6, \left(3 \cdot \beta\right)\right), \left(\color{blue}{\beta} \cdot \left(2 + \beta\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(6, \mathsf{*.f64}\left(3, \beta\right)\right), \left(\beta \cdot \left(2 + \beta\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(6, \mathsf{*.f64}\left(3, \beta\right)\right), \mathsf{*.f64}\left(\beta, \color{blue}{\left(2 + \beta\right)}\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(6, \mathsf{*.f64}\left(3, \beta\right)\right), \mathsf{*.f64}\left(\beta, \left(\beta + \color{blue}{2}\right)\right)\right)\right) \]

      14. +-lowering-+.f64 80.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(6, \mathsf{*.f64}\left(3, \beta\right)\right), \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(\beta, \color{blue}{2}\right)\right)\right)\right) \]

   9. Simplified 80.5%

      \[\leadsto \color{blue}{\frac{\frac{\beta + 1}{\beta + 2}}{\left(6 + 3 \cdot \beta\right) + \beta \cdot \left(\beta + 2\right)}} \]

   if 3e16 < beta

   1. Initial program 75.6%

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

   3. Taylor expanded in beta around inf

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

      1. +-lowering-+.f64 81.9%

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

   5. Simplified 81.9%

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

   6. Taylor expanded in beta around -inf

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

      1. associate-*r* N/A

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

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

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

      3. mul-1-neg N/A

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

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

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

      8. associate-*r/ N/A

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

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

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

      10. distribute-lft-in N/A

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

      11. metadata-eval N/A

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

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

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

      13. mul-1-neg N/A

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

      14. neg-lowering-neg.f64 81.9%

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

   8. Simplified 81.9%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.8 \cdot 10^{-8}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\left(\alpha + 2\right) \cdot \left(\alpha + 2\right)}}{\left(\alpha + \beta\right) + 3}\\ \mathbf{elif}\;\beta \leq 3 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{1 + \beta}{2 + \beta}}{\left(6 + \beta \cdot 3\right) + \beta \cdot \left(2 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{2 + \left(\alpha + \beta\right)}}{\beta \cdot \left(\frac{\alpha - -3}{\beta} - -1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 3\\ \mathbf{if}\;\beta \leq 2.8 \cdot 10^{-8}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\left(\alpha + 2\right) \cdot \left(\alpha + 2\right)}}{t\_0}\\ \mathbf{elif}\;\beta \leq 3 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{1 + \beta}{2 + \beta}}{\left(6 + \beta \cdot 3\right) + \beta \cdot \left(2 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{t\_0}}{\alpha + \left(2 + \beta\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) 3.0)))
   (if (<= beta 2.8e-8)
     (/ (/ (+ 1.0 alpha) (* (+ alpha 2.0) (+ alpha 2.0))) t_0)
     (if (<= beta 3e+16)
       (/
        (/ (+ 1.0 beta) (+ 2.0 beta))
        (+ (+ 6.0 (* beta 3.0)) (* beta (+ 2.0 beta))))
       (/ (/ (+ 1.0 alpha) t_0) (+ alpha (+ 2.0 beta)))))))

C

double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + 3.0;
	double tmp;
	if (beta <= 2.8e-8) {
		tmp = ((1.0 + alpha) / ((alpha + 2.0) * (alpha + 2.0))) / t_0;
	} else if (beta <= 3e+16) {
		tmp = ((1.0 + beta) / (2.0 + beta)) / ((6.0 + (beta * 3.0)) + (beta * (2.0 + beta)));
	} else {
		tmp = ((1.0 + alpha) / t_0) / (alpha + (2.0 + beta));
	}
	return tmp;
}

Fortran

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 <= 2.8d-8) then
        tmp = ((1.0d0 + alpha) / ((alpha + 2.0d0) * (alpha + 2.0d0))) / t_0
    else if (beta <= 3d+16) then
        tmp = ((1.0d0 + beta) / (2.0d0 + beta)) / ((6.0d0 + (beta * 3.0d0)) + (beta * (2.0d0 + beta)))
    else
        tmp = ((1.0d0 + alpha) / t_0) / (alpha + (2.0d0 + beta))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + 3.0;
	double tmp;
	if (beta <= 2.8e-8) {
		tmp = ((1.0 + alpha) / ((alpha + 2.0) * (alpha + 2.0))) / t_0;
	} else if (beta <= 3e+16) {
		tmp = ((1.0 + beta) / (2.0 + beta)) / ((6.0 + (beta * 3.0)) + (beta * (2.0 + beta)));
	} else {
		tmp = ((1.0 + alpha) / t_0) / (alpha + (2.0 + beta));
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (alpha + beta) + 3.0
	tmp = 0
	if beta <= 2.8e-8:
		tmp = ((1.0 + alpha) / ((alpha + 2.0) * (alpha + 2.0))) / t_0
	elif beta <= 3e+16:
		tmp = ((1.0 + beta) / (2.0 + beta)) / ((6.0 + (beta * 3.0)) + (beta * (2.0 + beta)))
	else:
		tmp = ((1.0 + alpha) / t_0) / (alpha + (2.0 + beta))
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(alpha + beta) + 3.0)
	tmp = 0.0
	if (beta <= 2.8e-8)
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(Float64(alpha + 2.0) * Float64(alpha + 2.0))) / t_0);
	elseif (beta <= 3e+16)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(2.0 + beta)) / Float64(Float64(6.0 + Float64(beta * 3.0)) + Float64(beta * Float64(2.0 + beta))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / t_0) / Float64(alpha + Float64(2.0 + beta)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (alpha + beta) + 3.0;
	tmp = 0.0;
	if (beta <= 2.8e-8)
		tmp = ((1.0 + alpha) / ((alpha + 2.0) * (alpha + 2.0))) / t_0;
	elseif (beta <= 3e+16)
		tmp = ((1.0 + beta) / (2.0 + beta)) / ((6.0 + (beta * 3.0)) + (beta * (2.0 + beta)));
	else
		tmp = ((1.0 + alpha) / t_0) / (alpha + (2.0 + beta));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision]}, If[LessEqual[beta, 2.8e-8], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(N[(alpha + 2.0), $MachinePrecision] * N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[beta, 3e+16], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] / N[(N[(6.0 + N[(beta * 3.0), $MachinePrecision]), $MachinePrecision] + N[(beta * N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if beta < 2.7999999999999999e-8

   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. metadata-eval N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. *-rgt-identity N/A

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

      5. distribute-lft-in N/A

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

      6. +-commutative N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. *-lft-identity N/A

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

      10. distribute-rgt-in N/A

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

      11. +-commutative N/A

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

      12. associate-/l* N/A

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

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

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

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

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

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

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

      16. +-commutative N/A

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

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

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

      18. associate-+l+ N/A

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

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

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

      20. +-lowering-+.f64 99.9%

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

   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%

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

   7. Taylor expanded in beta around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

      6. +-lowering-+.f64 98.7%

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

   9. Simplified 98.7%

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

   if 2.7999999999999999e-8 < beta < 3e16

   1. Initial program 99.2%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 98.7%

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

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

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

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

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

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

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

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

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

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), 6\right), \left(\left(\alpha + \color{blue}{\beta}\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), 6\right), \mathsf{*.f64}\left(\left(\alpha + \beta\right), \color{blue}{\left(\left(\alpha + \beta\right) + 2\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), 6\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \left(\color{blue}{\left(\alpha + \beta\right)} + 2\right)\right)\right)\right)\right) \]

      12. associate-+l+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), 6\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \left(\alpha + \color{blue}{\left(\beta + 2\right)}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), 6\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{+.f64}\left(\alpha, \color{blue}{\left(\beta + 2\right)}\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 99.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), 6\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, \color{blue}{2}\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 99.2%

      \[\leadsto \frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \color{blue}{\left(\left(\left(\alpha + \beta\right) \cdot 3 + 6\right) + \left(\alpha + \beta\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)}} \]

   7. Taylor expanded in alpha around 0

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

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{1 + \beta}{2 + \beta}}{\color{blue}{6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 + \beta}{2 + \beta}\right), \color{blue}{\left(6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + \beta\right), \left(2 + \beta\right)\right), \left(\color{blue}{6} + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\beta + 1\right), \left(2 + \beta\right)\right), \left(6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \left(2 + \beta\right)\right), \left(6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \left(\beta + 2\right)\right), \left(6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)\right) \]

      8. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(\left(6 + 3 \cdot \beta\right) + \color{blue}{\beta \cdot \left(2 + \beta\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\left(6 + 3 \cdot \beta\right), \color{blue}{\left(\beta \cdot \left(2 + \beta\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(6, \left(3 \cdot \beta\right)\right), \left(\color{blue}{\beta} \cdot \left(2 + \beta\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(6, \mathsf{*.f64}\left(3, \beta\right)\right), \left(\beta \cdot \left(2 + \beta\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(6, \mathsf{*.f64}\left(3, \beta\right)\right), \mathsf{*.f64}\left(\beta, \color{blue}{\left(2 + \beta\right)}\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(6, \mathsf{*.f64}\left(3, \beta\right)\right), \mathsf{*.f64}\left(\beta, \left(\beta + \color{blue}{2}\right)\right)\right)\right) \]

      14. +-lowering-+.f64 80.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(6, \mathsf{*.f64}\left(3, \beta\right)\right), \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(\beta, \color{blue}{2}\right)\right)\right)\right) \]

   9. Simplified 80.5%

      \[\leadsto \color{blue}{\frac{\frac{\beta + 1}{\beta + 2}}{\left(6 + 3 \cdot \beta\right) + \beta \cdot \left(\beta + 2\right)}} \]

   if 3e16 < beta

   1. Initial program 75.6%

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

   3. Taylor expanded in beta around inf

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

      1. +-lowering-+.f64 81.9%

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

   5. Simplified 81.9%

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

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

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

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

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

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

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

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

      6. metadata-eval N/A

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

      7. associate-+l+ N/A

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

      8. metadata-eval N/A

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

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

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

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

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

      11. metadata-eval N/A

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

      12. associate-+r+ N/A

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

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

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

      14. +-commutative N/A

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

      15. +-lowering-+.f64 81.9%

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

   7. Applied egg-rr 81.9%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.8 \cdot 10^{-8}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\left(\alpha + 2\right) \cdot \left(\alpha + 2\right)}}{\left(\alpha + \beta\right) + 3}\\ \mathbf{elif}\;\beta \leq 3 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{1 + \beta}{2 + \beta}}{\left(6 + \beta \cdot 3\right) + \beta \cdot \left(2 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 3}}{\alpha + \left(2 + \beta\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 3\\ \mathbf{if}\;\beta \leq 2.8 \cdot 10^{-8}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\left(\alpha + 2\right) \cdot \left(\alpha + 2\right)}}{t\_0}\\ \mathbf{elif}\;\beta \leq 5.5 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{1 + \beta}{2 + \beta}}{6 + \beta \cdot \left(\beta + 5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) 3.0)))
   (if (<= beta 2.8e-8)
     (/ (/ (+ 1.0 alpha) (* (+ alpha 2.0) (+ alpha 2.0))) t_0)
     (if (<= beta 5.5e+14)
       (/ (/ (+ 1.0 beta) (+ 2.0 beta)) (+ 6.0 (* beta (+ beta 5.0))))
       (/ (/ (+ 1.0 alpha) (+ alpha (+ 2.0 beta))) t_0)))))

C

double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + 3.0;
	double tmp;
	if (beta <= 2.8e-8) {
		tmp = ((1.0 + alpha) / ((alpha + 2.0) * (alpha + 2.0))) / t_0;
	} else if (beta <= 5.5e+14) {
		tmp = ((1.0 + beta) / (2.0 + beta)) / (6.0 + (beta * (beta + 5.0)));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) / t_0;
	}
	return tmp;
}

Fortran

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 <= 2.8d-8) then
        tmp = ((1.0d0 + alpha) / ((alpha + 2.0d0) * (alpha + 2.0d0))) / t_0
    else if (beta <= 5.5d+14) then
        tmp = ((1.0d0 + beta) / (2.0d0 + beta)) / (6.0d0 + (beta * (beta + 5.0d0)))
    else
        tmp = ((1.0d0 + alpha) / (alpha + (2.0d0 + beta))) / t_0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + 3.0;
	double tmp;
	if (beta <= 2.8e-8) {
		tmp = ((1.0 + alpha) / ((alpha + 2.0) * (alpha + 2.0))) / t_0;
	} else if (beta <= 5.5e+14) {
		tmp = ((1.0 + beta) / (2.0 + beta)) / (6.0 + (beta * (beta + 5.0)));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) / t_0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (alpha + beta) + 3.0
	tmp = 0
	if beta <= 2.8e-8:
		tmp = ((1.0 + alpha) / ((alpha + 2.0) * (alpha + 2.0))) / t_0
	elif beta <= 5.5e+14:
		tmp = ((1.0 + beta) / (2.0 + beta)) / (6.0 + (beta * (beta + 5.0)))
	else:
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) / t_0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(alpha + beta) + 3.0)
	tmp = 0.0
	if (beta <= 2.8e-8)
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(Float64(alpha + 2.0) * Float64(alpha + 2.0))) / t_0);
	elseif (beta <= 5.5e+14)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(2.0 + beta)) / Float64(6.0 + Float64(beta * Float64(beta + 5.0))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(2.0 + beta))) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (alpha + beta) + 3.0;
	tmp = 0.0;
	if (beta <= 2.8e-8)
		tmp = ((1.0 + alpha) / ((alpha + 2.0) * (alpha + 2.0))) / t_0;
	elseif (beta <= 5.5e+14)
		tmp = ((1.0 + beta) / (2.0 + beta)) / (6.0 + (beta * (beta + 5.0)));
	else
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision]}, If[LessEqual[beta, 2.8e-8], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(N[(alpha + 2.0), $MachinePrecision] * N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[beta, 5.5e+14], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] / N[(6.0 + N[(beta * N[(beta + 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if beta < 2.7999999999999999e-8

   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. metadata-eval N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. *-rgt-identity N/A

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

      5. distribute-lft-in N/A

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

      6. +-commutative N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. *-lft-identity N/A

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

      10. distribute-rgt-in N/A

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

      11. +-commutative N/A

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

      12. associate-/l* N/A

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

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

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

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

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

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

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

      16. +-commutative N/A

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

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

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

      18. associate-+l+ N/A

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

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

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

      20. +-lowering-+.f64 99.9%

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

   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%

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

   7. Taylor expanded in beta around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

      6. +-lowering-+.f64 98.7%

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

   9. Simplified 98.7%

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

   if 2.7999999999999999e-8 < beta < 5.5e14

   1. Initial program 99.2%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 98.7%

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

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

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

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

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

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

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

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

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

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), 6\right), \left(\left(\alpha + \color{blue}{\beta}\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), 6\right), \mathsf{*.f64}\left(\left(\alpha + \beta\right), \color{blue}{\left(\left(\alpha + \beta\right) + 2\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), 6\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \left(\color{blue}{\left(\alpha + \beta\right)} + 2\right)\right)\right)\right)\right) \]

      12. associate-+l+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), 6\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \left(\alpha + \color{blue}{\left(\beta + 2\right)}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), 6\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{+.f64}\left(\alpha, \color{blue}{\left(\beta + 2\right)}\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 99.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), 6\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, \color{blue}{2}\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 99.2%

      \[\leadsto \frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \color{blue}{\left(\left(\left(\alpha + \beta\right) \cdot 3 + 6\right) + \left(\alpha + \beta\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)}} \]

   7. Taylor expanded in alpha around 0

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

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{1 + \beta}{2 + \beta}}{\color{blue}{6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 + \beta}{2 + \beta}\right), \color{blue}{\left(6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + \beta\right), \left(2 + \beta\right)\right), \left(\color{blue}{6} + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\beta + 1\right), \left(2 + \beta\right)\right), \left(6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \left(2 + \beta\right)\right), \left(6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \left(\beta + 2\right)\right), \left(6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)\right) \]

      8. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(\left(6 + 3 \cdot \beta\right) + \color{blue}{\beta \cdot \left(2 + \beta\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\left(6 + 3 \cdot \beta\right), \color{blue}{\left(\beta \cdot \left(2 + \beta\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(6, \left(3 \cdot \beta\right)\right), \left(\color{blue}{\beta} \cdot \left(2 + \beta\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(6, \mathsf{*.f64}\left(3, \beta\right)\right), \left(\beta \cdot \left(2 + \beta\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(6, \mathsf{*.f64}\left(3, \beta\right)\right), \mathsf{*.f64}\left(\beta, \color{blue}{\left(2 + \beta\right)}\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(6, \mathsf{*.f64}\left(3, \beta\right)\right), \mathsf{*.f64}\left(\beta, \left(\beta + \color{blue}{2}\right)\right)\right)\right) \]

      14. +-lowering-+.f64 80.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(6, \mathsf{*.f64}\left(3, \beta\right)\right), \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(\beta, \color{blue}{2}\right)\right)\right)\right) \]

   9. Simplified 80.5%

      \[\leadsto \color{blue}{\frac{\frac{\beta + 1}{\beta + 2}}{\left(6 + 3 \cdot \beta\right) + \beta \cdot \left(\beta + 2\right)}} \]

   10. Taylor expanded in beta around 0

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \color{blue}{\left(6 + \beta \cdot \left(5 + \beta\right)\right)}\right) \]
   11. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(6, \color{blue}{\left(\beta \cdot \left(5 + \beta\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(6, \mathsf{*.f64}\left(\beta, \color{blue}{\left(5 + \beta\right)}\right)\right)\right) \]

       3. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(6, \mathsf{*.f64}\left(\beta, \left(\beta + \color{blue}{5}\right)\right)\right)\right) \]

       4. +-lowering-+.f64 80.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(6, \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(\beta, \color{blue}{5}\right)\right)\right)\right) \]

   12. Simplified 80.5%

       \[\leadsto \frac{\frac{\beta + 1}{\beta + 2}}{\color{blue}{6 + \beta \cdot \left(\beta + 5\right)}} \]

   if 5.5e14 < beta

   1. Initial program 75.6%

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

      1. metadata-eval N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. *-rgt-identity N/A

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

      5. distribute-lft-in N/A

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

      6. +-commutative N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. *-lft-identity N/A

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

      10. distribute-rgt-in N/A

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

      11. +-commutative N/A

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

      12. associate-/l* N/A

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

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

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

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

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

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

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

      16. +-commutative N/A

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

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

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

      18. associate-+l+ N/A

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

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

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

      20. +-lowering-+.f64 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. /-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%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

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

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

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

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

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

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

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

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

      10. +-lowering-+.f64 99.6%

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

   8. Applied egg-rr 99.6%

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

   9. Taylor expanded in beta around inf

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

       1. +-commutative N/A

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

       2. +-lowering-+.f64 81.9%

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

   11. Simplified 81.9%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.8 \cdot 10^{-8}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\left(\alpha + 2\right) \cdot \left(\alpha + 2\right)}}{\left(\alpha + \beta\right) + 3}\\ \mathbf{elif}\;\beta \leq 5.5 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{1 + \beta}{2 + \beta}}{6 + \beta \cdot \left(\beta + 5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\left(\alpha + \beta\right) + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 90.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.95 \cdot 10^{-86}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + 3}}{\left(\alpha + 2\right) \cdot \left(\alpha + 2\right)}\\ \mathbf{elif}\;\beta \leq 3.4 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{1 + \beta}{2 + \beta}}{6 + \beta \cdot \left(\beta + 5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\left(\alpha + \beta\right) + 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.95e-86)
   (/ (/ (+ 1.0 alpha) (+ alpha 3.0)) (* (+ alpha 2.0) (+ alpha 2.0)))
   (if (<= beta 3.4e+15)
     (/ (/ (+ 1.0 beta) (+ 2.0 beta)) (+ 6.0 (* beta (+ beta 5.0))))
     (/ (/ (+ 1.0 alpha) (+ alpha (+ 2.0 beta))) (+ (+ alpha beta) 3.0)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.95e-86) {
		tmp = ((1.0 + alpha) / (alpha + 3.0)) / ((alpha + 2.0) * (alpha + 2.0));
	} else if (beta <= 3.4e+15) {
		tmp = ((1.0 + beta) / (2.0 + beta)) / (6.0 + (beta * (beta + 5.0)));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) / ((alpha + beta) + 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.95d-86) then
        tmp = ((1.0d0 + alpha) / (alpha + 3.0d0)) / ((alpha + 2.0d0) * (alpha + 2.0d0))
    else if (beta <= 3.4d+15) then
        tmp = ((1.0d0 + beta) / (2.0d0 + beta)) / (6.0d0 + (beta * (beta + 5.0d0)))
    else
        tmp = ((1.0d0 + alpha) / (alpha + (2.0d0 + beta))) / ((alpha + beta) + 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.95e-86) {
		tmp = ((1.0 + alpha) / (alpha + 3.0)) / ((alpha + 2.0) * (alpha + 2.0));
	} else if (beta <= 3.4e+15) {
		tmp = ((1.0 + beta) / (2.0 + beta)) / (6.0 + (beta * (beta + 5.0)));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) / ((alpha + beta) + 3.0);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 1.95e-86:
		tmp = ((1.0 + alpha) / (alpha + 3.0)) / ((alpha + 2.0) * (alpha + 2.0))
	elif beta <= 3.4e+15:
		tmp = ((1.0 + beta) / (2.0 + beta)) / (6.0 + (beta * (beta + 5.0)))
	else:
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) / ((alpha + beta) + 3.0)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.95e-86)
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + 3.0)) / Float64(Float64(alpha + 2.0) * Float64(alpha + 2.0)));
	elseif (beta <= 3.4e+15)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(2.0 + beta)) / Float64(6.0 + Float64(beta * Float64(beta + 5.0))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(2.0 + beta))) / Float64(Float64(alpha + beta) + 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.95e-86)
		tmp = ((1.0 + alpha) / (alpha + 3.0)) / ((alpha + 2.0) * (alpha + 2.0));
	elseif (beta <= 3.4e+15)
		tmp = ((1.0 + beta) / (2.0 + beta)) / (6.0 + (beta * (beta + 5.0)));
	else
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) / ((alpha + beta) + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 1.95e-86], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + 2.0), $MachinePrecision] * N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 3.4e+15], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] / N[(6.0 + N[(beta * N[(beta + 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if beta < 1.9500000000000001e-86

   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 96.9%

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

   5. Taylor expanded in beta around 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 96.0%

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

   7. Simplified 96.0%

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

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

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

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

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

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

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

      6. +-commutative N/A

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

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

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

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

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

      9. +-commutative N/A

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

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

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

      11. +-commutative N/A

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

      12. +-lowering-+.f64 98.3%

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

   9. Applied egg-rr 98.3%

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

   if 1.9500000000000001e-86 < beta < 3.4e15

   1. Initial program 99.6%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 96.2%

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

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

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

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

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

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

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

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

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

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), 6\right), \left(\left(\alpha + \color{blue}{\beta}\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), 6\right), \mathsf{*.f64}\left(\left(\alpha + \beta\right), \color{blue}{\left(\left(\alpha + \beta\right) + 2\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), 6\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \left(\color{blue}{\left(\alpha + \beta\right)} + 2\right)\right)\right)\right)\right) \]

      12. associate-+l+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), 6\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \left(\alpha + \color{blue}{\left(\beta + 2\right)}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), 6\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{+.f64}\left(\alpha, \color{blue}{\left(\beta + 2\right)}\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 96.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), 6\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, \color{blue}{2}\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 96.5%

      \[\leadsto \frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \color{blue}{\left(\left(\left(\alpha + \beta\right) \cdot 3 + 6\right) + \left(\alpha + \beta\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)}} \]

   7. Taylor expanded in alpha around 0

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

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{1 + \beta}{2 + \beta}}{\color{blue}{6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 + \beta}{2 + \beta}\right), \color{blue}{\left(6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + \beta\right), \left(2 + \beta\right)\right), \left(\color{blue}{6} + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\beta + 1\right), \left(2 + \beta\right)\right), \left(6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \left(2 + \beta\right)\right), \left(6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \left(\beta + 2\right)\right), \left(6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)\right) \]

      8. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(\left(6 + 3 \cdot \beta\right) + \color{blue}{\beta \cdot \left(2 + \beta\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\left(6 + 3 \cdot \beta\right), \color{blue}{\left(\beta \cdot \left(2 + \beta\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(6, \left(3 \cdot \beta\right)\right), \left(\color{blue}{\beta} \cdot \left(2 + \beta\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(6, \mathsf{*.f64}\left(3, \beta\right)\right), \left(\beta \cdot \left(2 + \beta\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(6, \mathsf{*.f64}\left(3, \beta\right)\right), \mathsf{*.f64}\left(\beta, \color{blue}{\left(2 + \beta\right)}\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(6, \mathsf{*.f64}\left(3, \beta\right)\right), \mathsf{*.f64}\left(\beta, \left(\beta + \color{blue}{2}\right)\right)\right)\right) \]

      14. +-lowering-+.f64 76.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(6, \mathsf{*.f64}\left(3, \beta\right)\right), \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(\beta, \color{blue}{2}\right)\right)\right)\right) \]

   9. Simplified 76.4%

      \[\leadsto \color{blue}{\frac{\frac{\beta + 1}{\beta + 2}}{\left(6 + 3 \cdot \beta\right) + \beta \cdot \left(\beta + 2\right)}} \]

   10. Taylor expanded in beta around 0

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \color{blue}{\left(6 + \beta \cdot \left(5 + \beta\right)\right)}\right) \]
   11. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(6, \color{blue}{\left(\beta \cdot \left(5 + \beta\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(6, \mathsf{*.f64}\left(\beta, \color{blue}{\left(5 + \beta\right)}\right)\right)\right) \]

       3. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(6, \mathsf{*.f64}\left(\beta, \left(\beta + \color{blue}{5}\right)\right)\right)\right) \]

       4. +-lowering-+.f64 76.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(6, \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(\beta, \color{blue}{5}\right)\right)\right)\right) \]

   12. Simplified 76.4%

       \[\leadsto \frac{\frac{\beta + 1}{\beta + 2}}{\color{blue}{6 + \beta \cdot \left(\beta + 5\right)}} \]

   if 3.4e15 < beta

   1. Initial program 75.6%

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

      1. metadata-eval N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. *-rgt-identity N/A

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

      5. distribute-lft-in N/A

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

      6. +-commutative N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. *-lft-identity N/A

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

      10. distribute-rgt-in N/A

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

      11. +-commutative N/A

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

      12. associate-/l* N/A

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

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

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

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

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

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

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

      16. +-commutative N/A

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

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

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

      18. associate-+l+ N/A

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

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

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

      20. +-lowering-+.f64 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. /-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%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

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

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

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

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

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

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

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

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

      10. +-lowering-+.f64 99.6%

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

   8. Applied egg-rr 99.6%

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

   9. Taylor expanded in beta around inf

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

       1. +-commutative N/A

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

       2. +-lowering-+.f64 81.9%

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

   11. Simplified 81.9%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.95 \cdot 10^{-86}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + 3}}{\left(\alpha + 2\right) \cdot \left(\alpha + 2\right)}\\ \mathbf{elif}\;\beta \leq 3.4 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{1 + \beta}{2 + \beta}}{6 + \beta \cdot \left(\beta + 5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\left(\alpha + \beta\right) + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 90.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.5 \cdot 10^{-86}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + 3}}{\left(\alpha + 2\right) \cdot \left(\alpha + 2\right)}\\ \mathbf{elif}\;\beta \leq 3 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(2 + \beta\right)}}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 3}}{\alpha + \left(2 + \beta\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.5e-86)
   (/ (/ (+ 1.0 alpha) (+ alpha 3.0)) (* (+ alpha 2.0) (+ alpha 2.0)))
   (if (<= beta 3e+16)
     (/ (/ (+ 1.0 beta) (* (+ 2.0 beta) (+ 2.0 beta))) (+ beta 3.0))
     (/ (/ (+ 1.0 alpha) (+ (+ alpha beta) 3.0)) (+ alpha (+ 2.0 beta))))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.5e-86) {
		tmp = ((1.0 + alpha) / (alpha + 3.0)) / ((alpha + 2.0) * (alpha + 2.0));
	} else if (beta <= 3e+16) {
		tmp = ((1.0 + beta) / ((2.0 + beta) * (2.0 + beta))) / (beta + 3.0);
	} else {
		tmp = ((1.0 + alpha) / ((alpha + beta) + 3.0)) / (alpha + (2.0 + beta));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.5d-86) then
        tmp = ((1.0d0 + alpha) / (alpha + 3.0d0)) / ((alpha + 2.0d0) * (alpha + 2.0d0))
    else if (beta <= 3d+16) then
        tmp = ((1.0d0 + beta) / ((2.0d0 + beta) * (2.0d0 + beta))) / (beta + 3.0d0)
    else
        tmp = ((1.0d0 + alpha) / ((alpha + beta) + 3.0d0)) / (alpha + (2.0d0 + beta))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.5e-86) {
		tmp = ((1.0 + alpha) / (alpha + 3.0)) / ((alpha + 2.0) * (alpha + 2.0));
	} else if (beta <= 3e+16) {
		tmp = ((1.0 + beta) / ((2.0 + beta) * (2.0 + beta))) / (beta + 3.0);
	} else {
		tmp = ((1.0 + alpha) / ((alpha + beta) + 3.0)) / (alpha + (2.0 + beta));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 1.5e-86:
		tmp = ((1.0 + alpha) / (alpha + 3.0)) / ((alpha + 2.0) * (alpha + 2.0))
	elif beta <= 3e+16:
		tmp = ((1.0 + beta) / ((2.0 + beta) * (2.0 + beta))) / (beta + 3.0)
	else:
		tmp = ((1.0 + alpha) / ((alpha + beta) + 3.0)) / (alpha + (2.0 + beta))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.5e-86)
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + 3.0)) / Float64(Float64(alpha + 2.0) * Float64(alpha + 2.0)));
	elseif (beta <= 3e+16)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(Float64(2.0 + beta) * Float64(2.0 + beta))) / Float64(beta + 3.0));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(Float64(alpha + beta) + 3.0)) / Float64(alpha + Float64(2.0 + beta)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.5e-86)
		tmp = ((1.0 + alpha) / (alpha + 3.0)) / ((alpha + 2.0) * (alpha + 2.0));
	elseif (beta <= 3e+16)
		tmp = ((1.0 + beta) / ((2.0 + beta) * (2.0 + beta))) / (beta + 3.0);
	else
		tmp = ((1.0 + alpha) / ((alpha + beta) + 3.0)) / (alpha + (2.0 + beta));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 1.5e-86], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + 2.0), $MachinePrecision] * N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 3e+16], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(2.0 + beta), $MachinePrecision] * N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if beta < 1.5e-86

   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 96.9%

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

   5. Taylor expanded in beta around 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 96.0%

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

   7. Simplified 96.0%

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

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

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

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

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

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

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

      6. +-commutative N/A

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

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

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

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

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

      9. +-commutative N/A

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

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

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

      11. +-commutative N/A

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

      12. +-lowering-+.f64 98.3%

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

   9. Applied egg-rr 98.3%

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

   if 1.5e-86 < beta < 3e16

   1. Initial program 99.6%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 96.2%

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

   5. Taylor expanded in alpha around 0

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

      1. 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 76.2%

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

   7. Simplified 76.2%

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

   if 3e16 < beta

   1. Initial program 75.6%

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

   3. Taylor expanded in beta around inf

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

      1. +-lowering-+.f64 81.9%

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

   5. Simplified 81.9%

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

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

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

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

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

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

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

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

      6. metadata-eval N/A

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

      7. associate-+l+ N/A

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

      8. metadata-eval N/A

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

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

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

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

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

      11. metadata-eval N/A

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

      12. associate-+r+ N/A

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

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

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

      14. +-commutative N/A

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

      15. +-lowering-+.f64 81.9%

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

   7. Applied egg-rr 81.9%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.5 \cdot 10^{-86}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + 3}}{\left(\alpha + 2\right) \cdot \left(\alpha + 2\right)}\\ \mathbf{elif}\;\beta \leq 3 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(2 + \beta\right)}}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 3}}{\alpha + \left(2 + \beta\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.8% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ 2.0 beta))))
   (* (/ (/ (+ 1.0 beta) t_0) (+ (+ alpha beta) 3.0)) (/ (+ 1.0 alpha) t_0))))

C

double code(double alpha, double beta) {
	double t_0 = alpha + (2.0 + beta);
	return (((1.0 + beta) / t_0) / ((alpha + beta) + 3.0)) * ((1.0 + alpha) / t_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 + (2.0d0 + beta)
    code = (((1.0d0 + beta) / t_0) / ((alpha + beta) + 3.0d0)) * ((1.0d0 + alpha) / t_0)
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(alpha, beta)
	t_0 = alpha + (2.0 + beta);
	tmp = (((1.0 + beta) / t_0) / ((alpha + beta) + 3.0)) * ((1.0 + alpha) / t_0);
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(1.0 + beta), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 93.7%

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

   1. metadata-eval N/A

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

   2. +-commutative N/A

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

   3. associate-+l+ N/A

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

   4. *-rgt-identity N/A

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

   5. distribute-lft-in N/A

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

   6. +-commutative N/A

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

   7. associate-+r+ N/A

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

   8. +-commutative N/A

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

   9. *-lft-identity N/A

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

   10. distribute-rgt-in N/A

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

   11. +-commutative N/A

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

   12. associate-/l* N/A

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

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

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

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

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

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

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

   16. +-commutative N/A

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

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

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

   18. associate-+l+ N/A

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

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

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

   20. +-lowering-+.f64 99.8%

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

4. Applied egg-rr 99.8%

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

   1. associate-/l/ N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. times-frac N/A

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

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

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

6. Applied egg-rr 99.8%

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

Alternative 10: 92.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.3:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + 3}}{\left(\alpha + 2\right) \cdot \left(\alpha + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 3}}{\alpha + \left(2 + \beta\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.3)
   (/ (/ (+ 1.0 alpha) (+ alpha 3.0)) (* (+ alpha 2.0) (+ alpha 2.0)))
   (/ (/ (+ 1.0 alpha) (+ (+ alpha beta) 3.0)) (+ alpha (+ 2.0 beta)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.3) {
		tmp = ((1.0 + alpha) / (alpha + 3.0)) / ((alpha + 2.0) * (alpha + 2.0));
	} else {
		tmp = ((1.0 + alpha) / ((alpha + beta) + 3.0)) / (alpha + (2.0 + beta));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.3d0) then
        tmp = ((1.0d0 + alpha) / (alpha + 3.0d0)) / ((alpha + 2.0d0) * (alpha + 2.0d0))
    else
        tmp = ((1.0d0 + alpha) / ((alpha + beta) + 3.0d0)) / (alpha + (2.0d0 + beta))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.3) {
		tmp = ((1.0 + alpha) / (alpha + 3.0)) / ((alpha + 2.0) * (alpha + 2.0));
	} else {
		tmp = ((1.0 + alpha) / ((alpha + beta) + 3.0)) / (alpha + (2.0 + beta));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 1.3:
		tmp = ((1.0 + alpha) / (alpha + 3.0)) / ((alpha + 2.0) * (alpha + 2.0))
	else:
		tmp = ((1.0 + alpha) / ((alpha + beta) + 3.0)) / (alpha + (2.0 + beta))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.3)
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + 3.0)) / Float64(Float64(alpha + 2.0) * Float64(alpha + 2.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(Float64(alpha + beta) + 3.0)) / Float64(alpha + Float64(2.0 + beta)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.3)
		tmp = ((1.0 + alpha) / (alpha + 3.0)) / ((alpha + 2.0) * (alpha + 2.0));
	else
		tmp = ((1.0 + alpha) / ((alpha + beta) + 3.0)) / (alpha + (2.0 + beta));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 1.3], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + 2.0), $MachinePrecision] * N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.30000000000000004

   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 96.7%

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

   5. 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 94.2%

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

   7. Simplified 94.2%

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

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

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

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

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

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

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

      6. +-commutative N/A

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

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

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

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

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

      9. +-commutative N/A

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

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

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

      11. +-commutative N/A

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

      12. +-lowering-+.f64 96.7%

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

   9. Applied egg-rr 96.7%

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

   if 1.30000000000000004 < 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. Add Preprocessing

   3. Taylor expanded in beta around inf

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

      1. +-lowering-+.f64 78.5%

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

   5. Simplified 78.5%

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

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

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

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

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

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

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

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

      6. metadata-eval N/A

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

      7. associate-+l+ N/A

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

      8. metadata-eval N/A

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

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

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

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

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

      11. metadata-eval N/A

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

      12. associate-+r+ N/A

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

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

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

      14. +-commutative N/A

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

      15. +-lowering-+.f64 78.5%

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

   7. Applied egg-rr 78.5%

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

Alternative 11: 92.0% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.3)
   (/ (/ (+ 1.0 alpha) (+ alpha 3.0)) (* (+ alpha 2.0) (+ alpha 2.0)))
   (/ (/ (+ 1.0 alpha) (+ (+ alpha beta) 3.0)) beta)))

C

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

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 3.3d0) then
        tmp = ((1.0d0 + alpha) / (alpha + 3.0d0)) / ((alpha + 2.0d0) * (alpha + 2.0d0))
    else
        tmp = ((1.0d0 + alpha) / ((alpha + beta) + 3.0d0)) / beta
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.3)
		tmp = ((1.0 + alpha) / (alpha + 3.0)) / ((alpha + 2.0) * (alpha + 2.0));
	else
		tmp = ((1.0 + alpha) / ((alpha + beta) + 3.0)) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 3.3], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + 2.0), $MachinePrecision] * N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.2999999999999998

   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 96.7%

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

   5. 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 94.2%

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

   7. Simplified 94.2%

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

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

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

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

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

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

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

      6. +-commutative N/A

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

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

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

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

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

      9. +-commutative N/A

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

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

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

      11. +-commutative N/A

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

      12. +-lowering-+.f64 96.7%

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

   9. Applied egg-rr 96.7%

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

   if 3.2999999999999998 < 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. Add Preprocessing

   3. Taylor expanded in beta around inf

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

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

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

      2. +-lowering-+.f64 77.7%

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

   5. Simplified 77.7%

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

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

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

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

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

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

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

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

      6. metadata-eval N/A

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

      7. associate-+l+ N/A

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

      8. metadata-eval N/A

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

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

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

      10. +-lowering-+.f64 77.7%

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

   7. Applied egg-rr 77.7%

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

Alternative 12: 88.6% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.3)
   (/ (+ 1.0 alpha) (* (* (+ alpha 2.0) (+ alpha 2.0)) (+ alpha 3.0)))
   (/ (/ (+ 1.0 alpha) (+ (+ alpha beta) 3.0)) beta)))

C

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

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 3.3d0) then
        tmp = (1.0d0 + alpha) / (((alpha + 2.0d0) * (alpha + 2.0d0)) * (alpha + 3.0d0))
    else
        tmp = ((1.0d0 + alpha) / ((alpha + beta) + 3.0d0)) / beta
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.3)
		tmp = (1.0 + alpha) / (((alpha + 2.0) * (alpha + 2.0)) * (alpha + 3.0));
	else
		tmp = ((1.0 + alpha) / ((alpha + beta) + 3.0)) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 3.3], N[(N[(1.0 + alpha), $MachinePrecision] / N[(N[(N[(alpha + 2.0), $MachinePrecision] * N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision] * N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.2999999999999998

   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 96.7%

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

   5. 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 94.2%

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

   7. Simplified 94.2%

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

   if 3.2999999999999998 < 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. Add Preprocessing

   3. Taylor expanded in beta around inf

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

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

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

      2. +-lowering-+.f64 77.7%

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

   5. Simplified 77.7%

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

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

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

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

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

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

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

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

      6. metadata-eval N/A

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

      7. associate-+l+ N/A

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

      8. metadata-eval N/A

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

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

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

      10. +-lowering-+.f64 77.7%

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

   7. Applied egg-rr 77.7%

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

4. Final simplification 89.7%

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

Alternative 13: 71.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.7:\\ \;\;\;\;0.08333333333333333 + \beta \cdot \left(\beta \cdot \left(\beta \cdot 0.024691358024691357 + -0.011574074074074073\right) + -0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 3}}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.7)
   (+
    0.08333333333333333
    (*
     beta
     (+
      (* beta (+ (* beta 0.024691358024691357) -0.011574074074074073))
      -0.027777777777777776)))
   (/ (/ (+ 1.0 alpha) (+ (+ alpha beta) 3.0)) beta)))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.7) {
		tmp = 0.08333333333333333 + (beta * ((beta * ((beta * 0.024691358024691357) + -0.011574074074074073)) + -0.027777777777777776));
	} else {
		tmp = ((1.0 + alpha) / ((alpha + beta) + 3.0)) / beta;
	}
	return tmp;
}

Fortran

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 * ((beta * 0.024691358024691357d0) + (-0.011574074074074073d0))) + (-0.027777777777777776d0)))
    else
        tmp = ((1.0d0 + alpha) / ((alpha + beta) + 3.0d0)) / beta
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.7) {
		tmp = 0.08333333333333333 + (beta * ((beta * ((beta * 0.024691358024691357) + -0.011574074074074073)) + -0.027777777777777776));
	} else {
		tmp = ((1.0 + alpha) / ((alpha + beta) + 3.0)) / beta;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 1.7:
		tmp = 0.08333333333333333 + (beta * ((beta * ((beta * 0.024691358024691357) + -0.011574074074074073)) + -0.027777777777777776))
	else:
		tmp = ((1.0 + alpha) / ((alpha + beta) + 3.0)) / beta
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.7)
		tmp = Float64(0.08333333333333333 + Float64(beta * Float64(Float64(beta * Float64(Float64(beta * 0.024691358024691357) + -0.011574074074074073)) + -0.027777777777777776)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(Float64(alpha + beta) + 3.0)) / beta);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.7)
		tmp = 0.08333333333333333 + (beta * ((beta * ((beta * 0.024691358024691357) + -0.011574074074074073)) + -0.027777777777777776));
	else
		tmp = ((1.0 + alpha) / ((alpha + beta) + 3.0)) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 1.7], N[(0.08333333333333333 + N[(beta * N[(N[(beta * N[(N[(beta * 0.024691358024691357), $MachinePrecision] + -0.011574074074074073), $MachinePrecision]), $MachinePrecision] + -0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision]), $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 96.7%

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

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

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

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

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

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

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

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

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

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), 6\right), \left(\left(\alpha + \color{blue}{\beta}\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), 6\right), \mathsf{*.f64}\left(\left(\alpha + \beta\right), \color{blue}{\left(\left(\alpha + \beta\right) + 2\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), 6\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \left(\color{blue}{\left(\alpha + \beta\right)} + 2\right)\right)\right)\right)\right) \]

      12. associate-+l+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), 6\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \left(\alpha + \color{blue}{\left(\beta + 2\right)}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), 6\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{+.f64}\left(\alpha, \color{blue}{\left(\beta + 2\right)}\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 96.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), 6\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, \color{blue}{2}\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 96.7%

      \[\leadsto \frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \color{blue}{\left(\left(\left(\alpha + \beta\right) \cdot 3 + 6\right) + \left(\alpha + \beta\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)}} \]

   7. Taylor expanded in alpha around 0

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

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{1 + \beta}{2 + \beta}}{\color{blue}{6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 + \beta}{2 + \beta}\right), \color{blue}{\left(6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + \beta\right), \left(2 + \beta\right)\right), \left(\color{blue}{6} + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\beta + 1\right), \left(2 + \beta\right)\right), \left(6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \left(2 + \beta\right)\right), \left(6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \left(\beta + 2\right)\right), \left(6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)\right) \]

      8. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(\left(6 + 3 \cdot \beta\right) + \color{blue}{\beta \cdot \left(2 + \beta\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\left(6 + 3 \cdot \beta\right), \color{blue}{\left(\beta \cdot \left(2 + \beta\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(6, \left(3 \cdot \beta\right)\right), \left(\color{blue}{\beta} \cdot \left(2 + \beta\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(6, \mathsf{*.f64}\left(3, \beta\right)\right), \left(\beta \cdot \left(2 + \beta\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(6, \mathsf{*.f64}\left(3, \beta\right)\right), \mathsf{*.f64}\left(\beta, \color{blue}{\left(2 + \beta\right)}\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(6, \mathsf{*.f64}\left(3, \beta\right)\right), \mathsf{*.f64}\left(\beta, \left(\beta + \color{blue}{2}\right)\right)\right)\right) \]

      14. +-lowering-+.f64 69.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(6, \mathsf{*.f64}\left(3, \beta\right)\right), \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(\beta, \color{blue}{2}\right)\right)\right)\right) \]

   9. Simplified 69.7%

      \[\leadsto \color{blue}{\frac{\frac{\beta + 1}{\beta + 2}}{\left(6 + 3 \cdot \beta\right) + \beta \cdot \left(\beta + 2\right)}} \]

   10. Taylor expanded in beta around 0

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

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \color{blue}{\left(\beta \cdot \left(\beta \cdot \left(\frac{2}{81} \cdot \beta - \frac{5}{432}\right) - \frac{1}{36}\right)\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \color{blue}{\left(\beta \cdot \left(\frac{2}{81} \cdot \beta - \frac{5}{432}\right) - \frac{1}{36}\right)}\right)\right) \]

       3. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \left(\beta \cdot \left(\frac{2}{81} \cdot \beta - \frac{5}{432}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{36}\right)\right)}\right)\right)\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \left(\beta \cdot \left(\frac{2}{81} \cdot \beta - \frac{5}{432}\right) + \frac{-1}{36}\right)\right)\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(\left(\beta \cdot \left(\frac{2}{81} \cdot \beta - \frac{5}{432}\right)\right), \color{blue}{\frac{-1}{36}}\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\beta, \left(\frac{2}{81} \cdot \beta - \frac{5}{432}\right)\right), \frac{-1}{36}\right)\right)\right) \]

       7. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\beta, \left(\frac{2}{81} \cdot \beta + \left(\mathsf{neg}\left(\frac{5}{432}\right)\right)\right)\right), \frac{-1}{36}\right)\right)\right) \]

       8. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\beta, \left(\frac{2}{81} \cdot \beta + \frac{-5}{432}\right)\right), \frac{-1}{36}\right)\right)\right) \]

       9. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(\left(\frac{2}{81} \cdot \beta\right), \frac{-5}{432}\right)\right), \frac{-1}{36}\right)\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(\left(\beta \cdot \frac{2}{81}\right), \frac{-5}{432}\right)\right), \frac{-1}{36}\right)\right)\right) \]

       11. *-lowering-*.f64 69.4%

           \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\beta, \frac{2}{81}\right), \frac{-5}{432}\right)\right), \frac{-1}{36}\right)\right)\right) \]

   12. Simplified 69.4%

       \[\leadsto \color{blue}{0.08333333333333333 + \beta \cdot \left(\beta \cdot \left(\beta \cdot 0.024691358024691357 + -0.011574074074074073\right) + -0.027777777777777776\right)} \]

   if 1.69999999999999996 < 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. Add Preprocessing

   3. Taylor expanded in beta around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1 + \alpha}{\beta}\right)}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + \alpha\right), \beta\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)}, 1\right)\right) \]

      2. +-lowering-+.f64 77.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \beta\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(\alpha, \beta\right)}, \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

   5. Simplified 77.7%

      \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\beta}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   6. Step-by-step derivation

      1. associate-/l/ N/A

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \beta}} \]

      2. associate-/r* N/A

         \[\leadsto \frac{\frac{1 + \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}}{\color{blue}{\beta}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 + \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\right), \color{blue}{\beta}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + \alpha\right), \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)\right), \beta\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)\right), \beta\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left(\left(\left(\alpha + \beta\right) + 2\right) + 1\right)\right), \beta\right) \]

      7. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left(\left(\alpha + \beta\right) + \left(2 + 1\right)\right)\right), \beta\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left(\left(\alpha + \beta\right) + 3\right)\right), \beta\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{+.f64}\left(\left(\alpha + \beta\right), 3\right)\right), \beta\right) \]

      10. +-lowering-+.f64 77.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right)\right), \beta\right) \]

   7. Applied egg-rr 77.7%

      \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 3}}{\beta}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 71.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.1:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot \left(-0.027777777777777776 + \alpha \cdot \left(-0.011574074074074073 + \alpha \cdot 0.024691358024691357\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 3}}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.1)
   (+
    0.08333333333333333
    (*
     alpha
     (+
      -0.027777777777777776
      (* alpha (+ -0.011574074074074073 (* alpha 0.024691358024691357))))))
   (/ (/ (+ 1.0 alpha) (+ (+ alpha beta) 3.0)) beta)))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.1) {
		tmp = 0.08333333333333333 + (alpha * (-0.027777777777777776 + (alpha * (-0.011574074074074073 + (alpha * 0.024691358024691357)))));
	} else {
		tmp = ((1.0 + alpha) / ((alpha + beta) + 3.0)) / beta;
	}
	return tmp;
}

Fortran

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) + (alpha * ((-0.011574074074074073d0) + (alpha * 0.024691358024691357d0)))))
    else
        tmp = ((1.0d0 + alpha) / ((alpha + beta) + 3.0d0)) / beta
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.1) {
		tmp = 0.08333333333333333 + (alpha * (-0.027777777777777776 + (alpha * (-0.011574074074074073 + (alpha * 0.024691358024691357)))));
	} else {
		tmp = ((1.0 + alpha) / ((alpha + beta) + 3.0)) / beta;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.1:
		tmp = 0.08333333333333333 + (alpha * (-0.027777777777777776 + (alpha * (-0.011574074074074073 + (alpha * 0.024691358024691357)))))
	else:
		tmp = ((1.0 + alpha) / ((alpha + beta) + 3.0)) / beta
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.1)
		tmp = Float64(0.08333333333333333 + Float64(alpha * Float64(-0.027777777777777776 + Float64(alpha * Float64(-0.011574074074074073 + Float64(alpha * 0.024691358024691357))))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(Float64(alpha + beta) + 3.0)) / beta);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.1)
		tmp = 0.08333333333333333 + (alpha * (-0.027777777777777776 + (alpha * (-0.011574074074074073 + (alpha * 0.024691358024691357)))));
	else
		tmp = ((1.0 + alpha) / ((alpha + beta) + 3.0)) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.1], N[(0.08333333333333333 + N[(alpha * N[(-0.027777777777777776 + N[(alpha * N[(-0.011574074074074073 + N[(alpha * 0.024691358024691357), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] / 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 96.7%

      \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)\right)}} \]
   4. Add Preprocessing

   5. 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 94.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \alpha\right), \mathsf{+.f64}\left(2, \alpha\right)\right), \mathsf{+.f64}\left(3, \color{blue}{\alpha}\right)\right)\right) \]

   7. Simplified 94.2%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(\left(2 + \alpha\right) \cdot \left(2 + \alpha\right)\right) \cdot \left(3 + \alpha\right)}} \]

   8. Taylor expanded in alpha around 0

      \[\leadsto \color{blue}{\frac{1}{12} + \alpha \cdot \left(\alpha \cdot \left(\frac{2}{81} \cdot \alpha - \frac{5}{432}\right) - \frac{1}{36}\right)} \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \color{blue}{\left(\alpha \cdot \left(\alpha \cdot \left(\frac{2}{81} \cdot \alpha - \frac{5}{432}\right) - \frac{1}{36}\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\alpha, \color{blue}{\left(\alpha \cdot \left(\frac{2}{81} \cdot \alpha - \frac{5}{432}\right) - \frac{1}{36}\right)}\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\alpha, \left(\alpha \cdot \left(\frac{2}{81} \cdot \alpha - \frac{5}{432}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{36}\right)\right)}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\alpha, \left(\alpha \cdot \left(\frac{2}{81} \cdot \alpha - \frac{5}{432}\right) + \frac{-1}{36}\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\alpha, \mathsf{+.f64}\left(\left(\alpha \cdot \left(\frac{2}{81} \cdot \alpha - \frac{5}{432}\right)\right), \color{blue}{\frac{-1}{36}}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\alpha, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\alpha, \left(\frac{2}{81} \cdot \alpha - \frac{5}{432}\right)\right), \frac{-1}{36}\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\alpha, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\alpha, \left(\frac{2}{81} \cdot \alpha + \left(\mathsf{neg}\left(\frac{5}{432}\right)\right)\right)\right), \frac{-1}{36}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\alpha, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\alpha, \left(\frac{2}{81} \cdot \alpha + \frac{-5}{432}\right)\right), \frac{-1}{36}\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\alpha, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\alpha, \mathsf{+.f64}\left(\left(\frac{2}{81} \cdot \alpha\right), \frac{-5}{432}\right)\right), \frac{-1}{36}\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\alpha, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\alpha, \mathsf{+.f64}\left(\left(\alpha \cdot \frac{2}{81}\right), \frac{-5}{432}\right)\right), \frac{-1}{36}\right)\right)\right) \]

      11. *-lowering-*.f64 67.5%

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\alpha, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\alpha, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\alpha, \frac{2}{81}\right), \frac{-5}{432}\right)\right), \frac{-1}{36}\right)\right)\right) \]

   10. Simplified 67.5%

       \[\leadsto \color{blue}{0.08333333333333333 + \alpha \cdot \left(\alpha \cdot \left(\alpha \cdot 0.024691358024691357 + -0.011574074074074073\right) + -0.027777777777777776\right)} \]

   if 2.10000000000000009 < 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. Add Preprocessing

   3. Taylor expanded in beta around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1 + \alpha}{\beta}\right)}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + \alpha\right), \beta\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)}, 1\right)\right) \]

      2. +-lowering-+.f64 77.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \beta\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(\alpha, \beta\right)}, \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

   5. Simplified 77.7%

      \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\beta}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   6. Step-by-step derivation

      1. associate-/l/ N/A

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \beta}} \]

      2. associate-/r* N/A

         \[\leadsto \frac{\frac{1 + \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}}{\color{blue}{\beta}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 + \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\right), \color{blue}{\beta}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + \alpha\right), \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)\right), \beta\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)\right), \beta\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left(\left(\left(\alpha + \beta\right) + 2\right) + 1\right)\right), \beta\right) \]

      7. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left(\left(\alpha + \beta\right) + \left(2 + 1\right)\right)\right), \beta\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left(\left(\alpha + \beta\right) + 3\right)\right), \beta\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{+.f64}\left(\left(\alpha + \beta\right), 3\right)\right), \beta\right) \]

      10. +-lowering-+.f64 77.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right)\right), \beta\right) \]

   7. Applied egg-rr 77.7%

      \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 3}}{\beta}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.1:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot \left(-0.027777777777777776 + \alpha \cdot \left(-0.011574074074074073 + \alpha \cdot 0.024691358024691357\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 3}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 71.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.8:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{elif}\;\beta \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.8)
   (+ 0.08333333333333333 (* beta -0.027777777777777776))
   (if (<= beta 1.35e+154)
     (/ (+ 1.0 alpha) (* beta beta))
     (/ (/ alpha beta) beta))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.8) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else if (beta <= 1.35e+154) {
		tmp = (1.0 + alpha) / (beta * beta);
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.8d0) then
        tmp = 0.08333333333333333d0 + (beta * (-0.027777777777777776d0))
    else if (beta <= 1.35d+154) then
        tmp = (1.0d0 + alpha) / (beta * beta)
    else
        tmp = (alpha / beta) / beta
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.8) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else if (beta <= 1.35e+154) {
		tmp = (1.0 + alpha) / (beta * beta);
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.8:
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776)
	elif beta <= 1.35e+154:
		tmp = (1.0 + alpha) / (beta * beta)
	else:
		tmp = (alpha / beta) / beta
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.8)
		tmp = Float64(0.08333333333333333 + Float64(beta * -0.027777777777777776));
	elseif (beta <= 1.35e+154)
		tmp = Float64(Float64(1.0 + alpha) / Float64(beta * beta));
	else
		tmp = Float64(Float64(alpha / beta) / beta);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.8)
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	elseif (beta <= 1.35e+154)
		tmp = (1.0 + alpha) / (beta * beta);
	else
		tmp = (alpha / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.8], N[(0.08333333333333333 + N[(beta * -0.027777777777777776), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 1.35e+154], N[(N[(1.0 + alpha), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / beta), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if beta < 2.7999999999999998

   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 96.7%

      \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \left(\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(3 + \color{blue}{\left(\alpha + \beta\right)}\right)\right)\right)\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \left(3 \cdot \left(\left(\alpha + \beta\right) + 2\right) + \color{blue}{\left(\alpha + \beta\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)}\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\left(3 \cdot \left(\left(\alpha + \beta\right) + 2\right)\right), \color{blue}{\left(\left(\alpha + \beta\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)\right)}\right)\right)\right) \]

      4. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\left(3 \cdot \left(\alpha + \beta\right) + 3 \cdot 2\right), \left(\color{blue}{\left(\alpha + \beta\right)} \cdot \left(\left(\alpha + \beta\right) + 2\right)\right)\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\left(\left(\alpha + \beta\right) \cdot 3 + 3 \cdot 2\right), \left(\left(\color{blue}{\alpha} + \beta\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\left(\left(\alpha + \beta\right) \cdot 3\right), \left(3 \cdot 2\right)\right), \left(\color{blue}{\left(\alpha + \beta\right)} \cdot \left(\left(\alpha + \beta\right) + 2\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\alpha + \beta\right), 3\right), \left(3 \cdot 2\right)\right), \left(\left(\color{blue}{\alpha} + \beta\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), \left(3 \cdot 2\right)\right), \left(\left(\alpha + \beta\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)\right)\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), 6\right), \left(\left(\alpha + \color{blue}{\beta}\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), 6\right), \mathsf{*.f64}\left(\left(\alpha + \beta\right), \color{blue}{\left(\left(\alpha + \beta\right) + 2\right)}\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), 6\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \left(\color{blue}{\left(\alpha + \beta\right)} + 2\right)\right)\right)\right)\right) \]

      12. associate-+l+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), 6\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \left(\alpha + \color{blue}{\left(\beta + 2\right)}\right)\right)\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), 6\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{+.f64}\left(\alpha, \color{blue}{\left(\beta + 2\right)}\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 96.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), 6\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, \color{blue}{2}\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 96.7%

      \[\leadsto \frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \color{blue}{\left(\left(\left(\alpha + \beta\right) \cdot 3 + 6\right) + \left(\alpha + \beta\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)}} \]

   7. Taylor expanded in alpha around 0

      \[\leadsto \color{blue}{\frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)}} \]
   8. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{1 + \beta}{2 + \beta}}{\color{blue}{6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 + \beta}{2 + \beta}\right), \color{blue}{\left(6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + \beta\right), \left(2 + \beta\right)\right), \left(\color{blue}{6} + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\beta + 1\right), \left(2 + \beta\right)\right), \left(6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \left(2 + \beta\right)\right), \left(6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \left(\beta + 2\right)\right), \left(6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)\right) \]

      8. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(\left(6 + 3 \cdot \beta\right) + \color{blue}{\beta \cdot \left(2 + \beta\right)}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\left(6 + 3 \cdot \beta\right), \color{blue}{\left(\beta \cdot \left(2 + \beta\right)\right)}\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(6, \left(3 \cdot \beta\right)\right), \left(\color{blue}{\beta} \cdot \left(2 + \beta\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(6, \mathsf{*.f64}\left(3, \beta\right)\right), \left(\beta \cdot \left(2 + \beta\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(6, \mathsf{*.f64}\left(3, \beta\right)\right), \mathsf{*.f64}\left(\beta, \color{blue}{\left(2 + \beta\right)}\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(6, \mathsf{*.f64}\left(3, \beta\right)\right), \mathsf{*.f64}\left(\beta, \left(\beta + \color{blue}{2}\right)\right)\right)\right) \]

      14. +-lowering-+.f64 69.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(6, \mathsf{*.f64}\left(3, \beta\right)\right), \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(\beta, \color{blue}{2}\right)\right)\right)\right) \]

   9. Simplified 69.7%

      \[\leadsto \color{blue}{\frac{\frac{\beta + 1}{\beta + 2}}{\left(6 + 3 \cdot \beta\right) + \beta \cdot \left(\beta + 2\right)}} \]

   10. Taylor expanded in beta around 0

       \[\leadsto \color{blue}{\frac{1}{12} + \frac{-1}{36} \cdot \beta} \]
   11. 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 68.6%

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \color{blue}{\frac{-1}{36}}\right)\right) \]

   12. Simplified 68.6%

       \[\leadsto \color{blue}{0.08333333333333333 + \beta \cdot -0.027777777777777776} \]

   if 2.7999999999999998 < beta < 1.35000000000000003e154

   1. Initial program 91.9%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 57.6%

      \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \alpha\right), \color{blue}{\left({\beta}^{2}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left({\color{blue}{\beta}}^{2}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left(\beta \cdot \color{blue}{\beta}\right)\right) \]

      4. *-lowering-*.f64 67.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\beta, \color{blue}{\beta}\right)\right) \]

   7. Simplified 67.5%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]

   if 1.35000000000000003e154 < beta

   1. Initial program 60.0%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 50.6%

      \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \alpha\right), \color{blue}{\left({\beta}^{2}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left({\color{blue}{\beta}}^{2}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left(\beta \cdot \color{blue}{\beta}\right)\right) \]

      4. *-lowering-*.f64 78.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\beta, \color{blue}{\beta}\right)\right) \]

   7. Simplified 78.6%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]
   8. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{1 + \alpha}{\beta}}{\color{blue}{\beta}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 + \alpha}{\beta}\right), \color{blue}{\beta}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + \alpha\right), \beta\right), \beta\right) \]

      4. +-lowering-+.f64 88.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \beta\right), \beta\right) \]

   9. Applied egg-rr 88.9%

      \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\beta}}{\beta}} \]

   10. Taylor expanded in alpha around inf

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\color{blue}{\alpha}, \beta\right), \beta\right) \]
   11. Step-by-step derivation

   12. Simplified 88.9%

       \[\leadsto \frac{\frac{\color{blue}{\alpha}}{\beta}}{\beta} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 71.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.5:\\ \;\;\;\;0.08333333333333333 + \beta \cdot \left(-0.027777777777777776 + \beta \cdot -0.011574074074074073\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 3}}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.5)
   (+
    0.08333333333333333
    (* beta (+ -0.027777777777777776 (* beta -0.011574074074074073))))
   (/ (/ (+ 1.0 alpha) (+ (+ alpha beta) 3.0)) beta)))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.5) {
		tmp = 0.08333333333333333 + (beta * (-0.027777777777777776 + (beta * -0.011574074074074073)));
	} else {
		tmp = ((1.0 + alpha) / ((alpha + beta) + 3.0)) / beta;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.5d0) then
        tmp = 0.08333333333333333d0 + (beta * ((-0.027777777777777776d0) + (beta * (-0.011574074074074073d0))))
    else
        tmp = ((1.0d0 + alpha) / ((alpha + beta) + 3.0d0)) / beta
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.5) {
		tmp = 0.08333333333333333 + (beta * (-0.027777777777777776 + (beta * -0.011574074074074073)));
	} else {
		tmp = ((1.0 + alpha) / ((alpha + beta) + 3.0)) / beta;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 1.5:
		tmp = 0.08333333333333333 + (beta * (-0.027777777777777776 + (beta * -0.011574074074074073)))
	else:
		tmp = ((1.0 + alpha) / ((alpha + beta) + 3.0)) / beta
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.5)
		tmp = Float64(0.08333333333333333 + Float64(beta * Float64(-0.027777777777777776 + Float64(beta * -0.011574074074074073))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(Float64(alpha + beta) + 3.0)) / beta);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.5)
		tmp = 0.08333333333333333 + (beta * (-0.027777777777777776 + (beta * -0.011574074074074073)));
	else
		tmp = ((1.0 + alpha) / ((alpha + beta) + 3.0)) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 1.5], N[(0.08333333333333333 + N[(beta * N[(-0.027777777777777776 + N[(beta * -0.011574074074074073), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.5

   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 96.7%

      \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \left(\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(3 + \color{blue}{\left(\alpha + \beta\right)}\right)\right)\right)\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \left(3 \cdot \left(\left(\alpha + \beta\right) + 2\right) + \color{blue}{\left(\alpha + \beta\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)}\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\left(3 \cdot \left(\left(\alpha + \beta\right) + 2\right)\right), \color{blue}{\left(\left(\alpha + \beta\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)\right)}\right)\right)\right) \]

      4. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\left(3 \cdot \left(\alpha + \beta\right) + 3 \cdot 2\right), \left(\color{blue}{\left(\alpha + \beta\right)} \cdot \left(\left(\alpha + \beta\right) + 2\right)\right)\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\left(\left(\alpha + \beta\right) \cdot 3 + 3 \cdot 2\right), \left(\left(\color{blue}{\alpha} + \beta\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\left(\left(\alpha + \beta\right) \cdot 3\right), \left(3 \cdot 2\right)\right), \left(\color{blue}{\left(\alpha + \beta\right)} \cdot \left(\left(\alpha + \beta\right) + 2\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\alpha + \beta\right), 3\right), \left(3 \cdot 2\right)\right), \left(\left(\color{blue}{\alpha} + \beta\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), \left(3 \cdot 2\right)\right), \left(\left(\alpha + \beta\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)\right)\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), 6\right), \left(\left(\alpha + \color{blue}{\beta}\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), 6\right), \mathsf{*.f64}\left(\left(\alpha + \beta\right), \color{blue}{\left(\left(\alpha + \beta\right) + 2\right)}\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), 6\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \left(\color{blue}{\left(\alpha + \beta\right)} + 2\right)\right)\right)\right)\right) \]

      12. associate-+l+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), 6\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \left(\alpha + \color{blue}{\left(\beta + 2\right)}\right)\right)\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), 6\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{+.f64}\left(\alpha, \color{blue}{\left(\beta + 2\right)}\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 96.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), 6\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, \color{blue}{2}\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 96.7%

      \[\leadsto \frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \color{blue}{\left(\left(\left(\alpha + \beta\right) \cdot 3 + 6\right) + \left(\alpha + \beta\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)}} \]

   7. Taylor expanded in alpha around 0

      \[\leadsto \color{blue}{\frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)}} \]
   8. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{1 + \beta}{2 + \beta}}{\color{blue}{6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 + \beta}{2 + \beta}\right), \color{blue}{\left(6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + \beta\right), \left(2 + \beta\right)\right), \left(\color{blue}{6} + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\beta + 1\right), \left(2 + \beta\right)\right), \left(6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \left(2 + \beta\right)\right), \left(6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \left(\beta + 2\right)\right), \left(6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)\right) \]

      8. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(\left(6 + 3 \cdot \beta\right) + \color{blue}{\beta \cdot \left(2 + \beta\right)}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\left(6 + 3 \cdot \beta\right), \color{blue}{\left(\beta \cdot \left(2 + \beta\right)\right)}\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(6, \left(3 \cdot \beta\right)\right), \left(\color{blue}{\beta} \cdot \left(2 + \beta\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(6, \mathsf{*.f64}\left(3, \beta\right)\right), \left(\beta \cdot \left(2 + \beta\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(6, \mathsf{*.f64}\left(3, \beta\right)\right), \mathsf{*.f64}\left(\beta, \color{blue}{\left(2 + \beta\right)}\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(6, \mathsf{*.f64}\left(3, \beta\right)\right), \mathsf{*.f64}\left(\beta, \left(\beta + \color{blue}{2}\right)\right)\right)\right) \]

      14. +-lowering-+.f64 69.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(6, \mathsf{*.f64}\left(3, \beta\right)\right), \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(\beta, \color{blue}{2}\right)\right)\right)\right) \]

   9. Simplified 69.7%

      \[\leadsto \color{blue}{\frac{\frac{\beta + 1}{\beta + 2}}{\left(6 + 3 \cdot \beta\right) + \beta \cdot \left(\beta + 2\right)}} \]

   10. Taylor expanded in beta around 0

       \[\leadsto \color{blue}{\frac{1}{12} + \beta \cdot \left(\frac{-5}{432} \cdot \beta - \frac{1}{36}\right)} \]
   11. 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 69.0%

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\beta, \frac{-5}{432}\right), \frac{-1}{36}\right)\right)\right) \]

   12. Simplified 69.0%

       \[\leadsto \color{blue}{0.08333333333333333 + \beta \cdot \left(\beta \cdot -0.011574074074074073 + -0.027777777777777776\right)} \]

   if 1.5 < 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. Add Preprocessing

   3. Taylor expanded in beta around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1 + \alpha}{\beta}\right)}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + \alpha\right), \beta\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)}, 1\right)\right) \]

      2. +-lowering-+.f64 77.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \beta\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(\alpha, \beta\right)}, \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

   5. Simplified 77.7%

      \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\beta}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   6. Step-by-step derivation

      1. associate-/l/ N/A

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \beta}} \]

      2. associate-/r* N/A

         \[\leadsto \frac{\frac{1 + \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}}{\color{blue}{\beta}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 + \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\right), \color{blue}{\beta}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + \alpha\right), \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)\right), \beta\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)\right), \beta\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left(\left(\left(\alpha + \beta\right) + 2\right) + 1\right)\right), \beta\right) \]

      7. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left(\left(\alpha + \beta\right) + \left(2 + 1\right)\right)\right), \beta\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left(\left(\alpha + \beta\right) + 3\right)\right), \beta\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{+.f64}\left(\left(\alpha + \beta\right), 3\right)\right), \beta\right) \]

      10. +-lowering-+.f64 77.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right)\right), \beta\right) \]

   7. Applied egg-rr 77.7%

      \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 3}}{\beta}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.5:\\ \;\;\;\;0.08333333333333333 + \beta \cdot \left(-0.027777777777777776 + \beta \cdot -0.011574074074074073\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 3}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 70.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.8:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{elif}\;\beta \leq 2.2 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.8)
   (+ 0.08333333333333333 (* beta -0.027777777777777776))
   (if (<= beta 2.2e+154) (/ (/ 1.0 beta) beta) (/ (/ alpha beta) beta))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.8) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else if (beta <= 2.2e+154) {
		tmp = (1.0 / beta) / beta;
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.8d0) then
        tmp = 0.08333333333333333d0 + (beta * (-0.027777777777777776d0))
    else if (beta <= 2.2d+154) then
        tmp = (1.0d0 / beta) / beta
    else
        tmp = (alpha / beta) / beta
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.8) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else if (beta <= 2.2e+154) {
		tmp = (1.0 / beta) / beta;
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.8:
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776)
	elif beta <= 2.2e+154:
		tmp = (1.0 / beta) / beta
	else:
		tmp = (alpha / beta) / beta
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.8)
		tmp = Float64(0.08333333333333333 + Float64(beta * -0.027777777777777776));
	elseif (beta <= 2.2e+154)
		tmp = Float64(Float64(1.0 / beta) / beta);
	else
		tmp = Float64(Float64(alpha / beta) / beta);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.8)
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	elseif (beta <= 2.2e+154)
		tmp = (1.0 / beta) / beta;
	else
		tmp = (alpha / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.8], N[(0.08333333333333333 + N[(beta * -0.027777777777777776), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 2.2e+154], N[(N[(1.0 / beta), $MachinePrecision] / beta), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / beta), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if beta < 2.7999999999999998

   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 96.7%

      \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \left(\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(3 + \color{blue}{\left(\alpha + \beta\right)}\right)\right)\right)\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \left(3 \cdot \left(\left(\alpha + \beta\right) + 2\right) + \color{blue}{\left(\alpha + \beta\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)}\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\left(3 \cdot \left(\left(\alpha + \beta\right) + 2\right)\right), \color{blue}{\left(\left(\alpha + \beta\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)\right)}\right)\right)\right) \]

      4. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\left(3 \cdot \left(\alpha + \beta\right) + 3 \cdot 2\right), \left(\color{blue}{\left(\alpha + \beta\right)} \cdot \left(\left(\alpha + \beta\right) + 2\right)\right)\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\left(\left(\alpha + \beta\right) \cdot 3 + 3 \cdot 2\right), \left(\left(\color{blue}{\alpha} + \beta\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\left(\left(\alpha + \beta\right) \cdot 3\right), \left(3 \cdot 2\right)\right), \left(\color{blue}{\left(\alpha + \beta\right)} \cdot \left(\left(\alpha + \beta\right) + 2\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\alpha + \beta\right), 3\right), \left(3 \cdot 2\right)\right), \left(\left(\color{blue}{\alpha} + \beta\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), \left(3 \cdot 2\right)\right), \left(\left(\alpha + \beta\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)\right)\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), 6\right), \left(\left(\alpha + \color{blue}{\beta}\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), 6\right), \mathsf{*.f64}\left(\left(\alpha + \beta\right), \color{blue}{\left(\left(\alpha + \beta\right) + 2\right)}\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), 6\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \left(\color{blue}{\left(\alpha + \beta\right)} + 2\right)\right)\right)\right)\right) \]

      12. associate-+l+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), 6\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \left(\alpha + \color{blue}{\left(\beta + 2\right)}\right)\right)\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), 6\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{+.f64}\left(\alpha, \color{blue}{\left(\beta + 2\right)}\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 96.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), 6\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, \color{blue}{2}\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 96.7%

      \[\leadsto \frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \color{blue}{\left(\left(\left(\alpha + \beta\right) \cdot 3 + 6\right) + \left(\alpha + \beta\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)}} \]

   7. Taylor expanded in alpha around 0

      \[\leadsto \color{blue}{\frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)}} \]
   8. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{1 + \beta}{2 + \beta}}{\color{blue}{6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 + \beta}{2 + \beta}\right), \color{blue}{\left(6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + \beta\right), \left(2 + \beta\right)\right), \left(\color{blue}{6} + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\beta + 1\right), \left(2 + \beta\right)\right), \left(6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \left(2 + \beta\right)\right), \left(6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \left(\beta + 2\right)\right), \left(6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)\right) \]

      8. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(\left(6 + 3 \cdot \beta\right) + \color{blue}{\beta \cdot \left(2 + \beta\right)}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\left(6 + 3 \cdot \beta\right), \color{blue}{\left(\beta \cdot \left(2 + \beta\right)\right)}\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(6, \left(3 \cdot \beta\right)\right), \left(\color{blue}{\beta} \cdot \left(2 + \beta\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(6, \mathsf{*.f64}\left(3, \beta\right)\right), \left(\beta \cdot \left(2 + \beta\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(6, \mathsf{*.f64}\left(3, \beta\right)\right), \mathsf{*.f64}\left(\beta, \color{blue}{\left(2 + \beta\right)}\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(6, \mathsf{*.f64}\left(3, \beta\right)\right), \mathsf{*.f64}\left(\beta, \left(\beta + \color{blue}{2}\right)\right)\right)\right) \]

      14. +-lowering-+.f64 69.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(6, \mathsf{*.f64}\left(3, \beta\right)\right), \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(\beta, \color{blue}{2}\right)\right)\right)\right) \]

   9. Simplified 69.7%

      \[\leadsto \color{blue}{\frac{\frac{\beta + 1}{\beta + 2}}{\left(6 + 3 \cdot \beta\right) + \beta \cdot \left(\beta + 2\right)}} \]

   10. Taylor expanded in beta around 0

       \[\leadsto \color{blue}{\frac{1}{12} + \frac{-1}{36} \cdot \beta} \]
   11. 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 68.6%

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \color{blue}{\frac{-1}{36}}\right)\right) \]

   12. Simplified 68.6%

       \[\leadsto \color{blue}{0.08333333333333333 + \beta \cdot -0.027777777777777776} \]

   if 2.7999999999999998 < beta < 2.2000000000000001e154

   1. Initial program 91.9%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 57.6%

      \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \alpha\right), \color{blue}{\left({\beta}^{2}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left({\color{blue}{\beta}}^{2}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left(\beta \cdot \color{blue}{\beta}\right)\right) \]

      4. *-lowering-*.f64 67.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\beta, \color{blue}{\beta}\right)\right) \]

   7. Simplified 67.5%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]
   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. +-lowering-+.f64 67.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \beta\right), \beta\right) \]

   9. Applied egg-rr 67.8%

      \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\beta}}{\beta}} \]

   10. Taylor expanded in alpha around 0

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\color{blue}{1}, \beta\right), \beta\right) \]
   11. Step-by-step derivation

   12. Simplified 63.2%

       \[\leadsto \frac{\frac{\color{blue}{1}}{\beta}}{\beta} \]

   if 2.2000000000000001e154 < beta

   1. Initial program 60.0%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 50.6%

      \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \alpha\right), \color{blue}{\left({\beta}^{2}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left({\color{blue}{\beta}}^{2}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left(\beta \cdot \color{blue}{\beta}\right)\right) \]

      4. *-lowering-*.f64 78.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\beta, \color{blue}{\beta}\right)\right) \]

   7. Simplified 78.6%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]
   8. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{1 + \alpha}{\beta}}{\color{blue}{\beta}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 + \alpha}{\beta}\right), \color{blue}{\beta}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + \alpha\right), \beta\right), \beta\right) \]

      4. +-lowering-+.f64 88.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \beta\right), \beta\right) \]

   9. Applied egg-rr 88.9%

      \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\beta}}{\beta}} \]

   10. Taylor expanded in alpha around inf

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\color{blue}{\alpha}, \beta\right), \beta\right) \]
   11. Step-by-step derivation

   12. Simplified 88.9%

       \[\leadsto \frac{\frac{\color{blue}{\alpha}}{\beta}}{\beta} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 71.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.7:\\ \;\;\;\;0.08333333333333333 + \beta \cdot \left(-0.027777777777777776 + \beta \cdot -0.011574074074074073\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.7)
   (+
    0.08333333333333333
    (* beta (+ -0.027777777777777776 (* beta -0.011574074074074073))))
   (/ (/ (+ 1.0 alpha) beta) beta)))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.7) {
		tmp = 0.08333333333333333 + (beta * (-0.027777777777777776 + (beta * -0.011574074074074073)));
	} else {
		tmp = ((1.0 + alpha) / beta) / beta;
	}
	return tmp;
}

Fortran

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 * ((-0.027777777777777776d0) + (beta * (-0.011574074074074073d0))))
    else
        tmp = ((1.0d0 + alpha) / beta) / beta
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.7) {
		tmp = 0.08333333333333333 + (beta * (-0.027777777777777776 + (beta * -0.011574074074074073)));
	} else {
		tmp = ((1.0 + alpha) / beta) / beta;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 1.7:
		tmp = 0.08333333333333333 + (beta * (-0.027777777777777776 + (beta * -0.011574074074074073)))
	else:
		tmp = ((1.0 + alpha) / beta) / beta
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.7)
		tmp = Float64(0.08333333333333333 + Float64(beta * Float64(-0.027777777777777776 + Float64(beta * -0.011574074074074073))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / beta);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.7)
		tmp = 0.08333333333333333 + (beta * (-0.027777777777777776 + (beta * -0.011574074074074073)));
	else
		tmp = ((1.0 + alpha) / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 1.7], N[(0.08333333333333333 + N[(beta * N[(-0.027777777777777776 + N[(beta * -0.011574074074074073), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 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 96.7%

      \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \left(\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(3 + \color{blue}{\left(\alpha + \beta\right)}\right)\right)\right)\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \left(3 \cdot \left(\left(\alpha + \beta\right) + 2\right) + \color{blue}{\left(\alpha + \beta\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)}\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\left(3 \cdot \left(\left(\alpha + \beta\right) + 2\right)\right), \color{blue}{\left(\left(\alpha + \beta\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)\right)}\right)\right)\right) \]

      4. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\left(3 \cdot \left(\alpha + \beta\right) + 3 \cdot 2\right), \left(\color{blue}{\left(\alpha + \beta\right)} \cdot \left(\left(\alpha + \beta\right) + 2\right)\right)\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\left(\left(\alpha + \beta\right) \cdot 3 + 3 \cdot 2\right), \left(\left(\color{blue}{\alpha} + \beta\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\left(\left(\alpha + \beta\right) \cdot 3\right), \left(3 \cdot 2\right)\right), \left(\color{blue}{\left(\alpha + \beta\right)} \cdot \left(\left(\alpha + \beta\right) + 2\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\alpha + \beta\right), 3\right), \left(3 \cdot 2\right)\right), \left(\left(\color{blue}{\alpha} + \beta\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), \left(3 \cdot 2\right)\right), \left(\left(\alpha + \beta\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)\right)\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), 6\right), \left(\left(\alpha + \color{blue}{\beta}\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), 6\right), \mathsf{*.f64}\left(\left(\alpha + \beta\right), \color{blue}{\left(\left(\alpha + \beta\right) + 2\right)}\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), 6\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \left(\color{blue}{\left(\alpha + \beta\right)} + 2\right)\right)\right)\right)\right) \]

      12. associate-+l+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), 6\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \left(\alpha + \color{blue}{\left(\beta + 2\right)}\right)\right)\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), 6\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{+.f64}\left(\alpha, \color{blue}{\left(\beta + 2\right)}\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 96.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), 6\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, \color{blue}{2}\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 96.7%

      \[\leadsto \frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \color{blue}{\left(\left(\left(\alpha + \beta\right) \cdot 3 + 6\right) + \left(\alpha + \beta\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)}} \]

   7. Taylor expanded in alpha around 0

      \[\leadsto \color{blue}{\frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)}} \]
   8. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{1 + \beta}{2 + \beta}}{\color{blue}{6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 + \beta}{2 + \beta}\right), \color{blue}{\left(6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + \beta\right), \left(2 + \beta\right)\right), \left(\color{blue}{6} + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\beta + 1\right), \left(2 + \beta\right)\right), \left(6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \left(2 + \beta\right)\right), \left(6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \left(\beta + 2\right)\right), \left(6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)\right) \]

      8. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(\left(6 + 3 \cdot \beta\right) + \color{blue}{\beta \cdot \left(2 + \beta\right)}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\left(6 + 3 \cdot \beta\right), \color{blue}{\left(\beta \cdot \left(2 + \beta\right)\right)}\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(6, \left(3 \cdot \beta\right)\right), \left(\color{blue}{\beta} \cdot \left(2 + \beta\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(6, \mathsf{*.f64}\left(3, \beta\right)\right), \left(\beta \cdot \left(2 + \beta\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(6, \mathsf{*.f64}\left(3, \beta\right)\right), \mathsf{*.f64}\left(\beta, \color{blue}{\left(2 + \beta\right)}\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(6, \mathsf{*.f64}\left(3, \beta\right)\right), \mathsf{*.f64}\left(\beta, \left(\beta + \color{blue}{2}\right)\right)\right)\right) \]

      14. +-lowering-+.f64 69.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(6, \mathsf{*.f64}\left(3, \beta\right)\right), \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(\beta, \color{blue}{2}\right)\right)\right)\right) \]

   9. Simplified 69.7%

      \[\leadsto \color{blue}{\frac{\frac{\beta + 1}{\beta + 2}}{\left(6 + 3 \cdot \beta\right) + \beta \cdot \left(\beta + 2\right)}} \]

   10. Taylor expanded in beta around 0

       \[\leadsto \color{blue}{\frac{1}{12} + \beta \cdot \left(\frac{-5}{432} \cdot \beta - \frac{1}{36}\right)} \]
   11. 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 69.0%

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\beta, \frac{-5}{432}\right), \frac{-1}{36}\right)\right)\right) \]

   12. Simplified 69.0%

       \[\leadsto \color{blue}{0.08333333333333333 + \beta \cdot \left(\beta \cdot -0.011574074074074073 + -0.027777777777777776\right)} \]

   if 1.69999999999999996 < 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 54.4%

      \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \alpha\right), \color{blue}{\left({\beta}^{2}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left({\color{blue}{\beta}}^{2}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left(\beta \cdot \color{blue}{\beta}\right)\right) \]

      4. *-lowering-*.f64 72.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\beta, \color{blue}{\beta}\right)\right) \]

   7. Simplified 72.6%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]
   8. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{1 + \alpha}{\beta}}{\color{blue}{\beta}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 + \alpha}{\beta}\right), \color{blue}{\beta}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + \alpha\right), \beta\right), \beta\right) \]

      4. +-lowering-+.f64 77.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \beta\right), \beta\right) \]

   9. Applied egg-rr 77.4%

      \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\beta}}{\beta}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.7:\\ \;\;\;\;0.08333333333333333 + \beta \cdot \left(-0.027777777777777776 + \beta \cdot -0.011574074074074073\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 71.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.2:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot \left(-0.027777777777777776 + \alpha \cdot -0.011574074074074073\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.2)
   (+
    0.08333333333333333
    (* alpha (+ -0.027777777777777776 (* alpha -0.011574074074074073))))
   (/ (/ (+ 1.0 alpha) beta) beta)))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.2) {
		tmp = 0.08333333333333333 + (alpha * (-0.027777777777777776 + (alpha * -0.011574074074074073)));
	} else {
		tmp = ((1.0 + alpha) / beta) / beta;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 3.2d0) then
        tmp = 0.08333333333333333d0 + (alpha * ((-0.027777777777777776d0) + (alpha * (-0.011574074074074073d0))))
    else
        tmp = ((1.0d0 + alpha) / beta) / beta
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.2) {
		tmp = 0.08333333333333333 + (alpha * (-0.027777777777777776 + (alpha * -0.011574074074074073)));
	} else {
		tmp = ((1.0 + alpha) / beta) / beta;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 3.2:
		tmp = 0.08333333333333333 + (alpha * (-0.027777777777777776 + (alpha * -0.011574074074074073)))
	else:
		tmp = ((1.0 + alpha) / beta) / beta
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.2)
		tmp = Float64(0.08333333333333333 + Float64(alpha * Float64(-0.027777777777777776 + Float64(alpha * -0.011574074074074073))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / beta);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.2)
		tmp = 0.08333333333333333 + (alpha * (-0.027777777777777776 + (alpha * -0.011574074074074073)));
	else
		tmp = ((1.0 + alpha) / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 3.2], N[(0.08333333333333333 + N[(alpha * N[(-0.027777777777777776 + N[(alpha * -0.011574074074074073), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.2000000000000002

   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 96.7%

      \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)\right)}} \]
   4. Add Preprocessing

   5. 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 94.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \alpha\right), \mathsf{+.f64}\left(2, \alpha\right)\right), \mathsf{+.f64}\left(3, \color{blue}{\alpha}\right)\right)\right) \]

   7. Simplified 94.2%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(\left(2 + \alpha\right) \cdot \left(2 + \alpha\right)\right) \cdot \left(3 + \alpha\right)}} \]

   8. Taylor expanded in alpha around 0

      \[\leadsto \color{blue}{\frac{1}{12} + \alpha \cdot \left(\frac{-5}{432} \cdot \alpha - \frac{1}{36}\right)} \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \color{blue}{\left(\alpha \cdot \left(\frac{-5}{432} \cdot \alpha - \frac{1}{36}\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\alpha, \color{blue}{\left(\frac{-5}{432} \cdot \alpha - \frac{1}{36}\right)}\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\alpha, \left(\frac{-5}{432} \cdot \alpha + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{36}\right)\right)}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\alpha, \left(\frac{-5}{432} \cdot \alpha + \frac{-1}{36}\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\alpha, \mathsf{+.f64}\left(\left(\frac{-5}{432} \cdot \alpha\right), \color{blue}{\frac{-1}{36}}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\alpha, \mathsf{+.f64}\left(\left(\alpha \cdot \frac{-5}{432}\right), \frac{-1}{36}\right)\right)\right) \]

      7. *-lowering-*.f64 67.1%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\alpha, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\alpha, \frac{-5}{432}\right), \frac{-1}{36}\right)\right)\right) \]

   10. Simplified 67.1%

       \[\leadsto \color{blue}{0.08333333333333333 + \alpha \cdot \left(\alpha \cdot -0.011574074074074073 + -0.027777777777777776\right)} \]

   if 3.2000000000000002 < 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 54.4%

      \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \alpha\right), \color{blue}{\left({\beta}^{2}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left({\color{blue}{\beta}}^{2}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left(\beta \cdot \color{blue}{\beta}\right)\right) \]

      4. *-lowering-*.f64 72.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\beta, \color{blue}{\beta}\right)\right) \]

   7. Simplified 72.6%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]
   8. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{1 + \alpha}{\beta}}{\color{blue}{\beta}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 + \alpha}{\beta}\right), \color{blue}{\beta}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + \alpha\right), \beta\right), \beta\right) \]

      4. +-lowering-+.f64 77.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \beta\right), \beta\right) \]

   9. Applied egg-rr 77.4%

      \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\beta}}{\beta}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.2:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot \left(-0.027777777777777776 + \alpha \cdot -0.011574074074074073\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 71.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.8:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.8)
   (+ 0.08333333333333333 (* beta -0.027777777777777776))
   (/ (/ (+ 1.0 alpha) beta) beta)))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.8) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = ((1.0 + alpha) / beta) / beta;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.8d0) then
        tmp = 0.08333333333333333d0 + (beta * (-0.027777777777777776d0))
    else
        tmp = ((1.0d0 + alpha) / beta) / beta
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.8) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = ((1.0 + alpha) / beta) / beta;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.8:
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776)
	else:
		tmp = ((1.0 + alpha) / beta) / beta
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.8)
		tmp = Float64(0.08333333333333333 + Float64(beta * -0.027777777777777776));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / beta);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.8)
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	else
		tmp = ((1.0 + alpha) / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.8], 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 < 2.7999999999999998

   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 96.7%

      \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \left(\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(3 + \color{blue}{\left(\alpha + \beta\right)}\right)\right)\right)\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \left(3 \cdot \left(\left(\alpha + \beta\right) + 2\right) + \color{blue}{\left(\alpha + \beta\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)}\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\left(3 \cdot \left(\left(\alpha + \beta\right) + 2\right)\right), \color{blue}{\left(\left(\alpha + \beta\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)\right)}\right)\right)\right) \]

      4. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\left(3 \cdot \left(\alpha + \beta\right) + 3 \cdot 2\right), \left(\color{blue}{\left(\alpha + \beta\right)} \cdot \left(\left(\alpha + \beta\right) + 2\right)\right)\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\left(\left(\alpha + \beta\right) \cdot 3 + 3 \cdot 2\right), \left(\left(\color{blue}{\alpha} + \beta\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\left(\left(\alpha + \beta\right) \cdot 3\right), \left(3 \cdot 2\right)\right), \left(\color{blue}{\left(\alpha + \beta\right)} \cdot \left(\left(\alpha + \beta\right) + 2\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\alpha + \beta\right), 3\right), \left(3 \cdot 2\right)\right), \left(\left(\color{blue}{\alpha} + \beta\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), \left(3 \cdot 2\right)\right), \left(\left(\alpha + \beta\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)\right)\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), 6\right), \left(\left(\alpha + \color{blue}{\beta}\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), 6\right), \mathsf{*.f64}\left(\left(\alpha + \beta\right), \color{blue}{\left(\left(\alpha + \beta\right) + 2\right)}\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), 6\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \left(\color{blue}{\left(\alpha + \beta\right)} + 2\right)\right)\right)\right)\right) \]

      12. associate-+l+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), 6\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \left(\alpha + \color{blue}{\left(\beta + 2\right)}\right)\right)\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), 6\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{+.f64}\left(\alpha, \color{blue}{\left(\beta + 2\right)}\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 96.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), 6\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, \color{blue}{2}\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 96.7%

      \[\leadsto \frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \color{blue}{\left(\left(\left(\alpha + \beta\right) \cdot 3 + 6\right) + \left(\alpha + \beta\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)}} \]

   7. Taylor expanded in alpha around 0

      \[\leadsto \color{blue}{\frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)}} \]
   8. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{1 + \beta}{2 + \beta}}{\color{blue}{6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 + \beta}{2 + \beta}\right), \color{blue}{\left(6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + \beta\right), \left(2 + \beta\right)\right), \left(\color{blue}{6} + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\beta + 1\right), \left(2 + \beta\right)\right), \left(6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \left(2 + \beta\right)\right), \left(6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \left(\beta + 2\right)\right), \left(6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)\right) \]

      8. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(\left(6 + 3 \cdot \beta\right) + \color{blue}{\beta \cdot \left(2 + \beta\right)}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\left(6 + 3 \cdot \beta\right), \color{blue}{\left(\beta \cdot \left(2 + \beta\right)\right)}\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(6, \left(3 \cdot \beta\right)\right), \left(\color{blue}{\beta} \cdot \left(2 + \beta\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(6, \mathsf{*.f64}\left(3, \beta\right)\right), \left(\beta \cdot \left(2 + \beta\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(6, \mathsf{*.f64}\left(3, \beta\right)\right), \mathsf{*.f64}\left(\beta, \color{blue}{\left(2 + \beta\right)}\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(6, \mathsf{*.f64}\left(3, \beta\right)\right), \mathsf{*.f64}\left(\beta, \left(\beta + \color{blue}{2}\right)\right)\right)\right) \]

      14. +-lowering-+.f64 69.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(6, \mathsf{*.f64}\left(3, \beta\right)\right), \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(\beta, \color{blue}{2}\right)\right)\right)\right) \]

   9. Simplified 69.7%

      \[\leadsto \color{blue}{\frac{\frac{\beta + 1}{\beta + 2}}{\left(6 + 3 \cdot \beta\right) + \beta \cdot \left(\beta + 2\right)}} \]

   10. Taylor expanded in beta around 0

       \[\leadsto \color{blue}{\frac{1}{12} + \frac{-1}{36} \cdot \beta} \]
   11. 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 68.6%

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \color{blue}{\frac{-1}{36}}\right)\right) \]

   12. Simplified 68.6%

       \[\leadsto \color{blue}{0.08333333333333333 + \beta \cdot -0.027777777777777776} \]

   if 2.7999999999999998 < 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 54.4%

      \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \alpha\right), \color{blue}{\left({\beta}^{2}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left({\color{blue}{\beta}}^{2}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left(\beta \cdot \color{blue}{\beta}\right)\right) \]

      4. *-lowering-*.f64 72.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\beta, \color{blue}{\beta}\right)\right) \]

   7. Simplified 72.6%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]
   8. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{1 + \alpha}{\beta}}{\color{blue}{\beta}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 + \alpha}{\beta}\right), \color{blue}{\beta}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + \alpha\right), \beta\right), \beta\right) \]

      4. +-lowering-+.f64 77.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \beta\right), \beta\right) \]

   9. Applied egg-rr 77.4%

      \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\beta}}{\beta}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 69.7% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.8:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.8)
   (+ 0.08333333333333333 (* beta -0.027777777777777776))
   (/ (/ 1.0 beta) beta)))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.8) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = (1.0 / beta) / beta;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.8d0) then
        tmp = 0.08333333333333333d0 + (beta * (-0.027777777777777776d0))
    else
        tmp = (1.0d0 / beta) / beta
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.8) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = (1.0 / beta) / beta;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.8:
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776)
	else:
		tmp = (1.0 / beta) / beta
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.8)
		tmp = Float64(0.08333333333333333 + Float64(beta * -0.027777777777777776));
	else
		tmp = Float64(Float64(1.0 / beta) / beta);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.8)
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	else
		tmp = (1.0 / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.8], N[(0.08333333333333333 + N[(beta * -0.027777777777777776), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.7999999999999998

   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 96.7%

      \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \left(\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(3 + \color{blue}{\left(\alpha + \beta\right)}\right)\right)\right)\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \left(3 \cdot \left(\left(\alpha + \beta\right) + 2\right) + \color{blue}{\left(\alpha + \beta\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)}\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\left(3 \cdot \left(\left(\alpha + \beta\right) + 2\right)\right), \color{blue}{\left(\left(\alpha + \beta\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)\right)}\right)\right)\right) \]

      4. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\left(3 \cdot \left(\alpha + \beta\right) + 3 \cdot 2\right), \left(\color{blue}{\left(\alpha + \beta\right)} \cdot \left(\left(\alpha + \beta\right) + 2\right)\right)\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\left(\left(\alpha + \beta\right) \cdot 3 + 3 \cdot 2\right), \left(\left(\color{blue}{\alpha} + \beta\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\left(\left(\alpha + \beta\right) \cdot 3\right), \left(3 \cdot 2\right)\right), \left(\color{blue}{\left(\alpha + \beta\right)} \cdot \left(\left(\alpha + \beta\right) + 2\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\alpha + \beta\right), 3\right), \left(3 \cdot 2\right)\right), \left(\left(\color{blue}{\alpha} + \beta\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), \left(3 \cdot 2\right)\right), \left(\left(\alpha + \beta\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)\right)\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), 6\right), \left(\left(\alpha + \color{blue}{\beta}\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), 6\right), \mathsf{*.f64}\left(\left(\alpha + \beta\right), \color{blue}{\left(\left(\alpha + \beta\right) + 2\right)}\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), 6\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \left(\color{blue}{\left(\alpha + \beta\right)} + 2\right)\right)\right)\right)\right) \]

      12. associate-+l+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), 6\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \left(\alpha + \color{blue}{\left(\beta + 2\right)}\right)\right)\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), 6\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{+.f64}\left(\alpha, \color{blue}{\left(\beta + 2\right)}\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 96.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), 6\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, \color{blue}{2}\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 96.7%

      \[\leadsto \frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \color{blue}{\left(\left(\left(\alpha + \beta\right) \cdot 3 + 6\right) + \left(\alpha + \beta\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)}} \]

   7. Taylor expanded in alpha around 0

      \[\leadsto \color{blue}{\frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)}} \]
   8. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{1 + \beta}{2 + \beta}}{\color{blue}{6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 + \beta}{2 + \beta}\right), \color{blue}{\left(6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + \beta\right), \left(2 + \beta\right)\right), \left(\color{blue}{6} + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\beta + 1\right), \left(2 + \beta\right)\right), \left(6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \left(2 + \beta\right)\right), \left(6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \left(\beta + 2\right)\right), \left(6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)\right) \]

      8. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(\left(6 + 3 \cdot \beta\right) + \color{blue}{\beta \cdot \left(2 + \beta\right)}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\left(6 + 3 \cdot \beta\right), \color{blue}{\left(\beta \cdot \left(2 + \beta\right)\right)}\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(6, \left(3 \cdot \beta\right)\right), \left(\color{blue}{\beta} \cdot \left(2 + \beta\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(6, \mathsf{*.f64}\left(3, \beta\right)\right), \left(\beta \cdot \left(2 + \beta\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(6, \mathsf{*.f64}\left(3, \beta\right)\right), \mathsf{*.f64}\left(\beta, \color{blue}{\left(2 + \beta\right)}\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(6, \mathsf{*.f64}\left(3, \beta\right)\right), \mathsf{*.f64}\left(\beta, \left(\beta + \color{blue}{2}\right)\right)\right)\right) \]

      14. +-lowering-+.f64 69.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(6, \mathsf{*.f64}\left(3, \beta\right)\right), \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(\beta, \color{blue}{2}\right)\right)\right)\right) \]

   9. Simplified 69.7%

      \[\leadsto \color{blue}{\frac{\frac{\beta + 1}{\beta + 2}}{\left(6 + 3 \cdot \beta\right) + \beta \cdot \left(\beta + 2\right)}} \]

   10. Taylor expanded in beta around 0

       \[\leadsto \color{blue}{\frac{1}{12} + \frac{-1}{36} \cdot \beta} \]
   11. 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 68.6%

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \color{blue}{\frac{-1}{36}}\right)\right) \]

   12. Simplified 68.6%

       \[\leadsto \color{blue}{0.08333333333333333 + \beta \cdot -0.027777777777777776} \]

   if 2.7999999999999998 < 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 54.4%

      \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \alpha\right), \color{blue}{\left({\beta}^{2}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left({\color{blue}{\beta}}^{2}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left(\beta \cdot \color{blue}{\beta}\right)\right) \]

      4. *-lowering-*.f64 72.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\beta, \color{blue}{\beta}\right)\right) \]

   7. Simplified 72.6%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]
   8. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{1 + \alpha}{\beta}}{\color{blue}{\beta}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 + \alpha}{\beta}\right), \color{blue}{\beta}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + \alpha\right), \beta\right), \beta\right) \]

      4. +-lowering-+.f64 77.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \beta\right), \beta\right) \]

   9. Applied egg-rr 77.4%

      \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\beta}}{\beta}} \]

   10. Taylor expanded in alpha around 0

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\color{blue}{1}, \beta\right), \beta\right) \]
   11. Step-by-step derivation

   12. Simplified 70.2%

       \[\leadsto \frac{\frac{\color{blue}{1}}{\beta}}{\beta} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 69.6% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.8:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.8)
   (+ 0.08333333333333333 (* beta -0.027777777777777776))
   (/ 1.0 (* beta beta))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.8) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = 1.0 / (beta * beta);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.8d0) then
        tmp = 0.08333333333333333d0 + (beta * (-0.027777777777777776d0))
    else
        tmp = 1.0d0 / (beta * beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.8) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = 1.0 / (beta * beta);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.8:
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776)
	else:
		tmp = 1.0 / (beta * beta)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.8)
		tmp = Float64(0.08333333333333333 + Float64(beta * -0.027777777777777776));
	else
		tmp = Float64(1.0 / Float64(beta * beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.8)
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	else
		tmp = 1.0 / (beta * beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.8], 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 < 2.7999999999999998

   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 96.7%

      \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \left(\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(3 + \color{blue}{\left(\alpha + \beta\right)}\right)\right)\right)\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \left(3 \cdot \left(\left(\alpha + \beta\right) + 2\right) + \color{blue}{\left(\alpha + \beta\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)}\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\left(3 \cdot \left(\left(\alpha + \beta\right) + 2\right)\right), \color{blue}{\left(\left(\alpha + \beta\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)\right)}\right)\right)\right) \]

      4. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\left(3 \cdot \left(\alpha + \beta\right) + 3 \cdot 2\right), \left(\color{blue}{\left(\alpha + \beta\right)} \cdot \left(\left(\alpha + \beta\right) + 2\right)\right)\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\left(\left(\alpha + \beta\right) \cdot 3 + 3 \cdot 2\right), \left(\left(\color{blue}{\alpha} + \beta\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\left(\left(\alpha + \beta\right) \cdot 3\right), \left(3 \cdot 2\right)\right), \left(\color{blue}{\left(\alpha + \beta\right)} \cdot \left(\left(\alpha + \beta\right) + 2\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\alpha + \beta\right), 3\right), \left(3 \cdot 2\right)\right), \left(\left(\color{blue}{\alpha} + \beta\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), \left(3 \cdot 2\right)\right), \left(\left(\alpha + \beta\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)\right)\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), 6\right), \left(\left(\alpha + \color{blue}{\beta}\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), 6\right), \mathsf{*.f64}\left(\left(\alpha + \beta\right), \color{blue}{\left(\left(\alpha + \beta\right) + 2\right)}\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), 6\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \left(\color{blue}{\left(\alpha + \beta\right)} + 2\right)\right)\right)\right)\right) \]

      12. associate-+l+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), 6\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \left(\alpha + \color{blue}{\left(\beta + 2\right)}\right)\right)\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), 6\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{+.f64}\left(\alpha, \color{blue}{\left(\beta + 2\right)}\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 96.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), 6\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, \color{blue}{2}\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 96.7%

      \[\leadsto \frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \color{blue}{\left(\left(\left(\alpha + \beta\right) \cdot 3 + 6\right) + \left(\alpha + \beta\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)}} \]

   7. Taylor expanded in alpha around 0

      \[\leadsto \color{blue}{\frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)}} \]
   8. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{1 + \beta}{2 + \beta}}{\color{blue}{6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 + \beta}{2 + \beta}\right), \color{blue}{\left(6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + \beta\right), \left(2 + \beta\right)\right), \left(\color{blue}{6} + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\beta + 1\right), \left(2 + \beta\right)\right), \left(6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \left(2 + \beta\right)\right), \left(6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \left(\beta + 2\right)\right), \left(6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)\right) \]

      8. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(\left(6 + 3 \cdot \beta\right) + \color{blue}{\beta \cdot \left(2 + \beta\right)}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\left(6 + 3 \cdot \beta\right), \color{blue}{\left(\beta \cdot \left(2 + \beta\right)\right)}\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(6, \left(3 \cdot \beta\right)\right), \left(\color{blue}{\beta} \cdot \left(2 + \beta\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(6, \mathsf{*.f64}\left(3, \beta\right)\right), \left(\beta \cdot \left(2 + \beta\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(6, \mathsf{*.f64}\left(3, \beta\right)\right), \mathsf{*.f64}\left(\beta, \color{blue}{\left(2 + \beta\right)}\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(6, \mathsf{*.f64}\left(3, \beta\right)\right), \mathsf{*.f64}\left(\beta, \left(\beta + \color{blue}{2}\right)\right)\right)\right) \]

      14. +-lowering-+.f64 69.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(6, \mathsf{*.f64}\left(3, \beta\right)\right), \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(\beta, \color{blue}{2}\right)\right)\right)\right) \]

   9. Simplified 69.7%

      \[\leadsto \color{blue}{\frac{\frac{\beta + 1}{\beta + 2}}{\left(6 + 3 \cdot \beta\right) + \beta \cdot \left(\beta + 2\right)}} \]

   10. Taylor expanded in beta around 0

       \[\leadsto \color{blue}{\frac{1}{12} + \frac{-1}{36} \cdot \beta} \]
   11. 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 68.6%

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \color{blue}{\frac{-1}{36}}\right)\right) \]

   12. Simplified 68.6%

       \[\leadsto \color{blue}{0.08333333333333333 + \beta \cdot -0.027777777777777776} \]

   if 2.7999999999999998 < 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 54.4%

      \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \alpha\right), \color{blue}{\left({\beta}^{2}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left({\color{blue}{\beta}}^{2}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left(\beta \cdot \color{blue}{\beta}\right)\right) \]

      4. *-lowering-*.f64 72.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\beta, \color{blue}{\beta}\right)\right) \]

   7. Simplified 72.6%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]

   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 70.1%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\beta, \color{blue}{\beta}\right)\right) \]

   10. Simplified 70.1%

       \[\leadsto \color{blue}{\frac{1}{\beta \cdot \beta}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 46.7% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.8:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.8)
   (+ 0.08333333333333333 (* beta -0.027777777777777776))
   (/ 0.3333333333333333 beta)))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.8) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = 0.3333333333333333 / beta;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.8d0) then
        tmp = 0.08333333333333333d0 + (beta * (-0.027777777777777776d0))
    else
        tmp = 0.3333333333333333d0 / beta
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.8) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = 0.3333333333333333 / beta;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.8:
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776)
	else:
		tmp = 0.3333333333333333 / beta
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.8)
		tmp = Float64(0.08333333333333333 + Float64(beta * -0.027777777777777776));
	else
		tmp = Float64(0.3333333333333333 / beta);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.8)
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	else
		tmp = 0.3333333333333333 / beta;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.8], N[(0.08333333333333333 + N[(beta * -0.027777777777777776), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.7999999999999998

   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 96.7%

      \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \left(\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(3 + \color{blue}{\left(\alpha + \beta\right)}\right)\right)\right)\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \left(3 \cdot \left(\left(\alpha + \beta\right) + 2\right) + \color{blue}{\left(\alpha + \beta\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)}\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\left(3 \cdot \left(\left(\alpha + \beta\right) + 2\right)\right), \color{blue}{\left(\left(\alpha + \beta\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)\right)}\right)\right)\right) \]

      4. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\left(3 \cdot \left(\alpha + \beta\right) + 3 \cdot 2\right), \left(\color{blue}{\left(\alpha + \beta\right)} \cdot \left(\left(\alpha + \beta\right) + 2\right)\right)\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\left(\left(\alpha + \beta\right) \cdot 3 + 3 \cdot 2\right), \left(\left(\color{blue}{\alpha} + \beta\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\left(\left(\alpha + \beta\right) \cdot 3\right), \left(3 \cdot 2\right)\right), \left(\color{blue}{\left(\alpha + \beta\right)} \cdot \left(\left(\alpha + \beta\right) + 2\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\alpha + \beta\right), 3\right), \left(3 \cdot 2\right)\right), \left(\left(\color{blue}{\alpha} + \beta\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), \left(3 \cdot 2\right)\right), \left(\left(\alpha + \beta\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)\right)\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), 6\right), \left(\left(\alpha + \color{blue}{\beta}\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), 6\right), \mathsf{*.f64}\left(\left(\alpha + \beta\right), \color{blue}{\left(\left(\alpha + \beta\right) + 2\right)}\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), 6\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \left(\color{blue}{\left(\alpha + \beta\right)} + 2\right)\right)\right)\right)\right) \]

      12. associate-+l+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), 6\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \left(\alpha + \color{blue}{\left(\beta + 2\right)}\right)\right)\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), 6\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{+.f64}\left(\alpha, \color{blue}{\left(\beta + 2\right)}\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 96.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 3\right), 6\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, \color{blue}{2}\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 96.7%

      \[\leadsto \frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \color{blue}{\left(\left(\left(\alpha + \beta\right) \cdot 3 + 6\right) + \left(\alpha + \beta\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)}} \]

   7. Taylor expanded in alpha around 0

      \[\leadsto \color{blue}{\frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)}} \]
   8. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{1 + \beta}{2 + \beta}}{\color{blue}{6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 + \beta}{2 + \beta}\right), \color{blue}{\left(6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + \beta\right), \left(2 + \beta\right)\right), \left(\color{blue}{6} + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\beta + 1\right), \left(2 + \beta\right)\right), \left(6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \left(2 + \beta\right)\right), \left(6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \left(\beta + 2\right)\right), \left(6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(6 + \left(3 \cdot \beta + \beta \cdot \left(2 + \beta\right)\right)\right)\right) \]

      8. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(\left(6 + 3 \cdot \beta\right) + \color{blue}{\beta \cdot \left(2 + \beta\right)}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\left(6 + 3 \cdot \beta\right), \color{blue}{\left(\beta \cdot \left(2 + \beta\right)\right)}\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(6, \left(3 \cdot \beta\right)\right), \left(\color{blue}{\beta} \cdot \left(2 + \beta\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(6, \mathsf{*.f64}\left(3, \beta\right)\right), \left(\beta \cdot \left(2 + \beta\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(6, \mathsf{*.f64}\left(3, \beta\right)\right), \mathsf{*.f64}\left(\beta, \color{blue}{\left(2 + \beta\right)}\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(6, \mathsf{*.f64}\left(3, \beta\right)\right), \mathsf{*.f64}\left(\beta, \left(\beta + \color{blue}{2}\right)\right)\right)\right) \]

      14. +-lowering-+.f64 69.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(6, \mathsf{*.f64}\left(3, \beta\right)\right), \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(\beta, \color{blue}{2}\right)\right)\right)\right) \]

   9. Simplified 69.7%

      \[\leadsto \color{blue}{\frac{\frac{\beta + 1}{\beta + 2}}{\left(6 + 3 \cdot \beta\right) + \beta \cdot \left(\beta + 2\right)}} \]

   10. Taylor expanded in beta around 0

       \[\leadsto \color{blue}{\frac{1}{12} + \frac{-1}{36} \cdot \beta} \]
   11. 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 68.6%

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \color{blue}{\frac{-1}{36}}\right)\right) \]

   12. Simplified 68.6%

       \[\leadsto \color{blue}{0.08333333333333333 + \beta \cdot -0.027777777777777776} \]

   if 2.7999999999999998 < 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. Add Preprocessing

   3. Taylor expanded in beta around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1 + \alpha}{\beta}\right)}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + \alpha\right), \beta\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)}, 1\right)\right) \]

      2. +-lowering-+.f64 77.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \beta\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(\alpha, \beta\right)}, \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

   5. Simplified 77.7%

      \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\beta}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   6. Step-by-step derivation

      1. flip-+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 \cdot 1 - \alpha \cdot \alpha}{1 - \alpha}\right), \beta\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(\alpha, \beta\right)}, \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      2. div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(1 \cdot 1 - \alpha \cdot \alpha\right) \cdot \frac{1}{1 - \alpha}\right), \beta\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(\alpha, \beta\right)}, \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 \cdot 1 - \alpha \cdot \alpha\right), \left(\frac{1}{1 - \alpha}\right)\right), \beta\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(\alpha, \beta\right)}, \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 - \alpha \cdot \alpha\right), \left(\frac{1}{1 - \alpha}\right)\right), \beta\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \left(\alpha \cdot \alpha\right)\right), \left(\frac{1}{1 - \alpha}\right)\right), \beta\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\alpha}, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\alpha, \alpha\right)\right), \left(\frac{1}{1 - \alpha}\right)\right), \beta\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\alpha, \alpha\right)\right), \mathsf{/.f64}\left(1, \left(1 - \alpha\right)\right)\right), \beta\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \color{blue}{\beta}\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      8. --lowering--.f64 71.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\alpha, \alpha\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \alpha\right)\right)\right), \beta\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

   7. Applied egg-rr 71.2%

      \[\leadsto \frac{\frac{\color{blue}{\left(1 - \alpha \cdot \alpha\right) \cdot \frac{1}{1 - \alpha}}}{\beta}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   8. Taylor expanded in alpha around 0

      \[\leadsto \color{blue}{\frac{1}{\beta \cdot \left(3 + \beta\right)}} \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\beta \cdot \left(3 + \beta\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\beta, \color{blue}{\left(3 + \beta\right)}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\beta, \left(\beta + \color{blue}{3}\right)\right)\right) \]

      4. +-lowering-+.f64 70.2%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(\beta, \color{blue}{3}\right)\right)\right) \]

   10. Simplified 70.2%

       \[\leadsto \color{blue}{\frac{1}{\beta \cdot \left(\beta + 3\right)}} \]

   11. Taylor expanded in beta around 0

       \[\leadsto \color{blue}{\frac{\frac{1}{3}}{\beta}} \]
   12. Step-by-step derivation

       1. /-lowering-/.f64 6.9%

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \color{blue}{\beta}\right) \]

   13. Simplified 6.9%

       \[\leadsto \color{blue}{\frac{0.3333333333333333}{\beta}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 24: 46.2% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.9:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.9)
   (+ 0.08333333333333333 (* alpha -0.027777777777777776))
   (/ 0.3333333333333333 beta)))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.9) {
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776);
	} else {
		tmp = 0.3333333333333333 / beta;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.9d0) then
        tmp = 0.08333333333333333d0 + (alpha * (-0.027777777777777776d0))
    else
        tmp = 0.3333333333333333d0 / beta
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.9) {
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776);
	} else {
		tmp = 0.3333333333333333 / beta;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.9:
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776)
	else:
		tmp = 0.3333333333333333 / beta
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.9)
		tmp = Float64(0.08333333333333333 + Float64(alpha * -0.027777777777777776));
	else
		tmp = Float64(0.3333333333333333 / beta);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.9)
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776);
	else
		tmp = 0.3333333333333333 / beta;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.9], N[(0.08333333333333333 + N[(alpha * -0.027777777777777776), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.89999999999999991

   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 96.7%

      \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)\right)}} \]
   4. Add Preprocessing

   5. 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 94.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \alpha\right), \mathsf{+.f64}\left(2, \alpha\right)\right), \mathsf{+.f64}\left(3, \color{blue}{\alpha}\right)\right)\right) \]

   7. Simplified 94.2%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(\left(2 + \alpha\right) \cdot \left(2 + \alpha\right)\right) \cdot \left(3 + \alpha\right)}} \]

   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 67.1%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\alpha, \color{blue}{\frac{-1}{36}}\right)\right) \]

   10. Simplified 67.1%

       \[\leadsto \color{blue}{0.08333333333333333 + \alpha \cdot -0.027777777777777776} \]

   if 2.89999999999999991 < 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. Add Preprocessing

   3. Taylor expanded in beta around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1 + \alpha}{\beta}\right)}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + \alpha\right), \beta\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)}, 1\right)\right) \]

      2. +-lowering-+.f64 77.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \beta\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(\alpha, \beta\right)}, \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

   5. Simplified 77.7%

      \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\beta}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   6. Step-by-step derivation

      1. flip-+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 \cdot 1 - \alpha \cdot \alpha}{1 - \alpha}\right), \beta\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(\alpha, \beta\right)}, \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      2. div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(1 \cdot 1 - \alpha \cdot \alpha\right) \cdot \frac{1}{1 - \alpha}\right), \beta\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(\alpha, \beta\right)}, \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 \cdot 1 - \alpha \cdot \alpha\right), \left(\frac{1}{1 - \alpha}\right)\right), \beta\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(\alpha, \beta\right)}, \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 - \alpha \cdot \alpha\right), \left(\frac{1}{1 - \alpha}\right)\right), \beta\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \left(\alpha \cdot \alpha\right)\right), \left(\frac{1}{1 - \alpha}\right)\right), \beta\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\alpha}, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\alpha, \alpha\right)\right), \left(\frac{1}{1 - \alpha}\right)\right), \beta\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\alpha, \alpha\right)\right), \mathsf{/.f64}\left(1, \left(1 - \alpha\right)\right)\right), \beta\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \color{blue}{\beta}\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      8. --lowering--.f64 71.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\alpha, \alpha\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \alpha\right)\right)\right), \beta\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

   7. Applied egg-rr 71.2%

      \[\leadsto \frac{\frac{\color{blue}{\left(1 - \alpha \cdot \alpha\right) \cdot \frac{1}{1 - \alpha}}}{\beta}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   8. Taylor expanded in alpha around 0

      \[\leadsto \color{blue}{\frac{1}{\beta \cdot \left(3 + \beta\right)}} \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\beta \cdot \left(3 + \beta\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\beta, \color{blue}{\left(3 + \beta\right)}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\beta, \left(\beta + \color{blue}{3}\right)\right)\right) \]

      4. +-lowering-+.f64 70.2%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(\beta, \color{blue}{3}\right)\right)\right) \]

   10. Simplified 70.2%

       \[\leadsto \color{blue}{\frac{1}{\beta \cdot \left(\beta + 3\right)}} \]

   11. Taylor expanded in beta around 0

       \[\leadsto \color{blue}{\frac{\frac{1}{3}}{\beta}} \]
   12. Step-by-step derivation

       1. /-lowering-/.f64 6.9%

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \color{blue}{\beta}\right) \]

   13. Simplified 6.9%

       \[\leadsto \color{blue}{\frac{0.3333333333333333}{\beta}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 25: 45.3% accurate, 35.0× speedup

Math

\[\begin{array}{l} \\ 0.08333333333333333 \end{array} \]

FPCore

(FPCore (alpha beta) :precision binary64 0.08333333333333333)

C

double code(double alpha, double beta) {
	return 0.08333333333333333;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = 0.08333333333333333d0
end function

Java

public static double code(double alpha, double beta) {
	return 0.08333333333333333;
}

Python

def code(alpha, beta):
	return 0.08333333333333333

Julia

function code(alpha, beta)
	return 0.08333333333333333
end

MATLAB

function tmp = code(alpha, beta)
	tmp = 0.08333333333333333;
end

Wolfram

code[alpha_, beta_] := 0.08333333333333333

Derivation

1. Initial program 93.7%

   \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
2. Step-by-step derivation

   1. associate-/l/ 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 85.2%

   \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in beta around 0

   \[\leadsto \color{blue}{\frac{1 + \alpha}{{\left(2 + \alpha\right)}^{2} \cdot \left(3 + \alpha\right)}} \]
6. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(1 + \alpha\right), \color{blue}{\left({\left(2 + \alpha\right)}^{2} \cdot \left(3 + \alpha\right)\right)}\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left(\color{blue}{{\left(2 + \alpha\right)}^{2}} \cdot \left(3 + \alpha\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\left({\left(2 + \alpha\right)}^{2}\right), \color{blue}{\left(3 + \alpha\right)}\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\left(\left(2 + \alpha\right) \cdot \left(2 + \alpha\right)\right), \left(\color{blue}{3} + \alpha\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 + \alpha\right), \left(2 + \alpha\right)\right), \left(\color{blue}{3} + \alpha\right)\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \alpha\right), \left(2 + \alpha\right)\right), \left(3 + \alpha\right)\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \alpha\right), \mathsf{+.f64}\left(2, \alpha\right)\right), \left(3 + \alpha\right)\right)\right) \]

   8. +-lowering-+.f64 74.5%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \alpha\right), \mathsf{+.f64}\left(2, \alpha\right)\right), \mathsf{+.f64}\left(3, \color{blue}{\alpha}\right)\right)\right) \]

7. Simplified 74.5%

   \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(\left(2 + \alpha\right) \cdot \left(2 + \alpha\right)\right) \cdot \left(3 + \alpha\right)}} \]

8. Taylor expanded in alpha around 0

   \[\leadsto \color{blue}{\frac{1}{12}} \]
9. Step-by-step derivation

10. Simplified 50.2%

    \[\leadsto \color{blue}{0.08333333333333333} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024161 
(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)))