Octave 3.8, jcobi/3

Percentage Accurate: 94.8% → 99.8%

Time: 13.3s

Alternatives: 20

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: 94.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.9%

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

3. Simplified 82.7%

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

   1. times-frac 94.3%

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

   2. +-commutative 94.3%

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

6. Applied egg-rr 94.3%

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

   1. associate-*l/ 94.3%

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

   2. associate-+r+ 94.3%

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

   3. associate-/r* 99.8%

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

   4. associate-+r+ 99.8%

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

8. Applied egg-rr 99.8%

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

Alternative 2: 92.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 5.2 \cdot 10^{+84}:\\ \;\;\;\;\frac{\left(\alpha + 1\right) \cdot \left(1 + \beta\right)}{t\_0 \cdot \left(t\_0 \cdot \left(\alpha + \left(\beta + 3\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\alpha + 1\right) \cdot \frac{1 - \frac{4 + \alpha \cdot 2}{\beta}}{\beta}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 5.2e+84)
     (/ (* (+ alpha 1.0) (+ 1.0 beta)) (* t_0 (* t_0 (+ alpha (+ beta 3.0)))))
     (/
      (* (+ alpha 1.0) (/ (- 1.0 (/ (+ 4.0 (* alpha 2.0)) beta)) beta))
      t_0))))

C

double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 5.2e+84) {
		tmp = ((alpha + 1.0) * (1.0 + beta)) / (t_0 * (t_0 * (alpha + (beta + 3.0))));
	} else {
		tmp = ((alpha + 1.0) * ((1.0 - ((4.0 + (alpha * 2.0)) / beta)) / beta)) / t_0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    if (beta <= 5.2d+84) then
        tmp = ((alpha + 1.0d0) * (1.0d0 + beta)) / (t_0 * (t_0 * (alpha + (beta + 3.0d0))))
    else
        tmp = ((alpha + 1.0d0) * ((1.0d0 - ((4.0d0 + (alpha * 2.0d0)) / beta)) / beta)) / t_0
    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 <= 5.2e+84) {
		tmp = ((alpha + 1.0) * (1.0 + beta)) / (t_0 * (t_0 * (alpha + (beta + 3.0))));
	} else {
		tmp = ((alpha + 1.0) * ((1.0 - ((4.0 + (alpha * 2.0)) / beta)) / beta)) / t_0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 5.2e+84:
		tmp = ((alpha + 1.0) * (1.0 + beta)) / (t_0 * (t_0 * (alpha + (beta + 3.0))))
	else:
		tmp = ((alpha + 1.0) * ((1.0 - ((4.0 + (alpha * 2.0)) / beta)) / beta)) / t_0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 5.2e+84)
		tmp = Float64(Float64(Float64(alpha + 1.0) * Float64(1.0 + beta)) / Float64(t_0 * Float64(t_0 * Float64(alpha + Float64(beta + 3.0)))));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) * Float64(Float64(1.0 - Float64(Float64(4.0 + Float64(alpha * 2.0)) / beta)) / beta)) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 5.2e+84)
		tmp = ((alpha + 1.0) * (1.0 + beta)) / (t_0 * (t_0 * (alpha + (beta + 3.0))));
	else
		tmp = ((alpha + 1.0) * ((1.0 - ((4.0 + (alpha * 2.0)) / beta)) / beta)) / t_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, 5.2e+84], N[(N[(N[(alpha + 1.0), $MachinePrecision] * N[(1.0 + beta), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[(t$95$0 * N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] * N[(N[(1.0 - N[(N[(4.0 + N[(alpha * 2.0), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.2000000000000002e84

   1. Initial program 99.8%

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

   3. Simplified 92.6%

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

   if 5.2000000000000002e84 < beta

   1. Initial program 69.4%

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

   3. Simplified 49.2%

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

      1. times-frac 79.2%

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

      2. +-commutative 79.2%

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

   6. Applied egg-rr 79.2%

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

      1. associate-*l/ 79.4%

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

      2. associate-+r+ 79.4%

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

      3. associate-/r* 99.8%

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

      4. associate-+r+ 99.8%

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in beta around inf 78.2%

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

       1. mul-1-neg 78.2%

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

   11. Simplified 78.2%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.2 \cdot 10^{+84}:\\ \;\;\;\;\frac{\left(\alpha + 1\right) \cdot \left(1 + \beta\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\alpha + 1\right) \cdot \frac{1 - \frac{4 + \alpha \cdot 2}{\beta}}{\beta}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 2.6 \cdot 10^{+85}:\\ \;\;\;\;\frac{\alpha + 1}{t\_0} \cdot \frac{1 + \beta}{t\_0 \cdot \left(\alpha + \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\alpha + 1\right) \cdot \frac{1 - \frac{4 + \alpha \cdot 2}{\beta}}{\beta}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 2.6e+85)
     (* (/ (+ alpha 1.0) t_0) (/ (+ 1.0 beta) (* t_0 (+ alpha (+ beta 3.0)))))
     (/
      (* (+ alpha 1.0) (/ (- 1.0 (/ (+ 4.0 (* alpha 2.0)) beta)) beta))
      t_0))))

C

double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 2.6e+85) {
		tmp = ((alpha + 1.0) / t_0) * ((1.0 + beta) / (t_0 * (alpha + (beta + 3.0))));
	} else {
		tmp = ((alpha + 1.0) * ((1.0 - ((4.0 + (alpha * 2.0)) / beta)) / beta)) / t_0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    if (beta <= 2.6d+85) then
        tmp = ((alpha + 1.0d0) / t_0) * ((1.0d0 + beta) / (t_0 * (alpha + (beta + 3.0d0))))
    else
        tmp = ((alpha + 1.0d0) * ((1.0d0 - ((4.0d0 + (alpha * 2.0d0)) / beta)) / beta)) / t_0
    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 <= 2.6e+85) {
		tmp = ((alpha + 1.0) / t_0) * ((1.0 + beta) / (t_0 * (alpha + (beta + 3.0))));
	} else {
		tmp = ((alpha + 1.0) * ((1.0 - ((4.0 + (alpha * 2.0)) / beta)) / beta)) / t_0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 2.6e+85:
		tmp = ((alpha + 1.0) / t_0) * ((1.0 + beta) / (t_0 * (alpha + (beta + 3.0))))
	else:
		tmp = ((alpha + 1.0) * ((1.0 - ((4.0 + (alpha * 2.0)) / beta)) / beta)) / t_0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 2.6e+85)
		tmp = Float64(Float64(Float64(alpha + 1.0) / t_0) * Float64(Float64(1.0 + beta) / Float64(t_0 * Float64(alpha + Float64(beta + 3.0)))));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) * Float64(Float64(1.0 - Float64(Float64(4.0 + Float64(alpha * 2.0)) / beta)) / beta)) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 2.6e+85)
		tmp = ((alpha + 1.0) / t_0) * ((1.0 + beta) / (t_0 * (alpha + (beta + 3.0))));
	else
		tmp = ((alpha + 1.0) * ((1.0 - ((4.0 + (alpha * 2.0)) / beta)) / beta)) / t_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, 2.6e+85], N[(N[(N[(alpha + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(N[(1.0 + beta), $MachinePrecision] / N[(t$95$0 * N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] * N[(N[(1.0 - N[(N[(4.0 + N[(alpha * 2.0), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.60000000000000011e85

   1. Initial program 99.8%

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

   3. Simplified 92.6%

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

      1. times-frac 98.7%

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

      2. +-commutative 98.7%

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

   6. Applied egg-rr 98.7%

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

   if 2.60000000000000011e85 < beta

   1. Initial program 69.4%

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

   3. Simplified 49.2%

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

      1. times-frac 79.2%

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

      2. +-commutative 79.2%

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

   6. Applied egg-rr 79.2%

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

      1. associate-*l/ 79.4%

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

      2. associate-+r+ 79.4%

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

      3. associate-/r* 99.8%

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

      4. associate-+r+ 99.8%

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in beta around inf 78.2%

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

       1. mul-1-neg 78.2%

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

   11. Simplified 78.2%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.6 \cdot 10^{+85}:\\ \;\;\;\;\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\alpha + 1\right) \cdot \frac{1 - \frac{4 + \alpha \cdot 2}{\beta}}{\beta}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5 \cdot 10^{+41}:\\ \;\;\;\;\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\alpha + 1\right) \cdot \frac{1 - \frac{4 + \alpha \cdot 2}{\beta}}{\beta}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5e+41)
   (/ (+ 1.0 beta) (* (+ beta 2.0) (* (+ beta 2.0) (+ alpha (+ beta 3.0)))))
   (/
    (* (+ alpha 1.0) (/ (- 1.0 (/ (+ 4.0 (* alpha 2.0)) beta)) beta))
    (+ alpha (+ beta 2.0)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5e+41) {
		tmp = (1.0 + beta) / ((beta + 2.0) * ((beta + 2.0) * (alpha + (beta + 3.0))));
	} else {
		tmp = ((alpha + 1.0) * ((1.0 - ((4.0 + (alpha * 2.0)) / beta)) / beta)) / (alpha + (beta + 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 5d+41) then
        tmp = (1.0d0 + beta) / ((beta + 2.0d0) * ((beta + 2.0d0) * (alpha + (beta + 3.0d0))))
    else
        tmp = ((alpha + 1.0d0) * ((1.0d0 - ((4.0d0 + (alpha * 2.0d0)) / beta)) / beta)) / (alpha + (beta + 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5e+41) {
		tmp = (1.0 + beta) / ((beta + 2.0) * ((beta + 2.0) * (alpha + (beta + 3.0))));
	} else {
		tmp = ((alpha + 1.0) * ((1.0 - ((4.0 + (alpha * 2.0)) / beta)) / beta)) / (alpha + (beta + 2.0));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 5e+41:
		tmp = (1.0 + beta) / ((beta + 2.0) * ((beta + 2.0) * (alpha + (beta + 3.0))))
	else:
		tmp = ((alpha + 1.0) * ((1.0 - ((4.0 + (alpha * 2.0)) / beta)) / beta)) / (alpha + (beta + 2.0))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 5e+41)
		tmp = Float64(Float64(1.0 + beta) / Float64(Float64(beta + 2.0) * Float64(Float64(beta + 2.0) * Float64(alpha + Float64(beta + 3.0)))));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) * Float64(Float64(1.0 - Float64(Float64(4.0 + Float64(alpha * 2.0)) / beta)) / beta)) / Float64(alpha + Float64(beta + 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 5e+41)
		tmp = (1.0 + beta) / ((beta + 2.0) * ((beta + 2.0) * (alpha + (beta + 3.0))));
	else
		tmp = ((alpha + 1.0) * ((1.0 - ((4.0 + (alpha * 2.0)) / beta)) / beta)) / (alpha + (beta + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 5e+41], N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(N[(beta + 2.0), $MachinePrecision] * N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] * N[(N[(1.0 - N[(N[(4.0 + N[(alpha * 2.0), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.00000000000000022e41

   1. Initial program 99.8%

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

      1. associate-/l/ 98.6%

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

      2. +-commutative 98.6%

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

      3. associate-+l+ 98.6%

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

      4. *-commutative 98.6%

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

      5. metadata-eval 98.6%

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

      6. associate-+l+ 98.6%

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

      7. metadata-eval 98.6%

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

      8. associate-+l+ 98.6%

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

      9. metadata-eval 98.6%

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

      10. metadata-eval 98.6%

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

      11. associate-+l+ 98.6%

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

   3. Simplified 98.6%

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

   5. Taylor expanded in alpha around 0 85.9%

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

      1. +-commutative 85.9%

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

   7. Simplified 85.9%

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

      1. *-un-lft-identity 85.9%

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

      2. associate-/l/ 85.9%

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

      3. *-commutative 85.9%

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

      4. associate-+r+ 85.9%

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

   9. Applied egg-rr 85.9%

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

       1. *-lft-identity 85.9%

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

       2. associate-*l* 85.9%

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

       3. associate-+r+ 85.9%

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

       4. +-commutative 85.9%

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

       5. +-commutative 85.9%

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

       6. +-commutative 85.9%

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

       7. *-commutative 85.9%

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

       8. +-commutative 85.9%

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

       9. +-commutative 85.9%

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

       10. +-commutative 85.9%

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

       11. +-commutative 85.9%

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

   11. Simplified 85.9%

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

   12. Taylor expanded in alpha around 0 70.2%

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

       1. +-commutative 70.2%

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

   14. Simplified 70.2%

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

   if 5.00000000000000022e41 < beta

   1. Initial program 73.8%

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

   3. Simplified 55.1%

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

      1. times-frac 82.2%

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

      2. +-commutative 82.2%

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

   6. Applied egg-rr 82.2%

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

      1. associate-*l/ 82.3%

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

      2. associate-+r+ 82.3%

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

      3. associate-/r* 99.7%

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

      4. associate-+r+ 99.7%

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

   8. Applied egg-rr 99.7%

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

   9. Taylor expanded in beta around inf 76.8%

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

       1. mul-1-neg 76.8%

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

   11. Simplified 76.8%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5 \cdot 10^{+41}:\\ \;\;\;\;\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\alpha + 1\right) \cdot \frac{1 - \frac{4 + \alpha \cdot 2}{\beta}}{\beta}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.5 \cdot 10^{+41}:\\ \;\;\;\;\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{4 + \alpha \cdot 2}{\beta}}{\beta} \cdot \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.5e+41)
   (/ (+ 1.0 beta) (* (+ beta 2.0) (* (+ beta 2.0) (+ alpha (+ beta 3.0)))))
   (*
    (/ (- 1.0 (/ (+ 4.0 (* alpha 2.0)) beta)) beta)
    (/ (+ alpha 1.0) (+ alpha (+ beta 2.0))))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.5e+41) {
		tmp = (1.0 + beta) / ((beta + 2.0) * ((beta + 2.0) * (alpha + (beta + 3.0))));
	} else {
		tmp = ((1.0 - ((4.0 + (alpha * 2.0)) / beta)) / beta) * ((alpha + 1.0) / (alpha + (beta + 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.5d+41) then
        tmp = (1.0d0 + beta) / ((beta + 2.0d0) * ((beta + 2.0d0) * (alpha + (beta + 3.0d0))))
    else
        tmp = ((1.0d0 - ((4.0d0 + (alpha * 2.0d0)) / beta)) / beta) * ((alpha + 1.0d0) / (alpha + (beta + 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.5e+41) {
		tmp = (1.0 + beta) / ((beta + 2.0) * ((beta + 2.0) * (alpha + (beta + 3.0))));
	} else {
		tmp = ((1.0 - ((4.0 + (alpha * 2.0)) / beta)) / beta) * ((alpha + 1.0) / (alpha + (beta + 2.0)));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.5e+41:
		tmp = (1.0 + beta) / ((beta + 2.0) * ((beta + 2.0) * (alpha + (beta + 3.0))))
	else:
		tmp = ((1.0 - ((4.0 + (alpha * 2.0)) / beta)) / beta) * ((alpha + 1.0) / (alpha + (beta + 2.0)))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.5e+41)
		tmp = Float64(Float64(1.0 + beta) / Float64(Float64(beta + 2.0) * Float64(Float64(beta + 2.0) * Float64(alpha + Float64(beta + 3.0)))));
	else
		tmp = Float64(Float64(Float64(1.0 - Float64(Float64(4.0 + Float64(alpha * 2.0)) / beta)) / beta) * Float64(Float64(alpha + 1.0) / Float64(alpha + Float64(beta + 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.5e+41)
		tmp = (1.0 + beta) / ((beta + 2.0) * ((beta + 2.0) * (alpha + (beta + 3.0))));
	else
		tmp = ((1.0 - ((4.0 + (alpha * 2.0)) / beta)) / beta) * ((alpha + 1.0) / (alpha + (beta + 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.5e+41], N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(N[(beta + 2.0), $MachinePrecision] * N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[(N[(4.0 + N[(alpha * 2.0), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision] * N[(N[(alpha + 1.0), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.50000000000000011e41

   1. Initial program 99.8%

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

      1. associate-/l/ 98.6%

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

      2. +-commutative 98.6%

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

      3. associate-+l+ 98.6%

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

      4. *-commutative 98.6%

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

      5. metadata-eval 98.6%

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

      6. associate-+l+ 98.6%

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

      7. metadata-eval 98.6%

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

      8. associate-+l+ 98.6%

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

      9. metadata-eval 98.6%

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

      10. metadata-eval 98.6%

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

      11. associate-+l+ 98.6%

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

   3. Simplified 98.6%

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

   5. Taylor expanded in alpha around 0 85.9%

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

      1. +-commutative 85.9%

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

   7. Simplified 85.9%

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

      1. *-un-lft-identity 85.9%

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

      2. associate-/l/ 85.9%

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

      3. *-commutative 85.9%

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

      4. associate-+r+ 85.9%

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

   9. Applied egg-rr 85.9%

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

       1. *-lft-identity 85.9%

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

       2. associate-*l* 85.9%

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

       3. associate-+r+ 85.9%

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

       4. +-commutative 85.9%

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

       5. +-commutative 85.9%

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

       6. +-commutative 85.9%

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

       7. *-commutative 85.9%

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

       8. +-commutative 85.9%

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

       9. +-commutative 85.9%

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

       10. +-commutative 85.9%

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

       11. +-commutative 85.9%

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

   11. Simplified 85.9%

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

   12. Taylor expanded in alpha around 0 70.2%

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

       1. +-commutative 70.2%

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

   14. Simplified 70.2%

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

   if 2.50000000000000011e41 < beta

   1. Initial program 73.8%

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

   3. Simplified 55.1%

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

      1. times-frac 82.2%

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

      2. +-commutative 82.2%

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

   6. Applied egg-rr 82.2%

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

   7. Taylor expanded in beta around inf 76.7%

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

      1. mul-1-neg 76.7%

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

   9. Simplified 76.7%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.5 \cdot 10^{+41}:\\ \;\;\;\;\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{4 + \alpha \cdot 2}{\beta}}{\beta} \cdot \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + \left(\alpha + \beta\right)\\ \mathbf{if}\;\beta \leq 6.5 \cdot 10^{+43}:\\ \;\;\;\;\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{t\_0}}{1 + t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (+ alpha beta))))
   (if (<= beta 6.5e+43)
     (/ (+ 1.0 beta) (* (+ beta 2.0) (* (+ beta 2.0) (+ alpha (+ beta 3.0)))))
     (/ (/ (+ alpha 1.0) t_0) (+ 1.0 t_0)))))

C

double code(double alpha, double beta) {
	double t_0 = 2.0 + (alpha + beta);
	double tmp;
	if (beta <= 6.5e+43) {
		tmp = (1.0 + beta) / ((beta + 2.0) * ((beta + 2.0) * (alpha + (beta + 3.0))));
	} else {
		tmp = ((alpha + 1.0) / t_0) / (1.0 + t_0);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 + (alpha + beta)
    if (beta <= 6.5d+43) then
        tmp = (1.0d0 + beta) / ((beta + 2.0d0) * ((beta + 2.0d0) * (alpha + (beta + 3.0d0))))
    else
        tmp = ((alpha + 1.0d0) / t_0) / (1.0d0 + t_0)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = 2.0 + (alpha + beta);
	double tmp;
	if (beta <= 6.5e+43) {
		tmp = (1.0 + beta) / ((beta + 2.0) * ((beta + 2.0) * (alpha + (beta + 3.0))));
	} else {
		tmp = ((alpha + 1.0) / t_0) / (1.0 + t_0);
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = 2.0 + (alpha + beta)
	tmp = 0
	if beta <= 6.5e+43:
		tmp = (1.0 + beta) / ((beta + 2.0) * ((beta + 2.0) * (alpha + (beta + 3.0))))
	else:
		tmp = ((alpha + 1.0) / t_0) / (1.0 + t_0)
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(alpha + beta))
	tmp = 0.0
	if (beta <= 6.5e+43)
		tmp = Float64(Float64(1.0 + beta) / Float64(Float64(beta + 2.0) * Float64(Float64(beta + 2.0) * Float64(alpha + Float64(beta + 3.0)))));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / t_0) / Float64(1.0 + t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = 2.0 + (alpha + beta);
	tmp = 0.0;
	if (beta <= 6.5e+43)
		tmp = (1.0 + beta) / ((beta + 2.0) * ((beta + 2.0) * (alpha + (beta + 3.0))));
	else
		tmp = ((alpha + 1.0) / t_0) / (1.0 + t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 6.5e+43], N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(N[(beta + 2.0), $MachinePrecision] * N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.4999999999999998e43

   1. Initial program 99.8%

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

      1. associate-/l/ 98.6%

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

      2. +-commutative 98.6%

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

      3. associate-+l+ 98.6%

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

      4. *-commutative 98.6%

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

      5. metadata-eval 98.6%

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

      6. associate-+l+ 98.6%

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

      7. metadata-eval 98.6%

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

      8. associate-+l+ 98.6%

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

      9. metadata-eval 98.6%

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

      10. metadata-eval 98.6%

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

      11. associate-+l+ 98.6%

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

   3. Simplified 98.6%

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

   5. Taylor expanded in alpha around 0 85.9%

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

      1. +-commutative 85.9%

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

   7. Simplified 85.9%

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

      1. *-un-lft-identity 85.9%

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

      2. associate-/l/ 85.9%

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

      3. *-commutative 85.9%

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

      4. associate-+r+ 85.9%

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

   9. Applied egg-rr 85.9%

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

       1. *-lft-identity 85.9%

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

       2. associate-*l* 85.9%

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

       3. associate-+r+ 85.9%

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

       4. +-commutative 85.9%

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

       5. +-commutative 85.9%

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

       6. +-commutative 85.9%

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

       7. *-commutative 85.9%

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

       8. +-commutative 85.9%

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

       9. +-commutative 85.9%

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

       10. +-commutative 85.9%

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

       11. +-commutative 85.9%

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

   11. Simplified 85.9%

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

   12. Taylor expanded in alpha around 0 70.2%

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

       1. +-commutative 70.2%

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

   14. Simplified 70.2%

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

   if 6.4999999999999998e43 < beta

   1. Initial program 73.8%

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

      1. div-inv 73.7%

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

      2. +-commutative 73.7%

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

      3. associate-+l+ 73.7%

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

      4. *-commutative 73.7%

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

      5. +-commutative 73.7%

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

      6. distribute-rgt1-in 73.7%

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

      7. fma-define 73.7%

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

      8. metadata-eval 73.7%

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

      9. associate-+r+ 73.7%

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

      10. metadata-eval 73.7%

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

      11. associate-+r+ 73.7%

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

   4. Applied egg-rr 73.7%

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

      1. associate-*l/ 73.8%

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

      2. associate-*r/ 73.8%

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

      3. *-rgt-identity 73.8%

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

      4. +-commutative 73.8%

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

      5. fma-undefine 73.8%

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

      6. *-commutative 73.8%

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

      7. +-commutative 73.8%

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

      8. associate-+r+ 73.8%

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

      9. +-commutative 73.8%

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

      10. *-rgt-identity 73.8%

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

      11. *-commutative 73.8%

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

      12. +-commutative 73.8%

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

      13. distribute-lft-in 73.8%

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

      14. +-commutative 73.8%

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

      15. *-commutative 73.8%

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

      16. associate-+r+ 73.8%

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

      17. +-commutative 73.8%

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

      18. +-commutative 73.8%

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

      19. associate-+r+ 73.8%

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

      20. +-commutative 73.8%

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

      21. +-commutative 73.8%

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

   6. Simplified 73.8%

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

   7. Taylor expanded in beta around inf 77.1%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.5 \cdot 10^{+43}:\\ \;\;\;\;\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{2 + \left(\alpha + \beta\right)}}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.2 \cdot 10^{+42}:\\ \;\;\;\;\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta \cdot \left(\frac{\alpha - -3}{\beta} - -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5.2e+42)
   (/ (+ 1.0 beta) (* (+ beta 2.0) (* (+ beta 2.0) (+ alpha (+ beta 3.0)))))
   (/ (/ (+ alpha 1.0) beta) (* beta (- (/ (- alpha -3.0) beta) -1.0)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.2e+42) {
		tmp = (1.0 + beta) / ((beta + 2.0) * ((beta + 2.0) * (alpha + (beta + 3.0))));
	} else {
		tmp = ((alpha + 1.0) / beta) / (beta * (((alpha - -3.0) / beta) - -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 5.2d+42) then
        tmp = (1.0d0 + beta) / ((beta + 2.0d0) * ((beta + 2.0d0) * (alpha + (beta + 3.0d0))))
    else
        tmp = ((alpha + 1.0d0) / beta) / (beta * (((alpha - (-3.0d0)) / beta) - (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.2e+42) {
		tmp = (1.0 + beta) / ((beta + 2.0) * ((beta + 2.0) * (alpha + (beta + 3.0))));
	} else {
		tmp = ((alpha + 1.0) / beta) / (beta * (((alpha - -3.0) / beta) - -1.0));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 5.2e+42:
		tmp = (1.0 + beta) / ((beta + 2.0) * ((beta + 2.0) * (alpha + (beta + 3.0))))
	else:
		tmp = ((alpha + 1.0) / beta) / (beta * (((alpha - -3.0) / beta) - -1.0))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 5.2e+42)
		tmp = Float64(Float64(1.0 + beta) / Float64(Float64(beta + 2.0) * Float64(Float64(beta + 2.0) * Float64(alpha + Float64(beta + 3.0)))));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / Float64(beta * Float64(Float64(Float64(alpha - -3.0) / beta) - -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 5.2e+42)
		tmp = (1.0 + beta) / ((beta + 2.0) * ((beta + 2.0) * (alpha + (beta + 3.0))));
	else
		tmp = ((alpha + 1.0) / beta) / (beta * (((alpha - -3.0) / beta) - -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 5.2e+42], N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(N[(beta + 2.0), $MachinePrecision] * N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(beta * N[(N[(N[(alpha - -3.0), $MachinePrecision] / beta), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.1999999999999998e42

   1. Initial program 99.8%

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

      1. associate-/l/ 98.6%

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

      2. +-commutative 98.6%

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

      3. associate-+l+ 98.6%

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

      4. *-commutative 98.6%

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

      5. metadata-eval 98.6%

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

      6. associate-+l+ 98.6%

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

      7. metadata-eval 98.6%

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

      8. associate-+l+ 98.6%

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

      9. metadata-eval 98.6%

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

      10. metadata-eval 98.6%

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

      11. associate-+l+ 98.6%

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

   3. Simplified 98.6%

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

   5. Taylor expanded in alpha around 0 85.9%

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

      1. +-commutative 85.9%

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

   7. Simplified 85.9%

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

      1. *-un-lft-identity 85.9%

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

      2. associate-/l/ 85.9%

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

      3. *-commutative 85.9%

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

      4. associate-+r+ 85.9%

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

   9. Applied egg-rr 85.9%

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

       1. *-lft-identity 85.9%

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

       2. associate-*l* 85.9%

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

       3. associate-+r+ 85.9%

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

       4. +-commutative 85.9%

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

       5. +-commutative 85.9%

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

       6. +-commutative 85.9%

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

       7. *-commutative 85.9%

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

       8. +-commutative 85.9%

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

       9. +-commutative 85.9%

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

       10. +-commutative 85.9%

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

       11. +-commutative 85.9%

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

   11. Simplified 85.9%

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

   12. Taylor expanded in alpha around 0 70.2%

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

       1. +-commutative 70.2%

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

   14. Simplified 70.2%

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

   if 5.1999999999999998e42 < beta

   1. Initial program 73.8%

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

   3. Taylor expanded in beta around inf 76.5%

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

   4. Taylor expanded in beta around -inf 76.5%

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

      1. mul-1-neg 76.5%

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

      2. distribute-rgt-neg-in 76.5%

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

      3. sub-neg 76.5%

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

      4. associate-*r/ 76.5%

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

      5. distribute-lft-in 76.5%

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

      6. metadata-eval 76.5%

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

      7. metadata-eval 76.5%

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

      8. mul-1-neg 76.5%

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

      9. unsub-neg 76.5%

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

      10. metadata-eval 76.5%

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

      11. metadata-eval 76.5%

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

   6. Simplified 76.5%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.2 \cdot 10^{+42}:\\ \;\;\;\;\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta \cdot \left(\frac{\alpha - -3}{\beta} - -1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.65 \cdot 10^{+39}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 2}}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta \cdot \left(\frac{\alpha - -3}{\beta} - -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.65e+39)
   (/ (/ (+ 1.0 beta) (+ beta 2.0)) (* (+ beta 2.0) (+ beta 3.0)))
   (/ (/ (+ alpha 1.0) beta) (* beta (- (/ (- alpha -3.0) beta) -1.0)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.65e+39) {
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0));
	} else {
		tmp = ((alpha + 1.0) / beta) / (beta * (((alpha - -3.0) / beta) - -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.65d+39) then
        tmp = ((1.0d0 + beta) / (beta + 2.0d0)) / ((beta + 2.0d0) * (beta + 3.0d0))
    else
        tmp = ((alpha + 1.0d0) / beta) / (beta * (((alpha - (-3.0d0)) / beta) - (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.65e+39) {
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0));
	} else {
		tmp = ((alpha + 1.0) / beta) / (beta * (((alpha - -3.0) / beta) - -1.0));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.65e+39:
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0))
	else:
		tmp = ((alpha + 1.0) / beta) / (beta * (((alpha - -3.0) / beta) - -1.0))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.65e+39)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(beta + 2.0)) / Float64(Float64(beta + 2.0) * Float64(beta + 3.0)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / Float64(beta * Float64(Float64(Float64(alpha - -3.0) / beta) - -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.65e+39)
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0));
	else
		tmp = ((alpha + 1.0) / beta) / (beta * (((alpha - -3.0) / beta) - -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.65e+39], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(beta * N[(N[(N[(alpha - -3.0), $MachinePrecision] / beta), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.64999999999999989e39

   1. Initial program 99.8%

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

      1. associate-/l/ 99.1%

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

      2. +-commutative 99.1%

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

      3. associate-+l+ 99.1%

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

      4. *-commutative 99.1%

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

      5. metadata-eval 99.1%

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

      6. associate-+l+ 99.1%

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

      7. metadata-eval 99.1%

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

      8. associate-+l+ 99.1%

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

      9. metadata-eval 99.1%

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

      10. metadata-eval 99.1%

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

      11. associate-+l+ 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in alpha around 0 86.2%

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

      1. +-commutative 86.2%

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

   7. Simplified 86.2%

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

   8. Taylor expanded in alpha around 0 68.4%

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

      1. +-commutative 68.4%

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

      2. +-commutative 68.4%

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

   10. Simplified 68.4%

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

   if 2.64999999999999989e39 < beta

   1. Initial program 74.5%

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

   3. Taylor expanded in beta around inf 75.8%

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

   4. Taylor expanded in beta around -inf 75.8%

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

      1. mul-1-neg 75.8%

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

      2. distribute-rgt-neg-in 75.8%

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

      3. sub-neg 75.8%

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

      4. associate-*r/ 75.8%

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

      5. distribute-lft-in 75.8%

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

      6. metadata-eval 75.8%

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

      7. metadata-eval 75.8%

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

      8. mul-1-neg 75.8%

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

      9. unsub-neg 75.8%

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

      10. metadata-eval 75.8%

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

      11. metadata-eval 75.8%

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

   6. Simplified 75.8%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.65 \cdot 10^{+39}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 2}}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta \cdot \left(\frac{\alpha - -3}{\beta} - -1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4 \cdot 10^{+39}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 2}}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4e+39)
   (/ (/ (+ 1.0 beta) (+ beta 2.0)) (* (+ beta 2.0) (+ beta 3.0)))
   (/ (/ (+ alpha 1.0) beta) (+ beta (+ alpha 3.0)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4e+39) {
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0));
	} else {
		tmp = ((alpha + 1.0) / beta) / (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 <= 4d+39) then
        tmp = ((1.0d0 + beta) / (beta + 2.0d0)) / ((beta + 2.0d0) * (beta + 3.0d0))
    else
        tmp = ((alpha + 1.0d0) / beta) / (beta + (alpha + 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4e+39) {
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0));
	} else {
		tmp = ((alpha + 1.0) / beta) / (beta + (alpha + 3.0));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 4e+39:
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0))
	else:
		tmp = ((alpha + 1.0) / beta) / (beta + (alpha + 3.0))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4e+39)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(beta + 2.0)) / Float64(Float64(beta + 2.0) * Float64(beta + 3.0)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / Float64(beta + Float64(alpha + 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4e+39)
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0));
	else
		tmp = ((alpha + 1.0) / beta) / (beta + (alpha + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 4e+39], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.99999999999999976e39

   1. Initial program 99.8%

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

      1. associate-/l/ 99.1%

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

      2. +-commutative 99.1%

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

      3. associate-+l+ 99.1%

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

      4. *-commutative 99.1%

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

      5. metadata-eval 99.1%

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

      6. associate-+l+ 99.1%

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

      7. metadata-eval 99.1%

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

      8. associate-+l+ 99.1%

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

      9. metadata-eval 99.1%

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

      10. metadata-eval 99.1%

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

      11. associate-+l+ 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in alpha around 0 86.2%

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

      1. +-commutative 86.2%

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

   7. Simplified 86.2%

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

   8. Taylor expanded in alpha around 0 68.4%

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

      1. +-commutative 68.4%

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

      2. +-commutative 68.4%

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

   10. Simplified 68.4%

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

   if 3.99999999999999976e39 < beta

   1. Initial program 74.5%

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

   3. Taylor expanded in beta around inf 75.8%

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

   4. Taylor expanded in alpha around 0 75.8%

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

      1. associate-+r+ 75.8%

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

   6. Simplified 75.8%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4 \cdot 10^{+39}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 2}}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 72.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.9:\\ \;\;\;\;\frac{1}{\left(2 + \left(\alpha + \beta\right)\right) \cdot \left(6 - \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.9)
   (/ 1.0 (* (+ 2.0 (+ alpha beta)) (- 6.0 beta)))
   (/ (/ (+ alpha 1.0) beta) (+ beta (+ alpha 3.0)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.9) {
		tmp = 1.0 / ((2.0 + (alpha + beta)) * (6.0 - beta));
	} else {
		tmp = ((alpha + 1.0) / beta) / (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 <= 2.9d0) then
        tmp = 1.0d0 / ((2.0d0 + (alpha + beta)) * (6.0d0 - beta))
    else
        tmp = ((alpha + 1.0d0) / beta) / (beta + (alpha + 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.9) {
		tmp = 1.0 / ((2.0 + (alpha + beta)) * (6.0 - beta));
	} else {
		tmp = ((alpha + 1.0) / beta) / (beta + (alpha + 3.0));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.9:
		tmp = 1.0 / ((2.0 + (alpha + beta)) * (6.0 - beta))
	else:
		tmp = ((alpha + 1.0) / beta) / (beta + (alpha + 3.0))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.9)
		tmp = Float64(1.0 / Float64(Float64(2.0 + Float64(alpha + beta)) * Float64(6.0 - beta)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / Float64(beta + Float64(alpha + 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.9)
		tmp = 1.0 / ((2.0 + (alpha + beta)) * (6.0 - beta));
	else
		tmp = ((alpha + 1.0) / beta) / (beta + (alpha + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.9], N[(1.0 / N[(N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision] * N[(6.0 - beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.89999999999999991

   1. Initial program 99.9%

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

   3. Simplified 94.2%

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

      1. associate-+r+ 94.2%

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

      2. fma-undefine 94.2%

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

      3. *-commutative 94.2%

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

      4. associate-+l+ 94.2%

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

      5. +-commutative 94.2%

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

      6. associate-+l+ 94.2%

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

      7. *-commutative 94.2%

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

      8. associate-*r* 94.2%

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

      9. associate-+r+ 94.2%

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

      10. +-commutative 94.2%

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

      11. associate-/l/ 99.6%

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

      12. clear-num 99.6%

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

      13. inv-pow 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. unpow-1 99.6%

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

      2. associate-/l* 99.6%

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

      3. associate-+r+ 99.6%

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

      4. +-commutative 99.6%

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

      5. +-commutative 99.6%

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

      6. associate-+r+ 99.6%

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

      7. +-commutative 99.6%

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

      8. associate-+r+ 99.6%

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

      9. +-commutative 99.6%

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

      10. fma-undefine 99.6%

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

      11. *-commutative 99.6%

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

      12. +-commutative 99.6%

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

      13. associate-+r+ 99.6%

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

      14. +-commutative 99.6%

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

      15. *-rgt-identity 99.6%

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

      16. *-commutative 99.6%

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

      17. +-commutative 99.6%

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

      18. distribute-lft-in 99.6%

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

      19. +-commutative 99.6%

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

      20. *-commutative 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in alpha around 0 70.4%

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

   10. Taylor expanded in beta around 0 70.2%

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

       1. +-commutative 70.2%

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

       2. mul-1-neg 70.2%

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

   12. Simplified 70.2%

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

   if 2.89999999999999991 < beta

   1. Initial program 79.2%

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

   3. Taylor expanded in beta around inf 70.4%

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

   4. Taylor expanded in alpha around 0 70.4%

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

      1. associate-+r+ 70.4%

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

   6. Simplified 70.4%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.9:\\ \;\;\;\;\frac{1}{\left(2 + \left(\alpha + \beta\right)\right) \cdot \left(6 - \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 72.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.4:\\ \;\;\;\;\frac{1}{\left(2 + \left(\alpha + \beta\right)\right) \cdot 6}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5.4)
   (/ 1.0 (* (+ 2.0 (+ alpha beta)) 6.0))
   (/ (/ (+ alpha 1.0) beta) (+ beta (+ alpha 3.0)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.4) {
		tmp = 1.0 / ((2.0 + (alpha + beta)) * 6.0);
	} else {
		tmp = ((alpha + 1.0) / beta) / (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 <= 5.4d0) then
        tmp = 1.0d0 / ((2.0d0 + (alpha + beta)) * 6.0d0)
    else
        tmp = ((alpha + 1.0d0) / beta) / (beta + (alpha + 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.4) {
		tmp = 1.0 / ((2.0 + (alpha + beta)) * 6.0);
	} else {
		tmp = ((alpha + 1.0) / beta) / (beta + (alpha + 3.0));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 5.4:
		tmp = 1.0 / ((2.0 + (alpha + beta)) * 6.0)
	else:
		tmp = ((alpha + 1.0) / beta) / (beta + (alpha + 3.0))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 5.4)
		tmp = Float64(1.0 / Float64(Float64(2.0 + Float64(alpha + beta)) * 6.0));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / Float64(beta + Float64(alpha + 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 5.4)
		tmp = 1.0 / ((2.0 + (alpha + beta)) * 6.0);
	else
		tmp = ((alpha + 1.0) / beta) / (beta + (alpha + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 5.4], N[(1.0 / N[(N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision] * 6.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.4000000000000004

   1. Initial program 99.9%

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

   3. Simplified 94.2%

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

      1. associate-+r+ 94.2%

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

      2. fma-undefine 94.2%

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

      3. *-commutative 94.2%

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

      4. associate-+l+ 94.2%

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

      5. +-commutative 94.2%

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

      6. associate-+l+ 94.2%

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

      7. *-commutative 94.2%

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

      8. associate-*r* 94.2%

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

      9. associate-+r+ 94.2%

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

      10. +-commutative 94.2%

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

      11. associate-/l/ 99.6%

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

      12. clear-num 99.6%

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

      13. inv-pow 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. unpow-1 99.6%

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

      2. associate-/l* 99.6%

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

      3. associate-+r+ 99.6%

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

      4. +-commutative 99.6%

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

      5. +-commutative 99.6%

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

      6. associate-+r+ 99.6%

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

      7. +-commutative 99.6%

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

      8. associate-+r+ 99.6%

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

      9. +-commutative 99.6%

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

      10. fma-undefine 99.6%

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

      11. *-commutative 99.6%

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

      12. +-commutative 99.6%

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

      13. associate-+r+ 99.6%

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

      14. +-commutative 99.6%

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

      15. *-rgt-identity 99.6%

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

      16. *-commutative 99.6%

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

      17. +-commutative 99.6%

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

      18. distribute-lft-in 99.6%

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

      19. +-commutative 99.6%

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

      20. *-commutative 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in alpha around 0 70.4%

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

   10. Taylor expanded in beta around 0 69.9%

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

   if 5.4000000000000004 < beta

   1. Initial program 79.2%

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

   3. Taylor expanded in beta around inf 70.4%

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

   4. Taylor expanded in alpha around 0 70.4%

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

      1. associate-+r+ 70.4%

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

   6. Simplified 70.4%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.4:\\ \;\;\;\;\frac{1}{\left(2 + \left(\alpha + \beta\right)\right) \cdot 6}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 72.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.5:\\ \;\;\;\;\frac{1}{\left(2 + \left(\alpha + \beta\right)\right) \cdot 6}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta + 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5.5)
   (/ 1.0 (* (+ 2.0 (+ alpha beta)) 6.0))
   (/ (/ (+ alpha 1.0) beta) (+ beta 3.0))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.5) {
		tmp = 1.0 / ((2.0 + (alpha + beta)) * 6.0);
	} else {
		tmp = ((alpha + 1.0) / 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 <= 5.5d0) then
        tmp = 1.0d0 / ((2.0d0 + (alpha + beta)) * 6.0d0)
    else
        tmp = ((alpha + 1.0d0) / beta) / (beta + 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.5) {
		tmp = 1.0 / ((2.0 + (alpha + beta)) * 6.0);
	} else {
		tmp = ((alpha + 1.0) / beta) / (beta + 3.0);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 5.5:
		tmp = 1.0 / ((2.0 + (alpha + beta)) * 6.0)
	else:
		tmp = ((alpha + 1.0) / beta) / (beta + 3.0)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 5.5)
		tmp = Float64(1.0 / Float64(Float64(2.0 + Float64(alpha + beta)) * 6.0));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / Float64(beta + 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 5.5)
		tmp = 1.0 / ((2.0 + (alpha + beta)) * 6.0);
	else
		tmp = ((alpha + 1.0) / beta) / (beta + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 5.5], N[(1.0 / N[(N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision] * 6.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.5

   1. Initial program 99.9%

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

   3. Simplified 94.2%

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

      1. associate-+r+ 94.2%

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

      2. fma-undefine 94.2%

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

      3. *-commutative 94.2%

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

      4. associate-+l+ 94.2%

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

      5. +-commutative 94.2%

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

      6. associate-+l+ 94.2%

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

      7. *-commutative 94.2%

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

      8. associate-*r* 94.2%

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

      9. associate-+r+ 94.2%

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

      10. +-commutative 94.2%

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

      11. associate-/l/ 99.6%

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

      12. clear-num 99.6%

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

      13. inv-pow 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. unpow-1 99.6%

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

      2. associate-/l* 99.6%

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

      3. associate-+r+ 99.6%

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

      4. +-commutative 99.6%

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

      5. +-commutative 99.6%

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

      6. associate-+r+ 99.6%

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

      7. +-commutative 99.6%

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

      8. associate-+r+ 99.6%

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

      9. +-commutative 99.6%

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

      10. fma-undefine 99.6%

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

      11. *-commutative 99.6%

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

      12. +-commutative 99.6%

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

      13. associate-+r+ 99.6%

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

      14. +-commutative 99.6%

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

      15. *-rgt-identity 99.6%

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

      16. *-commutative 99.6%

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

      17. +-commutative 99.6%

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

      18. distribute-lft-in 99.6%

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

      19. +-commutative 99.6%

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

      20. *-commutative 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in alpha around 0 70.4%

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

   10. Taylor expanded in beta around 0 69.9%

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

   if 5.5 < beta

   1. Initial program 79.2%

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

   3. Taylor expanded in beta around inf 70.4%

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

   4. Taylor expanded in alpha around 0 70.1%

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

      1. +-commutative 70.1%

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

   6. Simplified 70.1%

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

4. Final simplification 69.9%

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

Alternative 13: 72.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 8.6:\\ \;\;\;\;\frac{1}{\left(2 + \left(\alpha + \beta\right)\right) \cdot 6}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 8.6)
   (/ 1.0 (* (+ 2.0 (+ alpha beta)) 6.0))
   (/ (/ (+ alpha 1.0) beta) beta)))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 8.6) {
		tmp = 1.0 / ((2.0 + (alpha + beta)) * 6.0);
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 8.6d0) then
        tmp = 1.0d0 / ((2.0d0 + (alpha + beta)) * 6.0d0)
    else
        tmp = ((alpha + 1.0d0) / beta) / beta
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 8.6) {
		tmp = 1.0 / ((2.0 + (alpha + beta)) * 6.0);
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 8.6:
		tmp = 1.0 / ((2.0 + (alpha + beta)) * 6.0)
	else:
		tmp = ((alpha + 1.0) / beta) / beta
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 8.6)
		tmp = Float64(1.0 / Float64(Float64(2.0 + Float64(alpha + beta)) * 6.0));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / beta);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 8.6)
		tmp = 1.0 / ((2.0 + (alpha + beta)) * 6.0);
	else
		tmp = ((alpha + 1.0) / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 8.6], N[(1.0 / N[(N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision] * 6.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 8.59999999999999964

   1. Initial program 99.9%

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

   3. Simplified 94.2%

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

      1. associate-+r+ 94.2%

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

      2. fma-undefine 94.2%

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

      3. *-commutative 94.2%

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

      4. associate-+l+ 94.2%

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

      5. +-commutative 94.2%

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

      6. associate-+l+ 94.2%

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

      7. *-commutative 94.2%

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

      8. associate-*r* 94.2%

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

      9. associate-+r+ 94.2%

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

      10. +-commutative 94.2%

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

      11. associate-/l/ 99.6%

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

      12. clear-num 99.6%

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

      13. inv-pow 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. unpow-1 99.6%

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

      2. associate-/l* 99.6%

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

      3. associate-+r+ 99.6%

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

      4. +-commutative 99.6%

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

      5. +-commutative 99.6%

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

      6. associate-+r+ 99.6%

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

      7. +-commutative 99.6%

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

      8. associate-+r+ 99.6%

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

      9. +-commutative 99.6%

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

      10. fma-undefine 99.6%

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

      11. *-commutative 99.6%

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

      12. +-commutative 99.6%

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

      13. associate-+r+ 99.6%

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

      14. +-commutative 99.6%

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

      15. *-rgt-identity 99.6%

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

      16. *-commutative 99.6%

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

      17. +-commutative 99.6%

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

      18. distribute-lft-in 99.6%

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

      19. +-commutative 99.6%

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

      20. *-commutative 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in alpha around 0 70.4%

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

   10. Taylor expanded in beta around 0 69.9%

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

   if 8.59999999999999964 < beta

   1. Initial program 79.2%

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

   3. Taylor expanded in beta around inf 70.4%

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

   4. Taylor expanded in alpha around 0 70.4%

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

      1. associate-+r+ 70.4%

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

   6. Simplified 70.4%

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

   7. Taylor expanded in beta around inf 70.0%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 8.6:\\ \;\;\;\;\frac{1}{\left(2 + \left(\alpha + \beta\right)\right) \cdot 6}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 71.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.7:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.7)
   (+ 0.08333333333333333 (* alpha -0.041666666666666664))
   (/ (/ (+ alpha 1.0) beta) beta)))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.7) {
		tmp = 0.08333333333333333 + (alpha * -0.041666666666666664);
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.7d0) then
        tmp = 0.08333333333333333d0 + (alpha * (-0.041666666666666664d0))
    else
        tmp = ((alpha + 1.0d0) / beta) / beta
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.7) {
		tmp = 0.08333333333333333 + (alpha * -0.041666666666666664);
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.7:
		tmp = 0.08333333333333333 + (alpha * -0.041666666666666664)
	else:
		tmp = ((alpha + 1.0) / beta) / beta
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.7)
		tmp = Float64(0.08333333333333333 + Float64(alpha * -0.041666666666666664));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / beta);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.7)
		tmp = 0.08333333333333333 + (alpha * -0.041666666666666664);
	else
		tmp = ((alpha + 1.0) / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.7], N[(0.08333333333333333 + N[(alpha * -0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.7000000000000002

   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 94.2%

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

      1. associate-+r+ 94.2%

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

      2. fma-undefine 94.2%

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

      3. *-commutative 94.2%

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

      4. associate-+l+ 94.2%

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

      5. +-commutative 94.2%

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

      6. associate-+l+ 94.2%

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

      7. *-commutative 94.2%

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

      8. associate-*r* 94.2%

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

      9. associate-+r+ 94.2%

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

      10. +-commutative 94.2%

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

      11. associate-/l/ 99.6%

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

      12. clear-num 99.6%

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

      13. inv-pow 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. unpow-1 99.6%

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

      2. associate-/l* 99.6%

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

      3. associate-+r+ 99.6%

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

      4. +-commutative 99.6%

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

      5. +-commutative 99.6%

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

      6. associate-+r+ 99.6%

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

      7. +-commutative 99.6%

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

      8. associate-+r+ 99.6%

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

      9. +-commutative 99.6%

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

      10. fma-undefine 99.6%

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

      11. *-commutative 99.6%

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

      12. +-commutative 99.6%

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

      13. associate-+r+ 99.6%

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

      14. +-commutative 99.6%

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

      15. *-rgt-identity 99.6%

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

      16. *-commutative 99.6%

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

      17. +-commutative 99.6%

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

      18. distribute-lft-in 99.6%

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

      19. +-commutative 99.6%

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

      20. *-commutative 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in alpha around 0 70.4%

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

   10. Taylor expanded in beta around 0 69.8%

       \[\leadsto \color{blue}{\frac{0.16666666666666666}{2 + \alpha}} \]

   11. Taylor expanded in alpha around 0 68.1%

       \[\leadsto \color{blue}{0.08333333333333333 + -0.041666666666666664 \cdot \alpha} \]
   12. Step-by-step derivation

       1. *-commutative 68.1%

          \[\leadsto 0.08333333333333333 + \color{blue}{\alpha \cdot -0.041666666666666664} \]

   13. Simplified 68.1%

       \[\leadsto \color{blue}{0.08333333333333333 + \alpha \cdot -0.041666666666666664} \]

   if 2.7000000000000002 < beta

   1. Initial program 79.2%

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

   3. Taylor expanded in beta around inf 70.4%

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

   4. Taylor expanded in alpha around 0 70.4%

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

      1. associate-+r+ 70.4%

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

   6. Simplified 70.4%

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

   7. Taylor expanded in beta around inf 70.0%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.7:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 69.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.1:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta + 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.1)
   (+ 0.08333333333333333 (* alpha -0.041666666666666664))
   (/ (/ 1.0 beta) (+ beta 3.0))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.1) {
		tmp = 0.08333333333333333 + (alpha * -0.041666666666666664);
	} else {
		tmp = (1.0 / 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 <= 2.1d0) then
        tmp = 0.08333333333333333d0 + (alpha * (-0.041666666666666664d0))
    else
        tmp = (1.0d0 / beta) / (beta + 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.1) {
		tmp = 0.08333333333333333 + (alpha * -0.041666666666666664);
	} else {
		tmp = (1.0 / beta) / (beta + 3.0);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.1:
		tmp = 0.08333333333333333 + (alpha * -0.041666666666666664)
	else:
		tmp = (1.0 / beta) / (beta + 3.0)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.1)
		tmp = Float64(0.08333333333333333 + Float64(alpha * -0.041666666666666664));
	else
		tmp = Float64(Float64(1.0 / beta) / Float64(beta + 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.1)
		tmp = 0.08333333333333333 + (alpha * -0.041666666666666664);
	else
		tmp = (1.0 / beta) / (beta + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.1], N[(0.08333333333333333 + N[(alpha * -0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.10000000000000009

   1. Initial program 99.9%

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

   3. Simplified 94.2%

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

      1. associate-+r+ 94.2%

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

      2. fma-undefine 94.2%

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

      3. *-commutative 94.2%

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

      4. associate-+l+ 94.2%

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

      5. +-commutative 94.2%

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

      6. associate-+l+ 94.2%

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

      7. *-commutative 94.2%

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

      8. associate-*r* 94.2%

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

      9. associate-+r+ 94.2%

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

      10. +-commutative 94.2%

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

      11. associate-/l/ 99.6%

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

      12. clear-num 99.6%

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

      13. inv-pow 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. unpow-1 99.6%

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

      2. associate-/l* 99.6%

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

      3. associate-+r+ 99.6%

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

      4. +-commutative 99.6%

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

      5. +-commutative 99.6%

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

      6. associate-+r+ 99.6%

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

      7. +-commutative 99.6%

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

      8. associate-+r+ 99.6%

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

      9. +-commutative 99.6%

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

      10. fma-undefine 99.6%

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

      11. *-commutative 99.6%

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

      12. +-commutative 99.6%

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

      13. associate-+r+ 99.6%

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

      14. +-commutative 99.6%

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

      15. *-rgt-identity 99.6%

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

      16. *-commutative 99.6%

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

      17. +-commutative 99.6%

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

      18. distribute-lft-in 99.6%

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

      19. +-commutative 99.6%

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

      20. *-commutative 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in alpha around 0 70.4%

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

   10. Taylor expanded in beta around 0 69.8%

       \[\leadsto \color{blue}{\frac{0.16666666666666666}{2 + \alpha}} \]

   11. Taylor expanded in alpha around 0 68.1%

       \[\leadsto \color{blue}{0.08333333333333333 + -0.041666666666666664 \cdot \alpha} \]
   12. Step-by-step derivation

       1. *-commutative 68.1%

          \[\leadsto 0.08333333333333333 + \color{blue}{\alpha \cdot -0.041666666666666664} \]

   13. Simplified 68.1%

       \[\leadsto \color{blue}{0.08333333333333333 + \alpha \cdot -0.041666666666666664} \]

   if 2.10000000000000009 < beta

   1. Initial program 79.2%

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

   3. Taylor expanded in beta around inf 70.4%

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

   4. Taylor expanded in alpha around 0 65.3%

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

      1. associate-/r* 66.6%

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

      2. +-commutative 66.6%

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

   6. Simplified 66.6%

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

Alternative 16: 69.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.05:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.05)
   (+ 0.08333333333333333 (* alpha -0.041666666666666664))
   (/ 1.0 (* beta (+ beta 3.0)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.05) {
		tmp = 0.08333333333333333 + (alpha * -0.041666666666666664);
	} else {
		tmp = 1.0 / (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 <= 2.05d0) then
        tmp = 0.08333333333333333d0 + (alpha * (-0.041666666666666664d0))
    else
        tmp = 1.0d0 / (beta * (beta + 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.05) {
		tmp = 0.08333333333333333 + (alpha * -0.041666666666666664);
	} else {
		tmp = 1.0 / (beta * (beta + 3.0));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.05:
		tmp = 0.08333333333333333 + (alpha * -0.041666666666666664)
	else:
		tmp = 1.0 / (beta * (beta + 3.0))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.05)
		tmp = Float64(0.08333333333333333 + Float64(alpha * -0.041666666666666664));
	else
		tmp = Float64(1.0 / Float64(beta * Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.05)
		tmp = 0.08333333333333333 + (alpha * -0.041666666666666664);
	else
		tmp = 1.0 / (beta * (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.05], N[(0.08333333333333333 + N[(alpha * -0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(beta * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.0499999999999998

   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 94.2%

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

      1. associate-+r+ 94.2%

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

      2. fma-undefine 94.2%

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

      3. *-commutative 94.2%

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

      4. associate-+l+ 94.2%

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

      5. +-commutative 94.2%

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

      6. associate-+l+ 94.2%

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

      7. *-commutative 94.2%

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

      8. associate-*r* 94.2%

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

      9. associate-+r+ 94.2%

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

      10. +-commutative 94.2%

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

      11. associate-/l/ 99.6%

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

      12. clear-num 99.6%

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

      13. inv-pow 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. unpow-1 99.6%

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

      2. associate-/l* 99.6%

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

      3. associate-+r+ 99.6%

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

      4. +-commutative 99.6%

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

      5. +-commutative 99.6%

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

      6. associate-+r+ 99.6%

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

      7. +-commutative 99.6%

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

      8. associate-+r+ 99.6%

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

      9. +-commutative 99.6%

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

      10. fma-undefine 99.6%

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

      11. *-commutative 99.6%

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

      12. +-commutative 99.6%

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

      13. associate-+r+ 99.6%

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

      14. +-commutative 99.6%

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

      15. *-rgt-identity 99.6%

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

      16. *-commutative 99.6%

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

      17. +-commutative 99.6%

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

      18. distribute-lft-in 99.6%

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

      19. +-commutative 99.6%

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

      20. *-commutative 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in alpha around 0 70.4%

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

   10. Taylor expanded in beta around 0 69.8%

       \[\leadsto \color{blue}{\frac{0.16666666666666666}{2 + \alpha}} \]

   11. Taylor expanded in alpha around 0 68.1%

       \[\leadsto \color{blue}{0.08333333333333333 + -0.041666666666666664 \cdot \alpha} \]
   12. Step-by-step derivation

       1. *-commutative 68.1%

          \[\leadsto 0.08333333333333333 + \color{blue}{\alpha \cdot -0.041666666666666664} \]

   13. Simplified 68.1%

       \[\leadsto \color{blue}{0.08333333333333333 + \alpha \cdot -0.041666666666666664} \]

   if 2.0499999999999998 < beta

   1. Initial program 79.2%

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

   3. Taylor expanded in beta around inf 70.4%

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

   4. Taylor expanded in alpha around 0 65.3%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.05:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 45.7% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.5:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{0.2}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.5)
   (+ 0.08333333333333333 (* alpha -0.041666666666666664))
   (/ 0.2 beta)))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.5) {
		tmp = 0.08333333333333333 + (alpha * -0.041666666666666664);
	} else {
		tmp = 0.2 / beta;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.5d0) then
        tmp = 0.08333333333333333d0 + (alpha * (-0.041666666666666664d0))
    else
        tmp = 0.2d0 / beta
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.5) {
		tmp = 0.08333333333333333 + (alpha * -0.041666666666666664);
	} else {
		tmp = 0.2 / beta;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 1.5:
		tmp = 0.08333333333333333 + (alpha * -0.041666666666666664)
	else:
		tmp = 0.2 / beta
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.5)
		tmp = Float64(0.08333333333333333 + Float64(alpha * -0.041666666666666664));
	else
		tmp = Float64(0.2 / beta);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.5)
		tmp = 0.08333333333333333 + (alpha * -0.041666666666666664);
	else
		tmp = 0.2 / beta;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 1.5], N[(0.08333333333333333 + N[(alpha * -0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(0.2 / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.5

   1. Initial program 99.9%

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

   3. Simplified 94.2%

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

      1. associate-+r+ 94.2%

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

      2. fma-undefine 94.2%

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

      3. *-commutative 94.2%

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

      4. associate-+l+ 94.2%

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

      5. +-commutative 94.2%

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

      6. associate-+l+ 94.2%

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

      7. *-commutative 94.2%

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

      8. associate-*r* 94.2%

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

      9. associate-+r+ 94.2%

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

      10. +-commutative 94.2%

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

      11. associate-/l/ 99.6%

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

      12. clear-num 99.6%

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

      13. inv-pow 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. unpow-1 99.6%

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

      2. associate-/l* 99.6%

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

      3. associate-+r+ 99.6%

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

      4. +-commutative 99.6%

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

      5. +-commutative 99.6%

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

      6. associate-+r+ 99.6%

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

      7. +-commutative 99.6%

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

      8. associate-+r+ 99.6%

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

      9. +-commutative 99.6%

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

      10. fma-undefine 99.6%

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

      11. *-commutative 99.6%

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

      12. +-commutative 99.6%

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

      13. associate-+r+ 99.6%

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

      14. +-commutative 99.6%

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

      15. *-rgt-identity 99.6%

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

      16. *-commutative 99.6%

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

      17. +-commutative 99.6%

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

      18. distribute-lft-in 99.6%

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

      19. +-commutative 99.6%

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

      20. *-commutative 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in alpha around 0 70.4%

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

   10. Taylor expanded in beta around 0 69.8%

       \[\leadsto \color{blue}{\frac{0.16666666666666666}{2 + \alpha}} \]

   11. Taylor expanded in alpha around 0 68.1%

       \[\leadsto \color{blue}{0.08333333333333333 + -0.041666666666666664 \cdot \alpha} \]
   12. Step-by-step derivation

       1. *-commutative 68.1%

          \[\leadsto 0.08333333333333333 + \color{blue}{\alpha \cdot -0.041666666666666664} \]

   13. Simplified 68.1%

       \[\leadsto \color{blue}{0.08333333333333333 + \alpha \cdot -0.041666666666666664} \]

   if 1.5 < beta

   1. Initial program 79.2%

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

   3. Simplified 60.2%

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

      1. associate-+r+ 60.2%

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

      2. fma-undefine 60.2%

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

      3. *-commutative 60.2%

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

      4. associate-+l+ 60.2%

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

      5. +-commutative 60.2%

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

      6. associate-+l+ 60.2%

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

      7. *-commutative 60.2%

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

      8. associate-*r* 60.2%

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

      9. associate-+r+ 60.2%

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

      10. +-commutative 60.2%

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

      11. associate-/l/ 73.3%

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

      12. clear-num 73.3%

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

      13. inv-pow 73.3%

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

   6. Applied egg-rr 73.3%

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

      1. unpow-1 73.3%

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

      2. associate-/l* 77.8%

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

      3. associate-+r+ 77.8%

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

      4. +-commutative 77.8%

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

      5. +-commutative 77.8%

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

      6. associate-+r+ 77.8%

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

      7. +-commutative 77.8%

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

      8. associate-+r+ 77.8%

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

      9. +-commutative 77.8%

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

      10. fma-undefine 77.8%

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

      11. *-commutative 77.8%

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

      12. +-commutative 77.8%

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

      13. associate-+r+ 77.8%

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

      14. +-commutative 77.8%

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

      15. *-rgt-identity 77.8%

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

      16. *-commutative 77.8%

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

      17. +-commutative 77.8%

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

      18. distribute-lft-in 77.8%

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

      19. +-commutative 77.8%

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

      20. *-commutative 77.8%

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

   8. Simplified 77.8%

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

   9. Taylor expanded in alpha around 0 71.2%

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

   10. Taylor expanded in beta around 0 7.7%

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

       1. *-commutative 7.7%

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

   12. Simplified 7.7%

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

   13. Taylor expanded in beta around inf 7.0%

       \[\leadsto \color{blue}{\frac{0.2}{\beta}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 46.1% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.4:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{0.2}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.4) 0.08333333333333333 (/ 0.2 beta)))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.4) {
		tmp = 0.08333333333333333;
	} else {
		tmp = 0.2 / beta;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.4d0) then
        tmp = 0.08333333333333333d0
    else
        tmp = 0.2d0 / beta
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.4) {
		tmp = 0.08333333333333333;
	} else {
		tmp = 0.2 / beta;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.4:
		tmp = 0.08333333333333333
	else:
		tmp = 0.2 / beta
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.4)
		tmp = 0.08333333333333333;
	else
		tmp = Float64(0.2 / beta);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.4)
		tmp = 0.08333333333333333;
	else
		tmp = 0.2 / beta;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.4], 0.08333333333333333, N[(0.2 / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.39999999999999991

   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 94.2%

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

      1. associate-+r+ 94.2%

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

      2. fma-undefine 94.2%

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

      3. *-commutative 94.2%

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

      4. associate-+l+ 94.2%

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

      5. +-commutative 94.2%

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

      6. associate-+l+ 94.2%

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

      7. *-commutative 94.2%

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

      8. associate-*r* 94.2%

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

      9. associate-+r+ 94.2%

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

      10. +-commutative 94.2%

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

      11. associate-/l/ 99.6%

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

      12. clear-num 99.6%

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

      13. inv-pow 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. unpow-1 99.6%

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

      2. associate-/l* 99.6%

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

      3. associate-+r+ 99.6%

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

      4. +-commutative 99.6%

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

      5. +-commutative 99.6%

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

      6. associate-+r+ 99.6%

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

      7. +-commutative 99.6%

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

      8. associate-+r+ 99.6%

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

      9. +-commutative 99.6%

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

      10. fma-undefine 99.6%

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

      11. *-commutative 99.6%

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

      12. +-commutative 99.6%

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

      13. associate-+r+ 99.6%

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

      14. +-commutative 99.6%

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

      15. *-rgt-identity 99.6%

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

      16. *-commutative 99.6%

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

      17. +-commutative 99.6%

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

      18. distribute-lft-in 99.6%

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

      19. +-commutative 99.6%

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

      20. *-commutative 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in alpha around 0 70.4%

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

   10. Taylor expanded in beta around 0 69.8%

       \[\leadsto \color{blue}{\frac{0.16666666666666666}{2 + \alpha}} \]

   11. Taylor expanded in alpha around 0 68.8%

       \[\leadsto \color{blue}{0.08333333333333333} \]

   if 2.39999999999999991 < beta

   1. Initial program 79.2%

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

   3. Simplified 60.2%

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

      1. associate-+r+ 60.2%

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

      2. fma-undefine 60.2%

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

      3. *-commutative 60.2%

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

      4. associate-+l+ 60.2%

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

      5. +-commutative 60.2%

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

      6. associate-+l+ 60.2%

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

      7. *-commutative 60.2%

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

      8. associate-*r* 60.2%

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

      9. associate-+r+ 60.2%

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

      10. +-commutative 60.2%

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

      11. associate-/l/ 73.3%

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

      12. clear-num 73.3%

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

      13. inv-pow 73.3%

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

   6. Applied egg-rr 73.3%

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

      1. unpow-1 73.3%

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

      2. associate-/l* 77.8%

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

      3. associate-+r+ 77.8%

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

      4. +-commutative 77.8%

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

      5. +-commutative 77.8%

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

      6. associate-+r+ 77.8%

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

      7. +-commutative 77.8%

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

      8. associate-+r+ 77.8%

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

      9. +-commutative 77.8%

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

      10. fma-undefine 77.8%

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

      11. *-commutative 77.8%

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

      12. +-commutative 77.8%

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

      13. associate-+r+ 77.8%

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

      14. +-commutative 77.8%

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

      15. *-rgt-identity 77.8%

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

      16. *-commutative 77.8%

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

      17. +-commutative 77.8%

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

      18. distribute-lft-in 77.8%

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

      19. +-commutative 77.8%

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

      20. *-commutative 77.8%

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

   8. Simplified 77.8%

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

   9. Taylor expanded in alpha around 0 71.2%

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

   10. Taylor expanded in beta around 0 7.7%

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

       1. *-commutative 7.7%

          \[\leadsto \frac{1}{\left(2 + \left(\beta + \alpha\right)\right) \cdot \frac{6 + \color{blue}{\beta \cdot 5}}{1 + \beta}} \]

   12. Simplified 7.7%

       \[\leadsto \frac{1}{\left(2 + \left(\beta + \alpha\right)\right) \cdot \frac{\color{blue}{6 + \beta \cdot 5}}{1 + \beta}} \]

   13. Taylor expanded in beta around inf 7.0%

       \[\leadsto \color{blue}{\frac{0.2}{\beta}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 46.2% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.16666666666666666}{\alpha + 2} \end{array} \]

FPCore

(FPCore (alpha beta) :precision binary64 (/ 0.16666666666666666 (+ alpha 2.0)))

C

double code(double alpha, double beta) {
	return 0.16666666666666666 / (alpha + 2.0);
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = 0.16666666666666666d0 / (alpha + 2.0d0)
end function

Java

public static double code(double alpha, double beta) {
	return 0.16666666666666666 / (alpha + 2.0);
}

Python

def code(alpha, beta):
	return 0.16666666666666666 / (alpha + 2.0)

Julia

function code(alpha, beta)
	return Float64(0.16666666666666666 / Float64(alpha + 2.0))
end

MATLAB

function tmp = code(alpha, beta)
	tmp = 0.16666666666666666 / (alpha + 2.0);
end

Wolfram

code[alpha_, beta_] := N[(0.16666666666666666 / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.9%

   \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

3. Simplified 82.7%

   \[\leadsto \color{blue}{\frac{\alpha + \left(\beta + \mathsf{fma}\left(\alpha, \beta, 1\right)\right)}{\left(\beta + \left(\alpha + 3\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-+r+ 82.7%

      \[\leadsto \frac{\color{blue}{\left(\alpha + \beta\right) + \mathsf{fma}\left(\alpha, \beta, 1\right)}}{\left(\beta + \left(\alpha + 3\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)} \]

   2. fma-undefine 82.7%

      \[\leadsto \frac{\left(\alpha + \beta\right) + \color{blue}{\left(\alpha \cdot \beta + 1\right)}}{\left(\beta + \left(\alpha + 3\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)} \]

   3. *-commutative 82.7%

      \[\leadsto \frac{\left(\alpha + \beta\right) + \left(\color{blue}{\beta \cdot \alpha} + 1\right)}{\left(\beta + \left(\alpha + 3\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)} \]

   4. associate-+l+ 82.7%

      \[\leadsto \frac{\color{blue}{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}}{\left(\beta + \left(\alpha + 3\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)} \]

   5. +-commutative 82.7%

      \[\leadsto \frac{\color{blue}{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}}{\left(\beta + \left(\alpha + 3\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)} \]

   6. associate-+l+ 82.7%

      \[\leadsto \frac{1 + \color{blue}{\left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)}}{\left(\beta + \left(\alpha + 3\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)} \]

   7. *-commutative 82.7%

      \[\leadsto \frac{1 + \left(\alpha + \left(\beta + \color{blue}{\alpha \cdot \beta}\right)\right)}{\left(\beta + \left(\alpha + 3\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)} \]

   8. associate-*r* 82.7%

      \[\leadsto \frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\color{blue}{\left(\left(\beta + \left(\alpha + 3\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}} \]

   9. associate-+r+ 82.7%

      \[\leadsto \frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\left(\color{blue}{\left(\left(\beta + \alpha\right) + 3\right)} \cdot \left(\alpha + \left(\beta + 2\right)\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]

   10. +-commutative 82.7%

       \[\leadsto \frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\left(\left(\color{blue}{\left(\alpha + \beta\right)} + 3\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]

   11. associate-/l/ 90.8%

       \[\leadsto \color{blue}{\frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}} \]

   12. clear-num 90.8%

       \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}}} \]

   13. inv-pow 90.8%

       \[\leadsto \color{blue}{{\left(\frac{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}\right)}^{-1}} \]

6. Applied egg-rr 90.8%

   \[\leadsto \color{blue}{{\left(\frac{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}\right)}^{-1}} \]
7. Step-by-step derivation

   1. unpow-1 90.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}}} \]

   2. associate-/l* 92.3%

      \[\leadsto \frac{1}{\color{blue}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \frac{\alpha + \left(\beta + 3\right)}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}}} \]

   3. associate-+r+ 92.3%

      \[\leadsto \frac{1}{\color{blue}{\left(\left(\alpha + \beta\right) + 2\right)} \cdot \frac{\alpha + \left(\beta + 3\right)}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}} \]

   4. +-commutative 92.3%

      \[\leadsto \frac{1}{\color{blue}{\left(2 + \left(\alpha + \beta\right)\right)} \cdot \frac{\alpha + \left(\beta + 3\right)}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}} \]

   5. +-commutative 92.3%

      \[\leadsto \frac{1}{\left(2 + \color{blue}{\left(\beta + \alpha\right)}\right) \cdot \frac{\alpha + \left(\beta + 3\right)}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}} \]

   6. associate-+r+ 92.3%

      \[\leadsto \frac{1}{\left(2 + \left(\beta + \alpha\right)\right) \cdot \frac{\color{blue}{\left(\alpha + \beta\right) + 3}}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}} \]

   7. +-commutative 92.3%

      \[\leadsto \frac{1}{\left(2 + \left(\beta + \alpha\right)\right) \cdot \frac{\color{blue}{3 + \left(\alpha + \beta\right)}}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}} \]

   8. associate-+r+ 92.3%

      \[\leadsto \frac{1}{\left(2 + \left(\beta + \alpha\right)\right) \cdot \frac{\color{blue}{\left(3 + \alpha\right) + \beta}}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}} \]

   9. +-commutative 92.3%

      \[\leadsto \frac{1}{\left(2 + \left(\beta + \alpha\right)\right) \cdot \frac{\left(3 + \alpha\right) + \beta}{\frac{1 + \mathsf{fma}\left(\color{blue}{1 + \alpha}, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}} \]

   10. fma-undefine 92.3%

       \[\leadsto \frac{1}{\left(2 + \left(\beta + \alpha\right)\right) \cdot \frac{\left(3 + \alpha\right) + \beta}{\frac{1 + \color{blue}{\left(\left(1 + \alpha\right) \cdot \beta + \alpha\right)}}{\alpha + \left(\beta + 2\right)}}} \]

   11. *-commutative 92.3%

       \[\leadsto \frac{1}{\left(2 + \left(\beta + \alpha\right)\right) \cdot \frac{\left(3 + \alpha\right) + \beta}{\frac{1 + \left(\color{blue}{\beta \cdot \left(1 + \alpha\right)} + \alpha\right)}{\alpha + \left(\beta + 2\right)}}} \]

   12. +-commutative 92.3%

       \[\leadsto \frac{1}{\left(2 + \left(\beta + \alpha\right)\right) \cdot \frac{\left(3 + \alpha\right) + \beta}{\frac{1 + \color{blue}{\left(\alpha + \beta \cdot \left(1 + \alpha\right)\right)}}{\alpha + \left(\beta + 2\right)}}} \]

   13. associate-+r+ 92.3%

       \[\leadsto \frac{1}{\left(2 + \left(\beta + \alpha\right)\right) \cdot \frac{\left(3 + \alpha\right) + \beta}{\frac{\color{blue}{\left(1 + \alpha\right) + \beta \cdot \left(1 + \alpha\right)}}{\alpha + \left(\beta + 2\right)}}} \]

   14. +-commutative 92.3%

       \[\leadsto \frac{1}{\left(2 + \left(\beta + \alpha\right)\right) \cdot \frac{\left(3 + \alpha\right) + \beta}{\frac{\color{blue}{\left(\alpha + 1\right)} + \beta \cdot \left(1 + \alpha\right)}{\alpha + \left(\beta + 2\right)}}} \]

   15. *-rgt-identity 92.3%

       \[\leadsto \frac{1}{\left(2 + \left(\beta + \alpha\right)\right) \cdot \frac{\left(3 + \alpha\right) + \beta}{\frac{\color{blue}{\left(\alpha + 1\right) \cdot 1} + \beta \cdot \left(1 + \alpha\right)}{\alpha + \left(\beta + 2\right)}}} \]

   16. *-commutative 92.3%

       \[\leadsto \frac{1}{\left(2 + \left(\beta + \alpha\right)\right) \cdot \frac{\left(3 + \alpha\right) + \beta}{\frac{\left(\alpha + 1\right) \cdot 1 + \color{blue}{\left(1 + \alpha\right) \cdot \beta}}{\alpha + \left(\beta + 2\right)}}} \]

   17. +-commutative 92.3%

       \[\leadsto \frac{1}{\left(2 + \left(\beta + \alpha\right)\right) \cdot \frac{\left(3 + \alpha\right) + \beta}{\frac{\left(\alpha + 1\right) \cdot 1 + \color{blue}{\left(\alpha + 1\right)} \cdot \beta}{\alpha + \left(\beta + 2\right)}}} \]

   18. distribute-lft-in 92.3%

       \[\leadsto \frac{1}{\left(2 + \left(\beta + \alpha\right)\right) \cdot \frac{\left(3 + \alpha\right) + \beta}{\frac{\color{blue}{\left(\alpha + 1\right) \cdot \left(1 + \beta\right)}}{\alpha + \left(\beta + 2\right)}}} \]

   19. +-commutative 92.3%

       \[\leadsto \frac{1}{\left(2 + \left(\beta + \alpha\right)\right) \cdot \frac{\left(3 + \alpha\right) + \beta}{\frac{\color{blue}{\left(1 + \alpha\right)} \cdot \left(1 + \beta\right)}{\alpha + \left(\beta + 2\right)}}} \]

   20. *-commutative 92.3%

       \[\leadsto \frac{1}{\left(2 + \left(\beta + \alpha\right)\right) \cdot \frac{\left(3 + \alpha\right) + \beta}{\frac{\color{blue}{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}}{\alpha + \left(\beta + 2\right)}}} \]

8. Simplified 92.3%

   \[\leadsto \color{blue}{\frac{1}{\left(2 + \left(\beta + \alpha\right)\right) \cdot \frac{\left(3 + \alpha\right) + \beta}{\frac{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}{2 + \left(\beta + \alpha\right)}}}} \]

9. Taylor expanded in alpha around 0 70.7%

   \[\leadsto \frac{1}{\left(2 + \left(\beta + \alpha\right)\right) \cdot \color{blue}{\frac{\left(2 + \beta\right) \cdot \left(3 + \beta\right)}{1 + \beta}}} \]

10. Taylor expanded in beta around 0 48.3%

    \[\leadsto \color{blue}{\frac{0.16666666666666666}{2 + \alpha}} \]

11. Final simplification 48.3%

    \[\leadsto \frac{0.16666666666666666}{\alpha + 2} \]
12. Add Preprocessing

Alternative 20: 45.2% accurate, 35.0× speedup

Math

\[\begin{array}{l} \\ 0.08333333333333333 \end{array} \]

FPCore

(FPCore (alpha beta) :precision binary64 0.08333333333333333)

C

double code(double alpha, double beta) {
	return 0.08333333333333333;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = 0.08333333333333333d0
end function

Java

public static double code(double alpha, double beta) {
	return 0.08333333333333333;
}

Python

def code(alpha, beta):
	return 0.08333333333333333

Julia

function code(alpha, beta)
	return 0.08333333333333333
end

MATLAB

function tmp = code(alpha, beta)
	tmp = 0.08333333333333333;
end

Wolfram

code[alpha_, beta_] := 0.08333333333333333

Derivation

1. Initial program 92.9%

   \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

3. Simplified 82.7%

   \[\leadsto \color{blue}{\frac{\alpha + \left(\beta + \mathsf{fma}\left(\alpha, \beta, 1\right)\right)}{\left(\beta + \left(\alpha + 3\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-+r+ 82.7%

      \[\leadsto \frac{\color{blue}{\left(\alpha + \beta\right) + \mathsf{fma}\left(\alpha, \beta, 1\right)}}{\left(\beta + \left(\alpha + 3\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)} \]

   2. fma-undefine 82.7%

      \[\leadsto \frac{\left(\alpha + \beta\right) + \color{blue}{\left(\alpha \cdot \beta + 1\right)}}{\left(\beta + \left(\alpha + 3\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)} \]

   3. *-commutative 82.7%

      \[\leadsto \frac{\left(\alpha + \beta\right) + \left(\color{blue}{\beta \cdot \alpha} + 1\right)}{\left(\beta + \left(\alpha + 3\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)} \]

   4. associate-+l+ 82.7%

      \[\leadsto \frac{\color{blue}{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}}{\left(\beta + \left(\alpha + 3\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)} \]

   5. +-commutative 82.7%

      \[\leadsto \frac{\color{blue}{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}}{\left(\beta + \left(\alpha + 3\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)} \]

   6. associate-+l+ 82.7%

      \[\leadsto \frac{1 + \color{blue}{\left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)}}{\left(\beta + \left(\alpha + 3\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)} \]

   7. *-commutative 82.7%

      \[\leadsto \frac{1 + \left(\alpha + \left(\beta + \color{blue}{\alpha \cdot \beta}\right)\right)}{\left(\beta + \left(\alpha + 3\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)} \]

   8. associate-*r* 82.7%

      \[\leadsto \frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\color{blue}{\left(\left(\beta + \left(\alpha + 3\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}} \]

   9. associate-+r+ 82.7%

      \[\leadsto \frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\left(\color{blue}{\left(\left(\beta + \alpha\right) + 3\right)} \cdot \left(\alpha + \left(\beta + 2\right)\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]

   10. +-commutative 82.7%

       \[\leadsto \frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\left(\left(\color{blue}{\left(\alpha + \beta\right)} + 3\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]

   11. associate-/l/ 90.8%

       \[\leadsto \color{blue}{\frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}} \]

   12. clear-num 90.8%

       \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}}} \]

   13. inv-pow 90.8%

       \[\leadsto \color{blue}{{\left(\frac{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}\right)}^{-1}} \]

6. Applied egg-rr 90.8%

   \[\leadsto \color{blue}{{\left(\frac{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}\right)}^{-1}} \]
7. Step-by-step derivation

   1. unpow-1 90.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}}} \]

   2. associate-/l* 92.3%

      \[\leadsto \frac{1}{\color{blue}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \frac{\alpha + \left(\beta + 3\right)}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}}} \]

   3. associate-+r+ 92.3%

      \[\leadsto \frac{1}{\color{blue}{\left(\left(\alpha + \beta\right) + 2\right)} \cdot \frac{\alpha + \left(\beta + 3\right)}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}} \]

   4. +-commutative 92.3%

      \[\leadsto \frac{1}{\color{blue}{\left(2 + \left(\alpha + \beta\right)\right)} \cdot \frac{\alpha + \left(\beta + 3\right)}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}} \]

   5. +-commutative 92.3%

      \[\leadsto \frac{1}{\left(2 + \color{blue}{\left(\beta + \alpha\right)}\right) \cdot \frac{\alpha + \left(\beta + 3\right)}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}} \]

   6. associate-+r+ 92.3%

      \[\leadsto \frac{1}{\left(2 + \left(\beta + \alpha\right)\right) \cdot \frac{\color{blue}{\left(\alpha + \beta\right) + 3}}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}} \]

   7. +-commutative 92.3%

      \[\leadsto \frac{1}{\left(2 + \left(\beta + \alpha\right)\right) \cdot \frac{\color{blue}{3 + \left(\alpha + \beta\right)}}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}} \]

   8. associate-+r+ 92.3%

      \[\leadsto \frac{1}{\left(2 + \left(\beta + \alpha\right)\right) \cdot \frac{\color{blue}{\left(3 + \alpha\right) + \beta}}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}} \]

   9. +-commutative 92.3%

      \[\leadsto \frac{1}{\left(2 + \left(\beta + \alpha\right)\right) \cdot \frac{\left(3 + \alpha\right) + \beta}{\frac{1 + \mathsf{fma}\left(\color{blue}{1 + \alpha}, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}} \]

   10. fma-undefine 92.3%

       \[\leadsto \frac{1}{\left(2 + \left(\beta + \alpha\right)\right) \cdot \frac{\left(3 + \alpha\right) + \beta}{\frac{1 + \color{blue}{\left(\left(1 + \alpha\right) \cdot \beta + \alpha\right)}}{\alpha + \left(\beta + 2\right)}}} \]

   11. *-commutative 92.3%

       \[\leadsto \frac{1}{\left(2 + \left(\beta + \alpha\right)\right) \cdot \frac{\left(3 + \alpha\right) + \beta}{\frac{1 + \left(\color{blue}{\beta \cdot \left(1 + \alpha\right)} + \alpha\right)}{\alpha + \left(\beta + 2\right)}}} \]

   12. +-commutative 92.3%

       \[\leadsto \frac{1}{\left(2 + \left(\beta + \alpha\right)\right) \cdot \frac{\left(3 + \alpha\right) + \beta}{\frac{1 + \color{blue}{\left(\alpha + \beta \cdot \left(1 + \alpha\right)\right)}}{\alpha + \left(\beta + 2\right)}}} \]

   13. associate-+r+ 92.3%

       \[\leadsto \frac{1}{\left(2 + \left(\beta + \alpha\right)\right) \cdot \frac{\left(3 + \alpha\right) + \beta}{\frac{\color{blue}{\left(1 + \alpha\right) + \beta \cdot \left(1 + \alpha\right)}}{\alpha + \left(\beta + 2\right)}}} \]

   14. +-commutative 92.3%

       \[\leadsto \frac{1}{\left(2 + \left(\beta + \alpha\right)\right) \cdot \frac{\left(3 + \alpha\right) + \beta}{\frac{\color{blue}{\left(\alpha + 1\right)} + \beta \cdot \left(1 + \alpha\right)}{\alpha + \left(\beta + 2\right)}}} \]

   15. *-rgt-identity 92.3%

       \[\leadsto \frac{1}{\left(2 + \left(\beta + \alpha\right)\right) \cdot \frac{\left(3 + \alpha\right) + \beta}{\frac{\color{blue}{\left(\alpha + 1\right) \cdot 1} + \beta \cdot \left(1 + \alpha\right)}{\alpha + \left(\beta + 2\right)}}} \]

   16. *-commutative 92.3%

       \[\leadsto \frac{1}{\left(2 + \left(\beta + \alpha\right)\right) \cdot \frac{\left(3 + \alpha\right) + \beta}{\frac{\left(\alpha + 1\right) \cdot 1 + \color{blue}{\left(1 + \alpha\right) \cdot \beta}}{\alpha + \left(\beta + 2\right)}}} \]

   17. +-commutative 92.3%

       \[\leadsto \frac{1}{\left(2 + \left(\beta + \alpha\right)\right) \cdot \frac{\left(3 + \alpha\right) + \beta}{\frac{\left(\alpha + 1\right) \cdot 1 + \color{blue}{\left(\alpha + 1\right)} \cdot \beta}{\alpha + \left(\beta + 2\right)}}} \]

   18. distribute-lft-in 92.3%

       \[\leadsto \frac{1}{\left(2 + \left(\beta + \alpha\right)\right) \cdot \frac{\left(3 + \alpha\right) + \beta}{\frac{\color{blue}{\left(\alpha + 1\right) \cdot \left(1 + \beta\right)}}{\alpha + \left(\beta + 2\right)}}} \]

   19. +-commutative 92.3%

       \[\leadsto \frac{1}{\left(2 + \left(\beta + \alpha\right)\right) \cdot \frac{\left(3 + \alpha\right) + \beta}{\frac{\color{blue}{\left(1 + \alpha\right)} \cdot \left(1 + \beta\right)}{\alpha + \left(\beta + 2\right)}}} \]

   20. *-commutative 92.3%

       \[\leadsto \frac{1}{\left(2 + \left(\beta + \alpha\right)\right) \cdot \frac{\left(3 + \alpha\right) + \beta}{\frac{\color{blue}{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}}{\alpha + \left(\beta + 2\right)}}} \]

8. Simplified 92.3%

   \[\leadsto \color{blue}{\frac{1}{\left(2 + \left(\beta + \alpha\right)\right) \cdot \frac{\left(3 + \alpha\right) + \beta}{\frac{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}{2 + \left(\beta + \alpha\right)}}}} \]

9. Taylor expanded in alpha around 0 70.7%

   \[\leadsto \frac{1}{\left(2 + \left(\beta + \alpha\right)\right) \cdot \color{blue}{\frac{\left(2 + \beta\right) \cdot \left(3 + \beta\right)}{1 + \beta}}} \]

10. Taylor expanded in beta around 0 48.3%

    \[\leadsto \color{blue}{\frac{0.16666666666666666}{2 + \alpha}} \]

11. Taylor expanded in alpha around 0 47.1%

    \[\leadsto \color{blue}{0.08333333333333333} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024145 
(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)))