Octave 3.8, jcobi/3

Percentage Accurate: 94.4% → 99.8%

Time: 32.5s

Alternatives: 18

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[alpha_, beta_] := N[(N[(N[(N[(1.0 + beta), $MachinePrecision] * N[(N[(1.0 + alpha), $MachinePrecision] / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.2%

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

   1. div-inv 95.2%

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

   2. +-commutative 95.2%

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

   3. *-commutative 95.2%

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

   4. associate-+r+ 95.2%

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

   5. +-commutative 95.2%

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

   6. distribute-rgt1-in 95.2%

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

   7. fma-define 95.2%

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

   8. metadata-eval 95.2%

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

   9. associate-+r+ 95.2%

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

   10. metadata-eval 95.2%

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

   11. associate-+r+ 95.2%

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

4. Applied egg-rr 95.2%

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

   1. associate-*l/ 95.2%

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

   2. associate-*r/ 95.2%

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

   3. *-rgt-identity 95.2%

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

   4. +-commutative 95.2%

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

   5. fma-undefine 95.2%

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

   6. +-commutative 95.2%

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

   7. *-commutative 95.2%

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

   8. +-commutative 95.2%

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

   9. associate-+r+ 95.2%

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

   10. distribute-lft1-in 95.2%

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

   11. +-commutative 95.2%

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

   12. +-commutative 95.2%

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

   13. +-commutative 95.2%

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

   14. +-commutative 95.2%

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

   15. +-commutative 95.2%

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

   16. +-commutative 95.2%

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

   17. +-commutative 95.2%

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

   18. +-commutative 95.2%

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

6. Simplified 95.2%

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

   1. associate-/l* 99.9%

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

   2. +-commutative 99.9%

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

   3. associate-+r+ 99.9%

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

   4. +-commutative 99.9%

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

   5. +-commutative 99.9%

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

8. Applied egg-rr 99.9%

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

9. Taylor expanded in alpha around 0 99.9%

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

    1. +-commutative 99.9%

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

11. Simplified 99.9%

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

12. Final simplification 99.9%

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

Alternative 2: 97.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 3\\ t_1 := 2 + \left(\beta + \alpha\right)\\ t_2 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 1.02 \cdot 10^{+152}:\\ \;\;\;\;\frac{\frac{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}{t\_2}}{t\_0 \cdot t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{t\_1} \cdot \frac{\frac{\beta}{t\_1}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) 3.0))
        (t_1 (+ 2.0 (+ beta alpha)))
        (t_2 (+ alpha (+ beta 2.0))))
   (if (<= beta 1.02e+152)
     (/ (/ (* (+ 1.0 beta) (+ 1.0 alpha)) t_2) (* t_0 t_2))
     (* (/ (+ 1.0 alpha) t_1) (/ (/ beta t_1) t_0)))))

C

double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 3.0;
	double t_1 = 2.0 + (beta + alpha);
	double t_2 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 1.02e+152) {
		tmp = (((1.0 + beta) * (1.0 + alpha)) / t_2) / (t_0 * t_2);
	} else {
		tmp = ((1.0 + alpha) / t_1) * ((beta / 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) :: t_2
    real(8) :: tmp
    t_0 = (beta + alpha) + 3.0d0
    t_1 = 2.0d0 + (beta + alpha)
    t_2 = alpha + (beta + 2.0d0)
    if (beta <= 1.02d+152) then
        tmp = (((1.0d0 + beta) * (1.0d0 + alpha)) / t_2) / (t_0 * t_2)
    else
        tmp = ((1.0d0 + alpha) / t_1) * ((beta / t_1) / t_0)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 3.0;
	double t_1 = 2.0 + (beta + alpha);
	double t_2 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 1.02e+152) {
		tmp = (((1.0 + beta) * (1.0 + alpha)) / t_2) / (t_0 * t_2);
	} else {
		tmp = ((1.0 + alpha) / t_1) * ((beta / t_1) / t_0);
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (beta + alpha) + 3.0
	t_1 = 2.0 + (beta + alpha)
	t_2 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 1.02e+152:
		tmp = (((1.0 + beta) * (1.0 + alpha)) / t_2) / (t_0 * t_2)
	else:
		tmp = ((1.0 + alpha) / t_1) * ((beta / t_1) / t_0)
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta + alpha) + 3.0)
	t_1 = Float64(2.0 + Float64(beta + alpha))
	t_2 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 1.02e+152)
		tmp = Float64(Float64(Float64(Float64(1.0 + beta) * Float64(1.0 + alpha)) / t_2) / Float64(t_0 * t_2));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / t_1) * Float64(Float64(beta / t_1) / t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (beta + alpha) + 3.0;
	t_1 = 2.0 + (beta + alpha);
	t_2 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 1.02e+152)
		tmp = (((1.0 + beta) * (1.0 + alpha)) / t_2) / (t_0 * t_2);
	else
		tmp = ((1.0 + alpha) / t_1) * ((beta / t_1) / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + 3.0), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1.02e+152], N[(N[(N[(N[(1.0 + beta), $MachinePrecision] * N[(1.0 + alpha), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] / N[(t$95$0 * t$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$1), $MachinePrecision] * N[(N[(beta / t$95$1), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.01999999999999999e152

   1. Initial program 99.4%

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

      1. associate-/l/ 98.2%

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

      2. +-commutative 98.2%

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

      3. associate-+l+ 98.2%

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

      4. *-commutative 98.2%

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

      5. metadata-eval 98.2%

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

      6. associate-+l+ 98.2%

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

      7. metadata-eval 98.2%

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

      8. +-commutative 98.2%

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

      9. +-commutative 98.2%

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

      10. +-commutative 98.2%

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

      11. metadata-eval 98.2%

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

      12. metadata-eval 98.2%

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

      13. associate-+l+ 98.2%

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

   3. Simplified 98.2%

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

      1. div-inv 98.2%

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

      2. +-commutative 98.2%

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

      3. associate-+r+ 98.2%

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

      4. *-commutative 98.2%

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

      5. associate-+r+ 98.2%

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

      6. metadata-eval 98.2%

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

      7. +-commutative 98.2%

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

      8. *-commutative 98.2%

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

      9. associate-+r+ 98.2%

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

      10. +-commutative 98.2%

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

      11. distribute-rgt1-in 98.2%

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

      12. fma-define 98.2%

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

      13. metadata-eval 98.2%

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

      14. associate-+r+ 98.2%

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

   6. Applied egg-rr 98.2%

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

      1. associate-*r/ 98.2%

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

      2. *-rgt-identity 98.2%

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

      3. +-commutative 98.2%

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

      4. fma-undefine 98.2%

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

      5. +-commutative 98.2%

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

      6. *-commutative 98.2%

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

      7. +-commutative 98.2%

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

      8. associate-+r+ 98.2%

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

      9. distribute-lft1-in 98.2%

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

      10. +-commutative 98.2%

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

      11. +-commutative 98.2%

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

      12. +-commutative 98.2%

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

      13. +-commutative 98.2%

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

      14. +-commutative 98.2%

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

      15. +-commutative 98.2%

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

      16. +-commutative 98.2%

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

      17. +-commutative 98.2%

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

      18. associate-+r+ 98.2%

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

      19. +-commutative 98.2%

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

      20. +-commutative 98.2%

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

   8. Simplified 98.2%

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

   if 1.01999999999999999e152 < beta

   1. Initial program 77.5%

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

      1. associate-/l/ 72.5%

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

      2. +-commutative 72.5%

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

      3. associate-+l+ 72.5%

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

      4. *-commutative 72.5%

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

      5. metadata-eval 72.5%

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

      6. associate-+l+ 72.5%

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

      7. metadata-eval 72.5%

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

      8. +-commutative 72.5%

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

      9. +-commutative 72.5%

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

      10. +-commutative 72.5%

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

      11. metadata-eval 72.5%

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

      12. metadata-eval 72.5%

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

      13. associate-+l+ 72.5%

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

   3. Simplified 72.5%

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

      1. div-inv 72.5%

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

      2. +-commutative 72.5%

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

      3. associate-+r+ 72.5%

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

      4. *-commutative 72.5%

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

      5. associate-+r+ 72.5%

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

      6. metadata-eval 72.5%

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

      7. +-commutative 72.5%

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

      8. *-commutative 72.5%

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

      9. associate-+r+ 72.5%

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

      10. +-commutative 72.5%

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

      11. distribute-rgt1-in 72.5%

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

      12. fma-define 72.5%

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

      13. metadata-eval 72.5%

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

      14. associate-+r+ 72.5%

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

   6. Applied egg-rr 72.5%

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

      1. associate-*r/ 72.5%

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

      2. *-rgt-identity 72.5%

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

      3. +-commutative 72.5%

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

      4. fma-undefine 72.5%

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

      5. +-commutative 72.5%

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

      6. *-commutative 72.5%

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

      7. +-commutative 72.5%

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

      8. associate-+r+ 72.5%

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

      9. distribute-lft1-in 72.5%

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

      10. +-commutative 72.5%

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

      11. +-commutative 72.5%

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

      12. +-commutative 72.5%

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

      13. +-commutative 72.5%

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

      14. +-commutative 72.5%

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

      15. +-commutative 72.5%

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

      16. +-commutative 72.5%

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

      17. +-commutative 72.5%

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

      18. associate-+r+ 72.5%

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

      19. +-commutative 72.5%

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

      20. +-commutative 72.5%

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

   8. Simplified 72.5%

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

   9. Taylor expanded in beta around inf 72.5%

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

       1. *-commutative 72.5%

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

   11. Simplified 72.5%

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

       1. *-un-lft-identity 72.5%

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

       2. associate-/l/ 70.8%

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

       3. *-commutative 70.8%

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

       4. associate-+l+ 70.8%

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

       5. associate-+r+ 70.8%

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

       6. +-commutative 70.8%

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

       7. +-commutative 70.8%

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

       8. associate-+l+ 70.8%

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

   13. Applied egg-rr 70.8%

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

       1. associate-*r/ 70.8%

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

       2. times-frac 72.5%

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

       3. *-commutative 72.5%

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

       4. associate-*r/ 85.9%

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

       5. *-commutative 85.9%

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

       6. associate-*r/ 85.9%

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

       7. *-commutative 85.9%

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

       8. *-lft-identity 85.9%

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

       9. times-frac 100.0%

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

       10. +-commutative 100.0%

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

       11. associate-+r+ 100.0%

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

       12. +-commutative 100.0%

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

       13. associate-+r+ 100.0%

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

       14. associate-+r+ 100.0%

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

       15. +-commutative 100.0%

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

   15. Simplified 100.0%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.02 \cdot 10^{+152}:\\ \;\;\;\;\frac{\frac{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}{\alpha + \left(\beta + 2\right)}}{\left(\left(\beta + \alpha\right) + 3\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{2 + \left(\beta + \alpha\right)} \cdot \frac{\frac{\beta}{2 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + \left(\beta + \alpha\right)\\ t_1 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 6.6 \cdot 10^{+18}:\\ \;\;\;\;\frac{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}{t\_1 \cdot \left(t\_1 \cdot \left(\alpha + \left(\beta + 3\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{t\_0} \cdot \frac{\frac{\beta}{t\_0}}{\left(\beta + \alpha\right) + 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (+ beta alpha))) (t_1 (+ alpha (+ beta 2.0))))
   (if (<= beta 6.6e+18)
     (/ (* (+ 1.0 beta) (+ 1.0 alpha)) (* t_1 (* t_1 (+ alpha (+ beta 3.0)))))
     (* (/ (+ 1.0 alpha) t_0) (/ (/ beta t_0) (+ (+ beta alpha) 3.0))))))

C

double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta + alpha);
	double t_1 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 6.6e+18) {
		tmp = ((1.0 + beta) * (1.0 + alpha)) / (t_1 * (t_1 * (alpha + (beta + 3.0))));
	} else {
		tmp = ((1.0 + alpha) / t_0) * ((beta / t_0) / ((beta + alpha) + 3.0));
	}
	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 = 2.0d0 + (beta + alpha)
    t_1 = alpha + (beta + 2.0d0)
    if (beta <= 6.6d+18) then
        tmp = ((1.0d0 + beta) * (1.0d0 + alpha)) / (t_1 * (t_1 * (alpha + (beta + 3.0d0))))
    else
        tmp = ((1.0d0 + alpha) / t_0) * ((beta / t_0) / ((beta + alpha) + 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta + alpha);
	double t_1 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 6.6e+18) {
		tmp = ((1.0 + beta) * (1.0 + alpha)) / (t_1 * (t_1 * (alpha + (beta + 3.0))));
	} else {
		tmp = ((1.0 + alpha) / t_0) * ((beta / t_0) / ((beta + alpha) + 3.0));
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = 2.0 + (beta + alpha)
	t_1 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 6.6e+18:
		tmp = ((1.0 + beta) * (1.0 + alpha)) / (t_1 * (t_1 * (alpha + (beta + 3.0))))
	else:
		tmp = ((1.0 + alpha) / t_0) * ((beta / t_0) / ((beta + alpha) + 3.0))
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(beta + alpha))
	t_1 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 6.6e+18)
		tmp = Float64(Float64(Float64(1.0 + beta) * Float64(1.0 + alpha)) / Float64(t_1 * Float64(t_1 * Float64(alpha + Float64(beta + 3.0)))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / t_0) * Float64(Float64(beta / t_0) / Float64(Float64(beta + alpha) + 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = 2.0 + (beta + alpha);
	t_1 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 6.6e+18)
		tmp = ((1.0 + beta) * (1.0 + alpha)) / (t_1 * (t_1 * (alpha + (beta + 3.0))));
	else
		tmp = ((1.0 + alpha) / t_0) * ((beta / t_0) / ((beta + alpha) + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 6.6e+18], N[(N[(N[(1.0 + beta), $MachinePrecision] * N[(1.0 + alpha), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * N[(t$95$1 * N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(N[(beta / t$95$0), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.6e18

   1. Initial program 99.9%

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

   3. Simplified 94.1%

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

   if 6.6e18 < beta

   1. Initial program 86.6%

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

      1. associate-/l/ 81.8%

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

      2. +-commutative 81.8%

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

      3. associate-+l+ 81.8%

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

      4. *-commutative 81.8%

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

      5. metadata-eval 81.8%

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

      6. associate-+l+ 81.8%

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

      7. metadata-eval 81.8%

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

      8. +-commutative 81.8%

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

      9. +-commutative 81.8%

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

      10. +-commutative 81.8%

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

      11. metadata-eval 81.8%

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

      12. metadata-eval 81.8%

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

      13. associate-+l+ 81.8%

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

   3. Simplified 81.8%

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

      1. div-inv 81.7%

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

      2. +-commutative 81.7%

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

      3. associate-+r+ 81.7%

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

      4. *-commutative 81.7%

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

      5. associate-+r+ 81.7%

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

      6. metadata-eval 81.7%

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

      7. +-commutative 81.7%

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

      8. *-commutative 81.7%

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

      9. associate-+r+ 81.7%

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

      10. +-commutative 81.7%

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

      11. distribute-rgt1-in 81.7%

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

      12. fma-define 81.7%

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

      13. metadata-eval 81.7%

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

      14. associate-+r+ 81.7%

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

   6. Applied egg-rr 81.7%

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

      1. associate-*r/ 81.8%

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

      2. *-rgt-identity 81.8%

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

      3. +-commutative 81.8%

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

      4. fma-undefine 81.8%

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

      5. +-commutative 81.8%

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

      6. *-commutative 81.8%

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

      7. +-commutative 81.8%

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

      8. associate-+r+ 81.8%

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

      9. distribute-lft1-in 81.8%

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

      10. +-commutative 81.8%

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

      11. +-commutative 81.8%

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

      12. +-commutative 81.8%

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

      13. +-commutative 81.8%

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

      14. +-commutative 81.8%

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

      15. +-commutative 81.8%

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

      16. +-commutative 81.8%

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

      17. +-commutative 81.8%

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

      18. associate-+r+ 81.8%

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

      19. +-commutative 81.8%

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

      20. +-commutative 81.8%

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

   8. Simplified 81.8%

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

   9. Taylor expanded in beta around inf 81.8%

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

       1. *-commutative 81.8%

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

   11. Simplified 81.8%

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

       1. *-un-lft-identity 81.8%

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

       2. associate-/l/ 59.9%

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

       3. *-commutative 59.9%

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

       4. associate-+l+ 59.9%

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

       5. associate-+r+ 59.9%

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

       6. +-commutative 59.9%

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

       7. +-commutative 59.9%

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

       8. associate-+l+ 59.9%

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

   13. Applied egg-rr 59.9%

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

       1. associate-*r/ 59.9%

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

       2. times-frac 81.7%

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

       3. *-commutative 81.7%

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

       4. associate-*r/ 90.1%

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

       5. *-commutative 90.1%

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

       6. associate-*r/ 90.2%

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

       7. *-commutative 90.2%

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

       8. *-lft-identity 90.2%

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

       9. times-frac 99.7%

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

       10. +-commutative 99.7%

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

       11. associate-+r+ 99.7%

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

       12. +-commutative 99.7%

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

       13. associate-+r+ 99.7%

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

       14. associate-+r+ 99.7%

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

       15. +-commutative 99.7%

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

   15. Simplified 99.7%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.6 \cdot 10^{+18}:\\ \;\;\;\;\frac{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{2 + \left(\beta + \alpha\right)} \cdot \frac{\frac{\beta}{2 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + \left(\beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 2 \cdot 10^{+18}:\\ \;\;\;\;\frac{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{t\_0} \cdot \frac{\frac{\beta}{t\_0}}{\left(\beta + \alpha\right) + 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (+ beta alpha))))
   (if (<= beta 2e+18)
     (/
      (* (+ 1.0 beta) (+ 1.0 alpha))
      (* (+ alpha (+ beta 2.0)) (* (+ beta 2.0) (+ beta 3.0))))
     (* (/ (+ 1.0 alpha) t_0) (/ (/ beta t_0) (+ (+ beta alpha) 3.0))))))

C

double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta + alpha);
	double tmp;
	if (beta <= 2e+18) {
		tmp = ((1.0 + beta) * (1.0 + alpha)) / ((alpha + (beta + 2.0)) * ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / t_0) * ((beta / t_0) / ((beta + alpha) + 3.0));
	}
	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 = 2.0d0 + (beta + alpha)
    if (beta <= 2d+18) then
        tmp = ((1.0d0 + beta) * (1.0d0 + alpha)) / ((alpha + (beta + 2.0d0)) * ((beta + 2.0d0) * (beta + 3.0d0)))
    else
        tmp = ((1.0d0 + alpha) / t_0) * ((beta / t_0) / ((beta + alpha) + 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta + alpha);
	double tmp;
	if (beta <= 2e+18) {
		tmp = ((1.0 + beta) * (1.0 + alpha)) / ((alpha + (beta + 2.0)) * ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / t_0) * ((beta / t_0) / ((beta + alpha) + 3.0));
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = 2.0 + (beta + alpha)
	tmp = 0
	if beta <= 2e+18:
		tmp = ((1.0 + beta) * (1.0 + alpha)) / ((alpha + (beta + 2.0)) * ((beta + 2.0) * (beta + 3.0)))
	else:
		tmp = ((1.0 + alpha) / t_0) * ((beta / t_0) / ((beta + alpha) + 3.0))
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(beta + alpha))
	tmp = 0.0
	if (beta <= 2e+18)
		tmp = Float64(Float64(Float64(1.0 + beta) * Float64(1.0 + alpha)) / Float64(Float64(alpha + Float64(beta + 2.0)) * Float64(Float64(beta + 2.0) * Float64(beta + 3.0))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / t_0) * Float64(Float64(beta / t_0) / Float64(Float64(beta + alpha) + 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = 2.0 + (beta + alpha);
	tmp = 0.0;
	if (beta <= 2e+18)
		tmp = ((1.0 + beta) * (1.0 + alpha)) / ((alpha + (beta + 2.0)) * ((beta + 2.0) * (beta + 3.0)));
	else
		tmp = ((1.0 + alpha) / t_0) * ((beta / t_0) / ((beta + alpha) + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2e+18], N[(N[(N[(1.0 + beta), $MachinePrecision] * N[(1.0 + alpha), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(N[(beta / t$95$0), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2e18

   1. Initial program 99.9%

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

   3. Simplified 94.1%

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

   5. Taylor expanded in alpha around 0 67.3%

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

      1. +-commutative 67.3%

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

      2. +-commutative 67.3%

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

   7. Simplified 67.3%

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

   if 2e18 < beta

   1. Initial program 86.6%

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

      1. associate-/l/ 81.8%

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

      2. +-commutative 81.8%

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

      3. associate-+l+ 81.8%

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

      4. *-commutative 81.8%

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

      5. metadata-eval 81.8%

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

      6. associate-+l+ 81.8%

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

      7. metadata-eval 81.8%

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

      8. +-commutative 81.8%

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

      9. +-commutative 81.8%

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

      10. +-commutative 81.8%

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

      11. metadata-eval 81.8%

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

      12. metadata-eval 81.8%

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

      13. associate-+l+ 81.8%

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

   3. Simplified 81.8%

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

      1. div-inv 81.7%

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

      2. +-commutative 81.7%

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

      3. associate-+r+ 81.7%

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

      4. *-commutative 81.7%

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

      5. associate-+r+ 81.7%

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

      6. metadata-eval 81.7%

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

      7. +-commutative 81.7%

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

      8. *-commutative 81.7%

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

      9. associate-+r+ 81.7%

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

      10. +-commutative 81.7%

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

      11. distribute-rgt1-in 81.7%

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

      12. fma-define 81.7%

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

      13. metadata-eval 81.7%

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

      14. associate-+r+ 81.7%

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

   6. Applied egg-rr 81.7%

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

      1. associate-*r/ 81.8%

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

      2. *-rgt-identity 81.8%

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

      3. +-commutative 81.8%

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

      4. fma-undefine 81.8%

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

      5. +-commutative 81.8%

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

      6. *-commutative 81.8%

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

      7. +-commutative 81.8%

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

      8. associate-+r+ 81.8%

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

      9. distribute-lft1-in 81.8%

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

      10. +-commutative 81.8%

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

      11. +-commutative 81.8%

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

      12. +-commutative 81.8%

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

      13. +-commutative 81.8%

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

      14. +-commutative 81.8%

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

      15. +-commutative 81.8%

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

      16. +-commutative 81.8%

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

      17. +-commutative 81.8%

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

      18. associate-+r+ 81.8%

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

      19. +-commutative 81.8%

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

      20. +-commutative 81.8%

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

   8. Simplified 81.8%

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

   9. Taylor expanded in beta around inf 81.8%

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

       1. *-commutative 81.8%

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

   11. Simplified 81.8%

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

       1. *-un-lft-identity 81.8%

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

       2. associate-/l/ 59.9%

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

       3. *-commutative 59.9%

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

       4. associate-+l+ 59.9%

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

       5. associate-+r+ 59.9%

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

       6. +-commutative 59.9%

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

       7. +-commutative 59.9%

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

       8. associate-+l+ 59.9%

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

   13. Applied egg-rr 59.9%

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

       1. associate-*r/ 59.9%

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

       2. times-frac 81.7%

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

       3. *-commutative 81.7%

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

       4. associate-*r/ 90.1%

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

       5. *-commutative 90.1%

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

       6. associate-*r/ 90.2%

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

       7. *-commutative 90.2%

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

       8. *-lft-identity 90.2%

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

       9. times-frac 99.7%

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

       10. +-commutative 99.7%

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

       11. associate-+r+ 99.7%

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

       12. +-commutative 99.7%

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

       13. associate-+r+ 99.7%

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

       14. associate-+r+ 99.7%

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

       15. +-commutative 99.7%

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

   15. Simplified 99.7%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2 \cdot 10^{+18}:\\ \;\;\;\;\frac{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{2 + \left(\beta + \alpha\right)} \cdot \frac{\frac{\beta}{2 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 7 \cdot 10^{+18}:\\ \;\;\;\;\frac{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}{t\_0 \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{t\_0}}{\left(\beta + \alpha\right) + 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 7e+18)
     (/ (* (+ 1.0 beta) (+ 1.0 alpha)) (* t_0 (* (+ beta 2.0) (+ beta 3.0))))
     (/ (/ (+ 1.0 alpha) t_0) (+ (+ beta alpha) 3.0)))))

C

double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 7e+18) {
		tmp = ((1.0 + beta) * (1.0 + alpha)) / (t_0 * ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / t_0) / ((beta + alpha) + 3.0);
	}
	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 + 2.0d0)
    if (beta <= 7d+18) then
        tmp = ((1.0d0 + beta) * (1.0d0 + alpha)) / (t_0 * ((beta + 2.0d0) * (beta + 3.0d0)))
    else
        tmp = ((1.0d0 + alpha) / t_0) / ((beta + alpha) + 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 7e+18) {
		tmp = ((1.0 + beta) * (1.0 + alpha)) / (t_0 * ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / t_0) / ((beta + alpha) + 3.0);
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 7e+18:
		tmp = ((1.0 + beta) * (1.0 + alpha)) / (t_0 * ((beta + 2.0) * (beta + 3.0)))
	else:
		tmp = ((1.0 + alpha) / t_0) / ((beta + alpha) + 3.0)
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 7e+18)
		tmp = Float64(Float64(Float64(1.0 + beta) * Float64(1.0 + alpha)) / Float64(t_0 * Float64(Float64(beta + 2.0) * Float64(beta + 3.0))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / t_0) / Float64(Float64(beta + alpha) + 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 7e+18)
		tmp = ((1.0 + beta) * (1.0 + alpha)) / (t_0 * ((beta + 2.0) * (beta + 3.0)));
	else
		tmp = ((1.0 + alpha) / t_0) / ((beta + alpha) + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 7e+18], N[(N[(N[(1.0 + beta), $MachinePrecision] * N[(1.0 + alpha), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 7e18

   1. Initial program 99.9%

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

   3. Simplified 94.1%

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

   5. Taylor expanded in alpha around 0 67.3%

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

      1. +-commutative 67.3%

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

      2. +-commutative 67.3%

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

   7. Simplified 67.3%

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

   if 7e18 < beta

   1. Initial program 86.6%

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

      1. div-inv 86.6%

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

      2. +-commutative 86.6%

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

      3. *-commutative 86.6%

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

      4. associate-+r+ 86.6%

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

      5. +-commutative 86.6%

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

      6. distribute-rgt1-in 86.6%

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

      7. fma-define 86.6%

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

      8. metadata-eval 86.6%

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

      9. associate-+r+ 86.6%

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

      10. metadata-eval 86.6%

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

      11. associate-+r+ 86.6%

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

   4. Applied egg-rr 86.6%

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

      1. associate-*l/ 86.6%

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

      2. associate-*r/ 86.6%

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

      3. *-rgt-identity 86.6%

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

      4. +-commutative 86.6%

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

      5. fma-undefine 86.6%

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

      6. +-commutative 86.6%

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

      7. *-commutative 86.6%

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

      8. +-commutative 86.6%

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

      9. associate-+r+ 86.6%

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

      10. distribute-lft1-in 86.6%

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

      11. +-commutative 86.6%

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

      12. +-commutative 86.6%

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

      13. +-commutative 86.6%

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

      14. +-commutative 86.6%

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

      15. +-commutative 86.6%

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

      16. +-commutative 86.6%

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

      17. +-commutative 86.6%

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

      18. +-commutative 86.6%

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

   6. Simplified 86.6%

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

      1. associate-/l* 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+r+ 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in alpha around 0 99.8%

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

       1. +-commutative 99.8%

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

   11. Simplified 99.8%

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

   12. Taylor expanded in beta around inf 85.1%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 7 \cdot 10^{+18}:\\ \;\;\;\;\frac{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}}{\left(\beta + \alpha\right) + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ t_1 := \left(\beta + \alpha\right) + 3\\ \mathbf{if}\;\beta \leq 3 \cdot 10^{+18}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 2}}{t\_1 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{t\_0}}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))) (t_1 (+ (+ beta alpha) 3.0)))
   (if (<= beta 3e+18)
     (/ (/ (+ 1.0 beta) (+ beta 2.0)) (* t_1 t_0))
     (/ (/ (+ 1.0 alpha) t_0) t_1))))

C

double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double t_1 = (beta + alpha) + 3.0;
	double tmp;
	if (beta <= 3e+18) {
		tmp = ((1.0 + beta) / (beta + 2.0)) / (t_1 * t_0);
	} else {
		tmp = ((1.0 + alpha) / t_0) / t_1;
	}
	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 + 2.0d0)
    t_1 = (beta + alpha) + 3.0d0
    if (beta <= 3d+18) then
        tmp = ((1.0d0 + beta) / (beta + 2.0d0)) / (t_1 * t_0)
    else
        tmp = ((1.0d0 + alpha) / t_0) / t_1
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double t_1 = (beta + alpha) + 3.0;
	double tmp;
	if (beta <= 3e+18) {
		tmp = ((1.0 + beta) / (beta + 2.0)) / (t_1 * t_0);
	} else {
		tmp = ((1.0 + alpha) / t_0) / t_1;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	t_1 = (beta + alpha) + 3.0
	tmp = 0
	if beta <= 3e+18:
		tmp = ((1.0 + beta) / (beta + 2.0)) / (t_1 * t_0)
	else:
		tmp = ((1.0 + alpha) / t_0) / t_1
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	t_1 = Float64(Float64(beta + alpha) + 3.0)
	tmp = 0.0
	if (beta <= 3e+18)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(beta + 2.0)) / Float64(t_1 * t_0));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / t_0) / t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	t_1 = (beta + alpha) + 3.0;
	tmp = 0.0;
	if (beta <= 3e+18)
		tmp = ((1.0 + beta) / (beta + 2.0)) / (t_1 * t_0);
	else
		tmp = ((1.0 + alpha) / t_0) / t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(beta + alpha), $MachinePrecision] + 3.0), $MachinePrecision]}, If[LessEqual[beta, 3e+18], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 3e18

   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/ 99.6%

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

      2. +-commutative 99.6%

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

      3. associate-+l+ 99.6%

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

      4. *-commutative 99.6%

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

      5. metadata-eval 99.6%

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

      6. associate-+l+ 99.6%

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

      7. metadata-eval 99.6%

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

      8. +-commutative 99.6%

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

      9. +-commutative 99.6%

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

      10. +-commutative 99.6%

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

      11. metadata-eval 99.6%

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

      12. metadata-eval 99.6%

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

      13. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in alpha around 0 83.5%

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

      1. +-commutative 83.5%

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

   7. Simplified 83.5%

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

   if 3e18 < beta

   1. Initial program 86.6%

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

      1. div-inv 86.6%

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

      2. +-commutative 86.6%

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

      3. *-commutative 86.6%

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

      4. associate-+r+ 86.6%

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

      5. +-commutative 86.6%

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

      6. distribute-rgt1-in 86.6%

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

      7. fma-define 86.6%

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

      8. metadata-eval 86.6%

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

      9. associate-+r+ 86.6%

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

      10. metadata-eval 86.6%

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

      11. associate-+r+ 86.6%

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

   4. Applied egg-rr 86.6%

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

      1. associate-*l/ 86.6%

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

      2. associate-*r/ 86.6%

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

      3. *-rgt-identity 86.6%

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

      4. +-commutative 86.6%

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

      5. fma-undefine 86.6%

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

      6. +-commutative 86.6%

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

      7. *-commutative 86.6%

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

      8. +-commutative 86.6%

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

      9. associate-+r+ 86.6%

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

      10. distribute-lft1-in 86.6%

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

      11. +-commutative 86.6%

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

      12. +-commutative 86.6%

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

      13. +-commutative 86.6%

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

      14. +-commutative 86.6%

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

      15. +-commutative 86.6%

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

      16. +-commutative 86.6%

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

      17. +-commutative 86.6%

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

      18. +-commutative 86.6%

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

   6. Simplified 86.6%

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

      1. associate-/l* 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+r+ 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in alpha around 0 99.8%

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

       1. +-commutative 99.8%

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

   11. Simplified 99.8%

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

   12. Taylor expanded in beta around inf 85.1%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3 \cdot 10^{+18}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 2}}{\left(\left(\beta + \alpha\right) + 3\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}}{\left(\beta + \alpha\right) + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.7:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + 2}}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}}{\left(\beta + \alpha\right) + 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.7)
   (/ (/ (+ 1.0 alpha) (+ alpha 2.0)) (* (+ alpha 2.0) (+ alpha 3.0)))
   (/ (/ (+ 1.0 alpha) (+ alpha (+ beta 2.0))) (+ (+ beta alpha) 3.0))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.7) {
		tmp = ((1.0 + alpha) / (alpha + 2.0)) / ((alpha + 2.0) * (alpha + 3.0));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) / ((beta + alpha) + 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 <= 1.7d0) then
        tmp = ((1.0d0 + alpha) / (alpha + 2.0d0)) / ((alpha + 2.0d0) * (alpha + 3.0d0))
    else
        tmp = ((1.0d0 + alpha) / (alpha + (beta + 2.0d0))) / ((beta + alpha) + 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.7) {
		tmp = ((1.0 + alpha) / (alpha + 2.0)) / ((alpha + 2.0) * (alpha + 3.0));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) / ((beta + alpha) + 3.0);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 1.7:
		tmp = ((1.0 + alpha) / (alpha + 2.0)) / ((alpha + 2.0) * (alpha + 3.0))
	else:
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) / ((beta + alpha) + 3.0)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.7)
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + 2.0)) / Float64(Float64(alpha + 2.0) * Float64(alpha + 3.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(beta + 2.0))) / Float64(Float64(beta + alpha) + 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.7)
		tmp = ((1.0 + alpha) / (alpha + 2.0)) / ((alpha + 2.0) * (alpha + 3.0));
	else
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) / ((beta + alpha) + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 1.7], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + 2.0), $MachinePrecision] * N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 3.0), $MachinePrecision]), $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/ 99.6%

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

      2. +-commutative 99.6%

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

      3. associate-+l+ 99.6%

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

      4. *-commutative 99.6%

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

      5. metadata-eval 99.6%

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

      6. associate-+l+ 99.6%

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

      7. metadata-eval 99.6%

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

      8. +-commutative 99.6%

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

      9. +-commutative 99.6%

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

      10. +-commutative 99.6%

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

      11. metadata-eval 99.6%

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

      12. metadata-eval 99.6%

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

      13. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in beta around 0 98.6%

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

   6. Taylor expanded in beta around 0 98.7%

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

      1. +-commutative 98.7%

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

      2. +-commutative 98.7%

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

   8. Simplified 98.7%

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

   if 1.69999999999999996 < beta

   1. Initial program 87.1%

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

      1. div-inv 87.1%

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

      2. +-commutative 87.1%

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

      3. *-commutative 87.1%

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

      4. associate-+r+ 87.1%

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

      5. +-commutative 87.1%

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

      6. distribute-rgt1-in 87.1%

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

      7. fma-define 87.1%

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

      8. metadata-eval 87.1%

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

      9. associate-+r+ 87.1%

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

      10. metadata-eval 87.1%

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

      11. associate-+r+ 87.1%

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

   4. Applied egg-rr 87.1%

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

      1. associate-*l/ 87.1%

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

      2. associate-*r/ 87.1%

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

      3. *-rgt-identity 87.1%

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

      4. +-commutative 87.1%

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

      5. fma-undefine 87.1%

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

      6. +-commutative 87.1%

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

      7. *-commutative 87.1%

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

      8. +-commutative 87.1%

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

      9. associate-+r+ 87.1%

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

      10. distribute-lft1-in 87.1%

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

      11. +-commutative 87.1%

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

      12. +-commutative 87.1%

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

      13. +-commutative 87.1%

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

      14. +-commutative 87.1%

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

      15. +-commutative 87.1%

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

      16. +-commutative 87.1%

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

      17. +-commutative 87.1%

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

      18. +-commutative 87.1%

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

   6. Simplified 87.1%

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

      1. associate-/l* 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+r+ 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in alpha around 0 99.8%

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

       1. +-commutative 99.8%

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

   11. Simplified 99.8%

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

   12. Taylor expanded in beta around inf 84.5%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.7:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + 2}}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}}{\left(\beta + \alpha\right) + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 92.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.4:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + 2}}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 3\right)}}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.4)
   (/ (/ (+ 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 <= 4.4) {
		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 <= 4.4d0) 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 <= 4.4) {
		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 <= 4.4:
		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 <= 4.4)
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + 2.0)) / Float64(Float64(alpha + 2.0) * Float64(alpha + 3.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(beta + 3.0))) / beta);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4.4)
		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, 4.4], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + 2.0), $MachinePrecision] * N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.4000000000000004

   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/ 99.6%

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

      2. +-commutative 99.6%

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

      3. associate-+l+ 99.6%

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

      4. *-commutative 99.6%

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

      5. metadata-eval 99.6%

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

      6. associate-+l+ 99.6%

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

      7. metadata-eval 99.6%

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

      8. +-commutative 99.6%

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

      9. +-commutative 99.6%

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

      10. +-commutative 99.6%

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

      11. metadata-eval 99.6%

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

      12. metadata-eval 99.6%

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

      13. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in beta around 0 98.6%

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

   6. Taylor expanded in beta around 0 98.7%

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

      1. +-commutative 98.7%

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

      2. +-commutative 98.7%

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

   8. Simplified 98.7%

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

   if 4.4000000000000004 < beta

   1. Initial program 87.1%

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

   3. Taylor expanded in beta around inf 84.0%

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

      1. *-un-lft-identity 84.0%

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

      2. associate-/l/ 83.7%

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

      3. metadata-eval 83.7%

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

      4. associate-+l+ 83.7%

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

      5. metadata-eval 83.7%

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

      6. associate-+r+ 83.7%

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

   5. Applied egg-rr 83.7%

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

      1. *-lft-identity 83.7%

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

      2. associate-/r* 84.0%

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

      3. +-commutative 84.0%

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

   7. Simplified 84.0%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.4:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + 2}}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 3\right)}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 83.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.6:\\ \;\;\;\;\frac{0.5}{\left(\left(\beta + \alpha\right) + 3\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 3\right)}}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5.6)
   (/ 0.5 (* (+ (+ beta alpha) 3.0) (+ alpha (+ beta 2.0))))
   (/ (/ (+ 1.0 alpha) (+ alpha (+ beta 3.0))) beta)))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.6) {
		tmp = 0.5 / (((beta + alpha) + 3.0) * (alpha + (beta + 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 <= 5.6d0) then
        tmp = 0.5d0 / (((beta + alpha) + 3.0d0) * (alpha + (beta + 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 <= 5.6) {
		tmp = 0.5 / (((beta + alpha) + 3.0) * (alpha + (beta + 2.0)));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (beta + 3.0))) / beta;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 5.6:
		tmp = 0.5 / (((beta + alpha) + 3.0) * (alpha + (beta + 2.0)))
	else:
		tmp = ((1.0 + alpha) / (alpha + (beta + 3.0))) / beta
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 5.6)
		tmp = Float64(0.5 / Float64(Float64(Float64(beta + alpha) + 3.0) * Float64(alpha + Float64(beta + 2.0))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(beta + 3.0))) / beta);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 5.6)
		tmp = 0.5 / (((beta + alpha) + 3.0) * (alpha + (beta + 2.0)));
	else
		tmp = ((1.0 + alpha) / (alpha + (beta + 3.0))) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 5.6], N[(0.5 / N[(N[(N[(beta + alpha), $MachinePrecision] + 3.0), $MachinePrecision] * N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.5999999999999996

   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/ 99.6%

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

      2. +-commutative 99.6%

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

      3. associate-+l+ 99.6%

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

      4. *-commutative 99.6%

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

      5. metadata-eval 99.6%

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

      6. associate-+l+ 99.6%

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

      7. metadata-eval 99.6%

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

      8. +-commutative 99.6%

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

      9. +-commutative 99.6%

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

      10. +-commutative 99.6%

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

      11. metadata-eval 99.6%

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

      12. metadata-eval 99.6%

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

      13. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in beta around 0 98.6%

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

   6. Taylor expanded in alpha around 0 82.6%

      \[\leadsto \frac{\color{blue}{0.5}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]

   if 5.5999999999999996 < beta

   1. Initial program 87.1%

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

   3. Taylor expanded in beta around inf 84.0%

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

      1. *-un-lft-identity 84.0%

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

      2. associate-/l/ 83.7%

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

      3. metadata-eval 83.7%

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

      4. associate-+l+ 83.7%

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

      5. metadata-eval 83.7%

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

      6. associate-+r+ 83.7%

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

   5. Applied egg-rr 83.7%

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

      1. *-lft-identity 83.7%

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

      2. associate-/r* 84.0%

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

      3. +-commutative 84.0%

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

   7. Simplified 84.0%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.6:\\ \;\;\;\;\frac{0.5}{\left(\left(\beta + \alpha\right) + 3\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 3\right)}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 71.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.4:\\ \;\;\;\;\frac{\frac{0.5}{\beta + 2}}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 3\right)}}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.4)
   (/ (/ 0.5 (+ beta 2.0)) (+ beta 3.0))
   (/ (/ (+ 1.0 alpha) (+ alpha (+ beta 3.0))) beta)))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.4) {
		tmp = (0.5 / (beta + 2.0)) / (beta + 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 <= 4.4d0) then
        tmp = (0.5d0 / (beta + 2.0d0)) / (beta + 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 <= 4.4) {
		tmp = (0.5 / (beta + 2.0)) / (beta + 3.0);
	} else {
		tmp = ((1.0 + alpha) / (alpha + (beta + 3.0))) / beta;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 4.4:
		tmp = (0.5 / (beta + 2.0)) / (beta + 3.0)
	else:
		tmp = ((1.0 + alpha) / (alpha + (beta + 3.0))) / beta
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.4)
		tmp = Float64(Float64(0.5 / Float64(beta + 2.0)) / Float64(beta + 3.0));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(beta + 3.0))) / beta);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4.4)
		tmp = (0.5 / (beta + 2.0)) / (beta + 3.0);
	else
		tmp = ((1.0 + alpha) / (alpha + (beta + 3.0))) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 4.4], N[(N[(0.5 / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.4000000000000004

   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/ 99.6%

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

      2. +-commutative 99.6%

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

      3. associate-+l+ 99.6%

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

      4. *-commutative 99.6%

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

      5. metadata-eval 99.6%

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

      6. associate-+l+ 99.6%

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

      7. metadata-eval 99.6%

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

      8. +-commutative 99.6%

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

      9. +-commutative 99.6%

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

      10. +-commutative 99.6%

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

      11. metadata-eval 99.6%

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

      12. metadata-eval 99.6%

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

      13. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in beta around 0 98.6%

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

   6. Taylor expanded in alpha around 0 66.6%

      \[\leadsto \color{blue}{\frac{0.5}{\left(2 + \beta\right) \cdot \left(3 + \beta\right)}} \]
   7. Step-by-step derivation

      1. associate-/r* 66.6%

         \[\leadsto \color{blue}{\frac{\frac{0.5}{2 + \beta}}{3 + \beta}} \]

   8. Simplified 66.6%

      \[\leadsto \color{blue}{\frac{\frac{0.5}{2 + \beta}}{3 + \beta}} \]

   if 4.4000000000000004 < beta

   1. Initial program 87.1%

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

   3. Taylor expanded in beta around inf 84.0%

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

      1. *-un-lft-identity 84.0%

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

      2. associate-/l/ 83.7%

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

      3. metadata-eval 83.7%

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

      4. associate-+l+ 83.7%

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

      5. metadata-eval 83.7%

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

      6. associate-+r+ 83.7%

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

   5. Applied egg-rr 83.7%

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

      1. *-lft-identity 83.7%

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

      2. associate-/r* 84.0%

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

      3. +-commutative 84.0%

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

   7. Simplified 84.0%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.4:\\ \;\;\;\;\frac{\frac{0.5}{\beta + 2}}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 3\right)}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 71.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.5:\\ \;\;\;\;\frac{\frac{0.5}{\beta + 2}}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.5)
   (/ (/ 0.5 (+ beta 2.0)) (+ beta 3.0))
   (/ (/ (+ 1.0 alpha) beta) (+ beta 3.0))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.5) {
		tmp = (0.5 / (beta + 2.0)) / (beta + 3.0);
	} else {
		tmp = ((1.0 + alpha) / beta) / (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 <= 4.5d0) then
        tmp = (0.5d0 / (beta + 2.0d0)) / (beta + 3.0d0)
    else
        tmp = ((1.0d0 + alpha) / beta) / (beta + 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.5) {
		tmp = (0.5 / (beta + 2.0)) / (beta + 3.0);
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 4.5:
		tmp = (0.5 / (beta + 2.0)) / (beta + 3.0)
	else:
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.5)
		tmp = Float64(Float64(0.5 / Float64(beta + 2.0)) / Float64(beta + 3.0));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(beta + 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4.5)
		tmp = (0.5 / (beta + 2.0)) / (beta + 3.0);
	else
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 4.5], N[(N[(0.5 / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.5

   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/ 99.6%

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

      2. +-commutative 99.6%

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

      3. associate-+l+ 99.6%

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

      4. *-commutative 99.6%

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

      5. metadata-eval 99.6%

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

      6. associate-+l+ 99.6%

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

      7. metadata-eval 99.6%

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

      8. +-commutative 99.6%

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

      9. +-commutative 99.6%

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

      10. +-commutative 99.6%

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

      11. metadata-eval 99.6%

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

      12. metadata-eval 99.6%

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

      13. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in beta around 0 98.6%

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

   6. Taylor expanded in alpha around 0 66.6%

      \[\leadsto \color{blue}{\frac{0.5}{\left(2 + \beta\right) \cdot \left(3 + \beta\right)}} \]
   7. Step-by-step derivation

      1. associate-/r* 66.6%

         \[\leadsto \color{blue}{\frac{\frac{0.5}{2 + \beta}}{3 + \beta}} \]

   8. Simplified 66.6%

      \[\leadsto \color{blue}{\frac{\frac{0.5}{2 + \beta}}{3 + \beta}} \]

   if 4.5 < beta

   1. Initial program 87.1%

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

   3. Taylor expanded in beta around inf 84.0%

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

   4. Taylor expanded in alpha around 0 83.8%

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

      1. +-commutative 83.8%

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

   6. Simplified 83.8%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.5:\\ \;\;\;\;\frac{\frac{0.5}{\beta + 2}}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 71.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 9.6:\\ \;\;\;\;\frac{\frac{0.5}{\beta + 2}}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 9.6)
   (/ (/ 0.5 (+ beta 2.0)) (+ beta 3.0))
   (/ (/ (+ 1.0 alpha) beta) beta)))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 9.6) {
		tmp = (0.5 / (beta + 2.0)) / (beta + 3.0);
	} 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 <= 9.6d0) then
        tmp = (0.5d0 / (beta + 2.0d0)) / (beta + 3.0d0)
    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 <= 9.6) {
		tmp = (0.5 / (beta + 2.0)) / (beta + 3.0);
	} else {
		tmp = ((1.0 + alpha) / beta) / beta;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 9.6:
		tmp = (0.5 / (beta + 2.0)) / (beta + 3.0)
	else:
		tmp = ((1.0 + alpha) / beta) / beta
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 9.6)
		tmp = Float64(Float64(0.5 / Float64(beta + 2.0)) / Float64(beta + 3.0));
	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 <= 9.6)
		tmp = (0.5 / (beta + 2.0)) / (beta + 3.0);
	else
		tmp = ((1.0 + alpha) / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 9.6], N[(N[(0.5 / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 9.59999999999999964

   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/ 99.6%

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

      2. +-commutative 99.6%

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

      3. associate-+l+ 99.6%

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

      4. *-commutative 99.6%

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

      5. metadata-eval 99.6%

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

      6. associate-+l+ 99.6%

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

      7. metadata-eval 99.6%

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

      8. +-commutative 99.6%

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

      9. +-commutative 99.6%

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

      10. +-commutative 99.6%

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

      11. metadata-eval 99.6%

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

      12. metadata-eval 99.6%

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

      13. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in beta around 0 98.6%

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

   6. Taylor expanded in alpha around 0 66.6%

      \[\leadsto \color{blue}{\frac{0.5}{\left(2 + \beta\right) \cdot \left(3 + \beta\right)}} \]
   7. Step-by-step derivation

      1. associate-/r* 66.6%

         \[\leadsto \color{blue}{\frac{\frac{0.5}{2 + \beta}}{3 + \beta}} \]

   8. Simplified 66.6%

      \[\leadsto \color{blue}{\frac{\frac{0.5}{2 + \beta}}{3 + \beta}} \]

   if 9.59999999999999964 < beta

   1. Initial program 87.1%

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

   3. Taylor expanded in beta around inf 84.0%

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

   4. Taylor expanded in alpha around -inf 73.9%

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

      1. associate-*r* 73.9%

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

      2. mul-1-neg 73.9%

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

      3. sub-neg 73.9%

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

      4. associate-*r/ 73.9%

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

      5. distribute-lft-in 73.9%

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

      6. metadata-eval 73.9%

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

      7. metadata-eval 73.9%

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

      8. mul-1-neg 73.9%

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

      9. unsub-neg 73.9%

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

      10. metadata-eval 73.9%

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

      11. metadata-eval 73.9%

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

   6. Simplified 73.9%

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

   7. Taylor expanded in beta around inf 83.8%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 9.6:\\ \;\;\;\;\frac{\frac{0.5}{\beta + 2}}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 30.3% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1 + \alpha}{\beta}}{\beta} \end{array} \]

FPCore

(FPCore (alpha beta) :precision binary64 (/ (/ (+ 1.0 alpha) beta) beta))

C

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

Fortran

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

Java

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

Python

def code(alpha, beta):
	return ((1.0 + alpha) / beta) / beta

Julia

function code(alpha, beta)
	return Float64(Float64(Float64(1.0 + alpha) / beta) / beta)
end

MATLAB

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

Wolfram

code[alpha_, beta_] := N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]

Derivation

1. Initial program 95.2%

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

3. Taylor expanded in beta around inf 32.5%

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

4. Taylor expanded in alpha around -inf 28.8%

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

   1. associate-*r* 28.8%

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

   2. mul-1-neg 28.8%

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

   3. sub-neg 28.8%

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

   4. associate-*r/ 28.8%

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

   5. distribute-lft-in 28.8%

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

   6. metadata-eval 28.8%

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

   7. metadata-eval 28.8%

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

   8. mul-1-neg 28.8%

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

   9. unsub-neg 28.8%

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

   10. metadata-eval 28.8%

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

   11. metadata-eval 28.8%

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

6. Simplified 28.8%

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

7. Taylor expanded in beta around inf 32.6%

   \[\leadsto \frac{\frac{1 + \alpha}{\beta}}{\color{blue}{\beta}} \]
8. Add Preprocessing

Alternative 14: 27.7% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{\beta + 3}}{\beta} \end{array} \]

FPCore

(FPCore (alpha beta) :precision binary64 (/ (/ 1.0 (+ beta 3.0)) beta))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[alpha_, beta_] := N[(N[(1.0 / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]

Derivation

1. Initial program 95.2%

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

3. Taylor expanded in beta around inf 32.5%

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

   1. *-un-lft-identity 32.5%

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

   2. associate-/l/ 32.4%

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

   3. metadata-eval 32.4%

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

   4. associate-+l+ 32.4%

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

   5. metadata-eval 32.4%

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

   6. associate-+r+ 32.4%

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

5. Applied egg-rr 32.4%

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

   1. *-un-lft-identity 32.4%

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

   2. associate-+r+ 32.4%

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

   3. metadata-eval 32.4%

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

   4. associate-+l+ 32.4%

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

   5. metadata-eval 32.4%

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

   6. associate-/r* 32.5%

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

   7. metadata-eval 32.5%

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

   8. associate-+l+ 32.5%

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

   9. metadata-eval 32.5%

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

   10. associate-+r+ 32.5%

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

7. Applied egg-rr 32.5%

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

8. Taylor expanded in alpha around 0 30.7%

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

9. Final simplification 30.7%

   \[\leadsto \frac{\frac{1}{\beta + 3}}{\beta} \]
10. Add Preprocessing

Alternative 15: 27.7% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{\beta}}{\beta + 3} \end{array} \]

FPCore

(FPCore (alpha beta) :precision binary64 (/ (/ 1.0 beta) (+ beta 3.0)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[alpha_, beta_] := N[(N[(1.0 / beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.2%

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

3. Taylor expanded in beta around inf 32.5%

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

   1. *-un-lft-identity 32.5%

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

   2. associate-/l/ 32.4%

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

   3. metadata-eval 32.4%

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

   4. associate-+l+ 32.4%

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

   5. metadata-eval 32.4%

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

   6. associate-+r+ 32.4%

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

5. Applied egg-rr 32.4%

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

6. Taylor expanded in alpha around 0 30.6%

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

   1. associate-/r* 30.7%

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

8. Simplified 30.7%

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

9. Final simplification 30.7%

   \[\leadsto \frac{\frac{1}{\beta}}{\beta + 3} \]
10. Add Preprocessing

Alternative 16: 27.5% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\beta \cdot \left(\beta + 3\right)} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[alpha_, beta_] := N[(1.0 / N[(beta * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.2%

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

3. Taylor expanded in beta around inf 32.5%

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

4. Taylor expanded in alpha around 0 30.6%

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

5. Final simplification 30.6%

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

Alternative 17: 27.9% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\beta \cdot \beta} \end{array} \]

FPCore

(FPCore (alpha beta) :precision binary64 (/ 1.0 (* beta beta)))

C

double code(double alpha, double beta) {
	return 1.0 / (beta * beta);
}

Fortran

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

Java

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

Python

def code(alpha, beta):
	return 1.0 / (beta * beta)

Julia

function code(alpha, beta)
	return Float64(1.0 / Float64(beta * beta))
end

MATLAB

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

Wolfram

code[alpha_, beta_] := N[(1.0 / N[(beta * beta), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.2%

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

3. Taylor expanded in beta around inf 32.5%

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

4. Taylor expanded in alpha around 0 30.6%

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

5. Taylor expanded in beta around inf 30.9%

   \[\leadsto \frac{1}{\beta \cdot \color{blue}{\beta}} \]
6. Add Preprocessing

Alternative 18: 4.3% accurate, 11.7× speedup

Math

\[\begin{array}{l} \\ \frac{0.3333333333333333}{\beta} \end{array} \]

FPCore

(FPCore (alpha beta) :precision binary64 (/ 0.3333333333333333 beta))

C

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

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = 0.3333333333333333d0 / beta
end function

Java

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

Python

def code(alpha, beta):
	return 0.3333333333333333 / beta

Julia

function code(alpha, beta)
	return Float64(0.3333333333333333 / beta)
end

MATLAB

function tmp = code(alpha, beta)
	tmp = 0.3333333333333333 / beta;
end

Wolfram

code[alpha_, beta_] := N[(0.3333333333333333 / beta), $MachinePrecision]

Derivation

1. Initial program 95.2%

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

3. Taylor expanded in beta around inf 32.5%

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

4. Taylor expanded in alpha around 0 30.6%

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

5. Taylor expanded in beta around 0 4.5%

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

Reproduce

herbie shell --seed 2024143 
(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)))