Octave 3.8, jcobi/3

Percentage Accurate: 93.7% → 99.8%

Time: 17.0s

Alternatives: 19

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: 93.7% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.3%

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

   1. associate-/l/ 92.0%

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

   2. +-commutative 92.0%

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

   3. +-commutative 92.0%

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

   4. associate-+r+ 92.0%

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

   5. associate-+r+ 92.0%

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

   6. associate-+r+ 92.0%

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

   7. distribute-rgt1-in 92.0%

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

   8. +-commutative 92.0%

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

   9. *-commutative 92.0%

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

   10. distribute-rgt1-in 92.0%

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

   11. +-commutative 92.0%

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

   12. metadata-eval 92.0%

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

   13. associate-+l+ 92.0%

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

3. Simplified 92.0%

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

   1. associate-+r+ 92.0%

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

   2. metadata-eval 92.0%

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

   3. associate-/l* 96.4%

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

   4. associate-+r+ 96.4%

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

   5. metadata-eval 96.4%

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

   6. metadata-eval 96.4%

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

   7. associate-+l+ 96.4%

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

   8. metadata-eval 96.4%

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

   9. times-frac 99.8%

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

   10. metadata-eval 99.8%

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

   11. associate-+r+ 99.8%

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

6. Applied egg-rr 99.8%

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

Alternative 2: 96.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ t_1 := \frac{\alpha + 1}{t\_0}\\ \mathbf{if}\;\beta \leq 5.8 \cdot 10^{+111}:\\ \;\;\;\;t\_1 \cdot \frac{1 + \beta}{t\_0 \cdot \left(\alpha + \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \frac{1 + -2 \cdot \frac{\alpha + 2}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))) (t_1 (/ (+ alpha 1.0) t_0)))
   (if (<= beta 5.8e+111)
     (* t_1 (/ (+ 1.0 beta) (* t_0 (+ alpha (+ beta 3.0)))))
     (* t_1 (/ (+ 1.0 (* -2.0 (/ (+ alpha 2.0) beta))) beta)))))

C

double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double t_1 = (alpha + 1.0) / t_0;
	double tmp;
	if (beta <= 5.8e+111) {
		tmp = t_1 * ((1.0 + beta) / (t_0 * (alpha + (beta + 3.0))));
	} else {
		tmp = t_1 * ((1.0 + (-2.0 * ((alpha + 2.0) / beta))) / beta);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    t_1 = (alpha + 1.0d0) / t_0
    if (beta <= 5.8d+111) then
        tmp = t_1 * ((1.0d0 + beta) / (t_0 * (alpha + (beta + 3.0d0))))
    else
        tmp = t_1 * ((1.0d0 + ((-2.0d0) * ((alpha + 2.0d0) / beta))) / beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double t_1 = (alpha + 1.0) / t_0;
	double tmp;
	if (beta <= 5.8e+111) {
		tmp = t_1 * ((1.0 + beta) / (t_0 * (alpha + (beta + 3.0))));
	} else {
		tmp = t_1 * ((1.0 + (-2.0 * ((alpha + 2.0) / beta))) / beta);
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	t_1 = (alpha + 1.0) / t_0
	tmp = 0
	if beta <= 5.8e+111:
		tmp = t_1 * ((1.0 + beta) / (t_0 * (alpha + (beta + 3.0))))
	else:
		tmp = t_1 * ((1.0 + (-2.0 * ((alpha + 2.0) / beta))) / beta)
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	t_1 = Float64(Float64(alpha + 1.0) / t_0)
	tmp = 0.0
	if (beta <= 5.8e+111)
		tmp = Float64(t_1 * Float64(Float64(1.0 + beta) / Float64(t_0 * Float64(alpha + Float64(beta + 3.0)))));
	else
		tmp = Float64(t_1 * Float64(Float64(1.0 + Float64(-2.0 * Float64(Float64(alpha + 2.0) / beta))) / beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	t_1 = (alpha + 1.0) / t_0;
	tmp = 0.0;
	if (beta <= 5.8e+111)
		tmp = t_1 * ((1.0 + beta) / (t_0 * (alpha + (beta + 3.0))));
	else
		tmp = t_1 * ((1.0 + (-2.0 * ((alpha + 2.0) / beta))) / beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(alpha + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[beta, 5.8e+111], N[(t$95$1 * N[(N[(1.0 + beta), $MachinePrecision] / N[(t$95$0 * N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(N[(1.0 + N[(-2.0 * N[(N[(alpha + 2.0), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.7999999999999999e111

   1. Initial program 97.9%

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

      1. associate-/l/ 97.0%

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

      2. +-commutative 97.0%

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

      3. +-commutative 97.0%

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

      4. associate-+r+ 97.0%

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

      5. associate-+r+ 97.0%

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

      6. associate-+r+ 97.0%

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

      7. distribute-rgt1-in 97.0%

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

      8. +-commutative 97.0%

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

      9. *-commutative 97.0%

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

      10. distribute-rgt1-in 97.0%

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

      11. +-commutative 97.0%

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

      12. metadata-eval 97.0%

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

      13. associate-+l+ 97.0%

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

   3. Simplified 97.0%

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

      1. associate-+r+ 97.0%

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

      2. metadata-eval 97.0%

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

      3. associate-/l/ 89.2%

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

      4. *-commutative 89.2%

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

      5. times-frac 99.0%

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

      6. +-commutative 99.0%

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

      7. associate-+l+ 99.0%

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

      8. metadata-eval 99.0%

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

      9. associate-+r+ 99.0%

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

   6. Applied egg-rr 99.0%

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

   if 5.7999999999999999e111 < beta

   1. Initial program 76.2%

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

      1. associate-/l/ 73.3%

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

      2. +-commutative 73.3%

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

      3. +-commutative 73.3%

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

      4. associate-+r+ 73.3%

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

      5. associate-+r+ 73.3%

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

      6. associate-+r+ 73.3%

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

      7. distribute-rgt1-in 73.3%

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

      8. +-commutative 73.3%

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

      9. *-commutative 73.3%

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

      10. distribute-rgt1-in 73.3%

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

      11. +-commutative 73.3%

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

      12. metadata-eval 73.3%

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

      13. associate-+l+ 73.3%

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

   3. Simplified 73.3%

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

      1. associate-+r+ 73.3%

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

      2. metadata-eval 73.3%

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

      3. associate-/l* 86.5%

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

      4. associate-+r+ 86.5%

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

      5. metadata-eval 86.5%

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

      6. metadata-eval 86.5%

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

      7. associate-+l+ 86.5%

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

      8. metadata-eval 86.5%

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

      9. times-frac 99.7%

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

      10. metadata-eval 99.7%

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

      11. associate-+r+ 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in beta around inf 94.8%

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

      1. mul-1-neg 94.8%

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

      2. *-commutative 94.8%

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

      3. distribute-neg-frac2 94.8%

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

      4. metadata-eval 94.8%

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

      5. distribute-rgt-in 94.8%

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

      6. neg-mul-1 94.8%

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

      7. times-frac 94.8%

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

      8. metadata-eval 94.8%

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

   9. Simplified 94.8%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.8 \cdot 10^{+111}:\\ \;\;\;\;\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + -2 \cdot \frac{\alpha + 2}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.8:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\alpha + 2\right) \cdot \left(\alpha + 2\right)}}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + -2 \cdot \frac{\alpha + 2}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.8)
   (/
    (/ (+ alpha 1.0) (* (+ alpha 2.0) (+ alpha 2.0)))
    (+ 1.0 (+ 2.0 (+ alpha beta))))
   (*
    (/ (+ alpha 1.0) (+ alpha (+ beta 2.0)))
    (/ (+ 1.0 (* -2.0 (/ (+ alpha 2.0) beta))) beta))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.8) {
		tmp = ((alpha + 1.0) / ((alpha + 2.0) * (alpha + 2.0))) / (1.0 + (2.0 + (alpha + beta)));
	} else {
		tmp = ((alpha + 1.0) / (alpha + (beta + 2.0))) * ((1.0 + (-2.0 * ((alpha + 2.0) / 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 <= 6.8d0) then
        tmp = ((alpha + 1.0d0) / ((alpha + 2.0d0) * (alpha + 2.0d0))) / (1.0d0 + (2.0d0 + (alpha + beta)))
    else
        tmp = ((alpha + 1.0d0) / (alpha + (beta + 2.0d0))) * ((1.0d0 + ((-2.0d0) * ((alpha + 2.0d0) / beta))) / beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.8) {
		tmp = ((alpha + 1.0) / ((alpha + 2.0) * (alpha + 2.0))) / (1.0 + (2.0 + (alpha + beta)));
	} else {
		tmp = ((alpha + 1.0) / (alpha + (beta + 2.0))) * ((1.0 + (-2.0 * ((alpha + 2.0) / beta))) / beta);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 6.8:
		tmp = ((alpha + 1.0) / ((alpha + 2.0) * (alpha + 2.0))) / (1.0 + (2.0 + (alpha + beta)))
	else:
		tmp = ((alpha + 1.0) / (alpha + (beta + 2.0))) * ((1.0 + (-2.0 * ((alpha + 2.0) / beta))) / beta)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.8)
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(Float64(alpha + 2.0) * Float64(alpha + 2.0))) / Float64(1.0 + Float64(2.0 + Float64(alpha + beta))));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(alpha + Float64(beta + 2.0))) * Float64(Float64(1.0 + Float64(-2.0 * Float64(Float64(alpha + 2.0) / beta))) / beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 6.8)
		tmp = ((alpha + 1.0) / ((alpha + 2.0) * (alpha + 2.0))) / (1.0 + (2.0 + (alpha + beta)));
	else
		tmp = ((alpha + 1.0) / (alpha + (beta + 2.0))) * ((1.0 + (-2.0 * ((alpha + 2.0) / beta))) / beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 6.8], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(N[(alpha + 2.0), $MachinePrecision] * N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(-2.0 * N[(N[(alpha + 2.0), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.79999999999999982

   1. Initial program 99.9%

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

   3. Taylor expanded in beta around 0 98.4%

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

      1. unpow2 98.4%

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

   5. Simplified 98.4%

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

   if 6.79999999999999982 < beta

   1. Initial program 81.8%

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

      1. associate-/l/ 79.1%

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

      2. +-commutative 79.1%

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

      3. +-commutative 79.1%

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

      4. associate-+r+ 79.1%

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

      5. associate-+r+ 79.1%

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

      6. associate-+r+ 79.1%

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

      7. distribute-rgt1-in 79.1%

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

      8. +-commutative 79.1%

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

      9. *-commutative 79.1%

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

      10. distribute-rgt1-in 79.1%

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

      11. +-commutative 79.1%

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

      12. metadata-eval 79.1%

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

      13. associate-+l+ 79.1%

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

   3. Simplified 79.1%

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

      1. associate-+r+ 79.1%

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

      2. metadata-eval 79.1%

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

      3. associate-/l* 91.0%

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

      4. associate-+r+ 91.0%

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

      5. metadata-eval 91.0%

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

      6. metadata-eval 91.0%

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

      7. associate-+l+ 91.0%

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

      8. metadata-eval 91.0%

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

      9. times-frac 99.7%

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

      10. metadata-eval 99.7%

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

      11. associate-+r+ 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in beta around inf 81.5%

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

      1. mul-1-neg 81.5%

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

      2. *-commutative 81.5%

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

      3. distribute-neg-frac2 81.5%

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

      4. metadata-eval 81.5%

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

      5. distribute-rgt-in 81.5%

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

      6. neg-mul-1 81.5%

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

      7. times-frac 81.5%

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

      8. metadata-eval 81.5%

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

   9. Simplified 81.5%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.8:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\alpha + 2\right) \cdot \left(\alpha + 2\right)}}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + -2 \cdot \frac{\alpha + 2}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 14500000000:\\ \;\;\;\;\frac{1 + \beta}{\left(\beta + 3\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + -2 \cdot \frac{\alpha + 2}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 14500000000.0)
   (/ (+ 1.0 beta) (* (+ beta 3.0) (* (+ beta 2.0) (+ beta 2.0))))
   (*
    (/ (+ alpha 1.0) (+ alpha (+ beta 2.0)))
    (/ (+ 1.0 (* -2.0 (/ (+ alpha 2.0) beta))) beta))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 14500000000.0) {
		tmp = (1.0 + beta) / ((beta + 3.0) * ((beta + 2.0) * (beta + 2.0)));
	} else {
		tmp = ((alpha + 1.0) / (alpha + (beta + 2.0))) * ((1.0 + (-2.0 * ((alpha + 2.0) / 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 <= 14500000000.0d0) then
        tmp = (1.0d0 + beta) / ((beta + 3.0d0) * ((beta + 2.0d0) * (beta + 2.0d0)))
    else
        tmp = ((alpha + 1.0d0) / (alpha + (beta + 2.0d0))) * ((1.0d0 + ((-2.0d0) * ((alpha + 2.0d0) / beta))) / beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 14500000000.0) {
		tmp = (1.0 + beta) / ((beta + 3.0) * ((beta + 2.0) * (beta + 2.0)));
	} else {
		tmp = ((alpha + 1.0) / (alpha + (beta + 2.0))) * ((1.0 + (-2.0 * ((alpha + 2.0) / beta))) / beta);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 14500000000.0:
		tmp = (1.0 + beta) / ((beta + 3.0) * ((beta + 2.0) * (beta + 2.0)))
	else:
		tmp = ((alpha + 1.0) / (alpha + (beta + 2.0))) * ((1.0 + (-2.0 * ((alpha + 2.0) / beta))) / beta)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 14500000000.0)
		tmp = Float64(Float64(1.0 + beta) / Float64(Float64(beta + 3.0) * Float64(Float64(beta + 2.0) * Float64(beta + 2.0))));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(alpha + Float64(beta + 2.0))) * Float64(Float64(1.0 + Float64(-2.0 * Float64(Float64(alpha + 2.0) / beta))) / beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 14500000000.0)
		tmp = (1.0 + beta) / ((beta + 3.0) * ((beta + 2.0) * (beta + 2.0)));
	else
		tmp = ((alpha + 1.0) / (alpha + (beta + 2.0))) * ((1.0 + (-2.0 * ((alpha + 2.0) / beta))) / beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 14500000000.0], N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(beta + 3.0), $MachinePrecision] * N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(-2.0 * N[(N[(alpha + 2.0), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.45e10

   1. Initial program 99.9%

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

      1. associate-/l/ 99.4%

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

      2. +-commutative 99.4%

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

      3. +-commutative 99.4%

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

      4. associate-+r+ 99.4%

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

      5. associate-+r+ 99.4%

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

      6. associate-+r+ 99.4%

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

      7. distribute-rgt1-in 99.4%

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

      8. +-commutative 99.4%

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

      9. *-commutative 99.4%

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

      10. distribute-rgt1-in 99.4%

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

      11. +-commutative 99.4%

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

      12. metadata-eval 99.4%

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

      13. associate-+l+ 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in alpha around 0 65.4%

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

      1. +-commutative 65.4%

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

      2. unpow2 65.4%

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

      3. +-commutative 65.4%

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

   7. Simplified 65.4%

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

   if 1.45e10 < beta

   1. Initial program 81.4%

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

      1. associate-/l/ 78.6%

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

      2. +-commutative 78.6%

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

      3. +-commutative 78.6%

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

      4. associate-+r+ 78.6%

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

      5. associate-+r+ 78.6%

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

      6. associate-+r+ 78.6%

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

      7. distribute-rgt1-in 78.6%

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

      8. +-commutative 78.6%

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

      9. *-commutative 78.6%

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

      10. distribute-rgt1-in 78.6%

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

      11. +-commutative 78.6%

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

      12. metadata-eval 78.6%

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

      13. associate-+l+ 78.6%

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

   3. Simplified 78.6%

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

      1. associate-+r+ 78.6%

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

      2. metadata-eval 78.6%

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

      3. associate-/l* 90.8%

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

      4. associate-+r+ 90.8%

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

      5. metadata-eval 90.8%

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

      6. metadata-eval 90.8%

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

      7. associate-+l+ 90.8%

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

      8. metadata-eval 90.8%

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

      9. times-frac 99.7%

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

      10. metadata-eval 99.7%

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

      11. associate-+r+ 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in beta around inf 82.8%

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

      1. mul-1-neg 82.8%

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

      2. *-commutative 82.8%

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

      3. distribute-neg-frac2 82.8%

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

      4. metadata-eval 82.8%

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

      5. distribute-rgt-in 82.8%

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

      6. neg-mul-1 82.8%

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

      7. times-frac 82.8%

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

      8. metadata-eval 82.8%

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

   9. Simplified 82.8%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 14500000000:\\ \;\;\;\;\frac{1 + \beta}{\left(\beta + 3\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + -2 \cdot \frac{\alpha + 2}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.6 \cdot 10^{+20}:\\ \;\;\;\;\frac{1 + \beta}{\left(\beta + 3\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + -2 \cdot \frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.6e+20)
   (/ (+ 1.0 beta) (* (+ beta 3.0) (* (+ beta 2.0) (+ beta 2.0))))
   (*
    (/ (+ alpha 1.0) (+ alpha (+ beta 2.0)))
    (/ (+ 1.0 (* -2.0 (/ alpha beta))) beta))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.6e+20) {
		tmp = (1.0 + beta) / ((beta + 3.0) * ((beta + 2.0) * (beta + 2.0)));
	} else {
		tmp = ((alpha + 1.0) / (alpha + (beta + 2.0))) * ((1.0 + (-2.0 * (alpha / beta))) / beta);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.6d+20) then
        tmp = (1.0d0 + beta) / ((beta + 3.0d0) * ((beta + 2.0d0) * (beta + 2.0d0)))
    else
        tmp = ((alpha + 1.0d0) / (alpha + (beta + 2.0d0))) * ((1.0d0 + ((-2.0d0) * (alpha / beta))) / beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.6e+20) {
		tmp = (1.0 + beta) / ((beta + 3.0) * ((beta + 2.0) * (beta + 2.0)));
	} else {
		tmp = ((alpha + 1.0) / (alpha + (beta + 2.0))) * ((1.0 + (-2.0 * (alpha / beta))) / beta);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.6e+20:
		tmp = (1.0 + beta) / ((beta + 3.0) * ((beta + 2.0) * (beta + 2.0)))
	else:
		tmp = ((alpha + 1.0) / (alpha + (beta + 2.0))) * ((1.0 + (-2.0 * (alpha / beta))) / beta)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.6e+20)
		tmp = Float64(Float64(1.0 + beta) / Float64(Float64(beta + 3.0) * Float64(Float64(beta + 2.0) * Float64(beta + 2.0))));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(alpha + Float64(beta + 2.0))) * Float64(Float64(1.0 + Float64(-2.0 * Float64(alpha / beta))) / beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.6e+20)
		tmp = (1.0 + beta) / ((beta + 3.0) * ((beta + 2.0) * (beta + 2.0)));
	else
		tmp = ((alpha + 1.0) / (alpha + (beta + 2.0))) * ((1.0 + (-2.0 * (alpha / beta))) / beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.6e+20], N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(beta + 3.0), $MachinePrecision] * N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(-2.0 * N[(alpha / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.6e20

   1. Initial program 99.9%

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

      1. associate-/l/ 99.4%

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

      2. +-commutative 99.4%

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

      3. +-commutative 99.4%

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

      4. associate-+r+ 99.4%

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

      5. associate-+r+ 99.4%

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

      6. associate-+r+ 99.4%

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

      7. distribute-rgt1-in 99.4%

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

      8. +-commutative 99.4%

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

      9. *-commutative 99.4%

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

      10. distribute-rgt1-in 99.4%

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

      11. +-commutative 99.4%

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

      12. metadata-eval 99.4%

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

      13. associate-+l+ 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in alpha around 0 65.5%

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

      1. +-commutative 65.5%

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

      2. unpow2 65.5%

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

      3. +-commutative 65.5%

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

   7. Simplified 65.5%

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

   if 2.6e20 < beta

   1. Initial program 80.1%

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

      1. associate-/l/ 77.1%

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

      2. +-commutative 77.1%

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

      3. +-commutative 77.1%

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

      4. associate-+r+ 77.1%

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

      5. associate-+r+ 77.1%

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

      6. associate-+r+ 77.1%

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

      7. distribute-rgt1-in 77.1%

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

      8. +-commutative 77.1%

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

      9. *-commutative 77.1%

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

      10. distribute-rgt1-in 77.1%

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

      11. +-commutative 77.1%

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

      12. metadata-eval 77.1%

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

      13. associate-+l+ 77.1%

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

   3. Simplified 77.1%

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

      1. associate-+r+ 77.1%

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

      2. metadata-eval 77.1%

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

      3. associate-/l* 90.2%

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

      4. associate-+r+ 90.2%

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

      5. metadata-eval 90.2%

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

      6. metadata-eval 90.2%

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

      7. associate-+l+ 90.2%

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

      8. metadata-eval 90.2%

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

      9. times-frac 99.7%

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

      10. metadata-eval 99.7%

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

      11. associate-+r+ 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in beta around inf 83.9%

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

      1. mul-1-neg 83.9%

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

      2. *-commutative 83.9%

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

      3. distribute-neg-frac2 83.9%

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

      4. metadata-eval 83.9%

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

      5. distribute-rgt-in 83.9%

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

      6. neg-mul-1 83.9%

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

      7. times-frac 83.9%

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

      8. metadata-eval 83.9%

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

   9. Simplified 83.9%

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

   10. Taylor expanded in alpha around inf 83.9%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.6 \cdot 10^{+20}:\\ \;\;\;\;\frac{1 + \beta}{\left(\beta + 3\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + -2 \cdot \frac{\alpha}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.7 \cdot 10^{+20}:\\ \;\;\;\;\frac{1 + \beta}{\left(\beta + 3\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta} + \frac{1}{\beta}}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.7e+20)
   (/ (+ 1.0 beta) (* (+ beta 3.0) (* (+ beta 2.0) (+ beta 2.0))))
   (/ (+ (/ alpha beta) (/ 1.0 beta)) (+ 1.0 (+ 2.0 (+ alpha beta))))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.7e+20) {
		tmp = (1.0 + beta) / ((beta + 3.0) * ((beta + 2.0) * (beta + 2.0)));
	} else {
		tmp = ((alpha / beta) + (1.0 / beta)) / (1.0 + (2.0 + (alpha + 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 <= 2.7d+20) then
        tmp = (1.0d0 + beta) / ((beta + 3.0d0) * ((beta + 2.0d0) * (beta + 2.0d0)))
    else
        tmp = ((alpha / beta) + (1.0d0 / beta)) / (1.0d0 + (2.0d0 + (alpha + beta)))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.7e+20) {
		tmp = (1.0 + beta) / ((beta + 3.0) * ((beta + 2.0) * (beta + 2.0)));
	} else {
		tmp = ((alpha / beta) + (1.0 / beta)) / (1.0 + (2.0 + (alpha + beta)));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.7e+20:
		tmp = (1.0 + beta) / ((beta + 3.0) * ((beta + 2.0) * (beta + 2.0)))
	else:
		tmp = ((alpha / beta) + (1.0 / beta)) / (1.0 + (2.0 + (alpha + beta)))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.7e+20)
		tmp = Float64(Float64(1.0 + beta) / Float64(Float64(beta + 3.0) * Float64(Float64(beta + 2.0) * Float64(beta + 2.0))));
	else
		tmp = Float64(Float64(Float64(alpha / beta) + Float64(1.0 / beta)) / Float64(1.0 + Float64(2.0 + Float64(alpha + beta))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.7e+20)
		tmp = (1.0 + beta) / ((beta + 3.0) * ((beta + 2.0) * (beta + 2.0)));
	else
		tmp = ((alpha / beta) + (1.0 / beta)) / (1.0 + (2.0 + (alpha + beta)));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.7e+20], N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(beta + 3.0), $MachinePrecision] * N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha / beta), $MachinePrecision] + N[(1.0 / beta), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.7e20

   1. Initial program 99.9%

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

      1. associate-/l/ 99.4%

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

      2. +-commutative 99.4%

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

      3. +-commutative 99.4%

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

      4. associate-+r+ 99.4%

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

      5. associate-+r+ 99.4%

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

      6. associate-+r+ 99.4%

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

      7. distribute-rgt1-in 99.4%

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

      8. +-commutative 99.4%

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

      9. *-commutative 99.4%

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

      10. distribute-rgt1-in 99.4%

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

      11. +-commutative 99.4%

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

      12. metadata-eval 99.4%

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

      13. associate-+l+ 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in alpha around 0 65.5%

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

      1. +-commutative 65.5%

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

      2. unpow2 65.5%

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

      3. +-commutative 65.5%

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

   7. Simplified 65.5%

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

   if 2.7e20 < beta

   1. Initial program 80.1%

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

   3. Taylor expanded in beta around inf 84.0%

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

   4. Taylor expanded in alpha around 0 84.0%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.7 \cdot 10^{+20}:\\ \;\;\;\;\frac{1 + \beta}{\left(\beta + 3\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta} + \frac{1}{\beta}}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.15 \cdot 10^{+20}:\\ \;\;\;\;\frac{1 + \beta}{\left(\beta + 3\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.15e+20)
   (/ (+ 1.0 beta) (* (+ beta 3.0) (* (+ beta 2.0) (+ beta 2.0))))
   (/ (/ (+ alpha 1.0) beta) (+ alpha (+ beta 3.0)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.15e+20) {
		tmp = (1.0 + beta) / ((beta + 3.0) * ((beta + 2.0) * (beta + 2.0)));
	} else {
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.15d+20) then
        tmp = (1.0d0 + beta) / ((beta + 3.0d0) * ((beta + 2.0d0) * (beta + 2.0d0)))
    else
        tmp = ((alpha + 1.0d0) / beta) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.15e+20) {
		tmp = (1.0 + beta) / ((beta + 3.0) * ((beta + 2.0) * (beta + 2.0)));
	} else {
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.15e+20:
		tmp = (1.0 + beta) / ((beta + 3.0) * ((beta + 2.0) * (beta + 2.0)))
	else:
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 3.0))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.15e+20)
		tmp = Float64(Float64(1.0 + beta) / Float64(Float64(beta + 3.0) * Float64(Float64(beta + 2.0) * Float64(beta + 2.0))));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.15e+20)
		tmp = (1.0 + beta) / ((beta + 3.0) * ((beta + 2.0) * (beta + 2.0)));
	else
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.15e+20], N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(beta + 3.0), $MachinePrecision] * N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.15e20

   1. Initial program 99.9%

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

      1. associate-/l/ 99.4%

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

      2. +-commutative 99.4%

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

      3. +-commutative 99.4%

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

      4. associate-+r+ 99.4%

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

      5. associate-+r+ 99.4%

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

      6. associate-+r+ 99.4%

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

      7. distribute-rgt1-in 99.4%

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

      8. +-commutative 99.4%

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

      9. *-commutative 99.4%

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

      10. distribute-rgt1-in 99.4%

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

      11. +-commutative 99.4%

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

      12. metadata-eval 99.4%

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

      13. associate-+l+ 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in alpha around 0 65.5%

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

      1. +-commutative 65.5%

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

      2. unpow2 65.5%

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

      3. +-commutative 65.5%

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

   7. Simplified 65.5%

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

   if 2.15e20 < beta

   1. Initial program 80.1%

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

   3. Taylor expanded in beta around inf 84.0%

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

      1. +-commutative 84.0%

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

      2. metadata-eval 84.0%

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

      3. associate-+l+ 84.0%

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

      4. metadata-eval 84.0%

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

      5. associate-+r+ 84.0%

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

   5. Applied egg-rr 84.0%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.15 \cdot 10^{+20}:\\ \;\;\;\;\frac{1 + \beta}{\left(\beta + 3\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 89.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 14.5:\\ \;\;\;\;\frac{\alpha + 1}{\left(\left(\alpha + 2\right) \cdot \left(\alpha + 2\right)\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 14.5)
   (/ (+ alpha 1.0) (* (* (+ alpha 2.0) (+ alpha 2.0)) (+ alpha 3.0)))
   (/ (/ (+ alpha 1.0) beta) (+ alpha (+ beta 3.0)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 14.5) {
		tmp = (alpha + 1.0) / (((alpha + 2.0) * (alpha + 2.0)) * (alpha + 3.0));
	} else {
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 14.5d0) then
        tmp = (alpha + 1.0d0) / (((alpha + 2.0d0) * (alpha + 2.0d0)) * (alpha + 3.0d0))
    else
        tmp = ((alpha + 1.0d0) / beta) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 14.5) {
		tmp = (alpha + 1.0) / (((alpha + 2.0) * (alpha + 2.0)) * (alpha + 3.0));
	} else {
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 14.5:
		tmp = (alpha + 1.0) / (((alpha + 2.0) * (alpha + 2.0)) * (alpha + 3.0))
	else:
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 3.0))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 14.5)
		tmp = Float64(Float64(alpha + 1.0) / Float64(Float64(Float64(alpha + 2.0) * Float64(alpha + 2.0)) * Float64(alpha + 3.0)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 14.5)
		tmp = (alpha + 1.0) / (((alpha + 2.0) * (alpha + 2.0)) * (alpha + 3.0));
	else
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 14.5], N[(N[(alpha + 1.0), $MachinePrecision] / N[(N[(N[(alpha + 2.0), $MachinePrecision] * N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision] * N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 14.5

   1. Initial program 99.9%

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

      1. associate-/l/ 99.4%

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

      2. +-commutative 99.4%

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

      3. +-commutative 99.4%

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

      4. associate-+r+ 99.4%

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

      5. associate-+r+ 99.4%

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

      6. associate-+r+ 99.4%

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

      7. distribute-rgt1-in 99.5%

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

      8. +-commutative 99.5%

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

      9. *-commutative 99.5%

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

      10. distribute-rgt1-in 99.5%

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

      11. +-commutative 99.5%

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

      12. metadata-eval 99.5%

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

      13. associate-+l+ 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in beta around 0 93.0%

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

      1. unpow2 93.0%

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

      2. +-commutative 93.0%

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

   7. Simplified 93.0%

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

   if 14.5 < beta

   1. Initial program 81.8%

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

   3. Taylor expanded in beta around inf 81.1%

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

      1. +-commutative 81.1%

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

      2. metadata-eval 81.1%

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

      3. associate-+l+ 81.1%

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

      4. metadata-eval 81.1%

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

      5. associate-+r+ 81.1%

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

   5. Applied egg-rr 81.1%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 14.5:\\ \;\;\;\;\frac{\alpha + 1}{\left(\left(\alpha + 2\right) \cdot \left(\alpha + 2\right)\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.85:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{elif}\;\beta \leq 4 \cdot 10^{+154}:\\ \;\;\;\;\frac{\alpha + 1}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.85)
   (+ 0.08333333333333333 (* beta -0.027777777777777776))
   (if (<= beta 4e+154)
     (/ (+ alpha 1.0) (* beta beta))
     (/ (/ alpha beta) beta))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.85) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else if (beta <= 4e+154) {
		tmp = (alpha + 1.0) / (beta * beta);
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.85d0) then
        tmp = 0.08333333333333333d0 + (beta * (-0.027777777777777776d0))
    else if (beta <= 4d+154) then
        tmp = (alpha + 1.0d0) / (beta * beta)
    else
        tmp = (alpha / beta) / beta
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.85) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else if (beta <= 4e+154) {
		tmp = (alpha + 1.0) / (beta * beta);
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.85:
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776)
	elif beta <= 4e+154:
		tmp = (alpha + 1.0) / (beta * beta)
	else:
		tmp = (alpha / beta) / beta
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.85)
		tmp = Float64(0.08333333333333333 + Float64(beta * -0.027777777777777776));
	elseif (beta <= 4e+154)
		tmp = Float64(Float64(alpha + 1.0) / Float64(beta * beta));
	else
		tmp = Float64(Float64(alpha / beta) / beta);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.85)
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	elseif (beta <= 4e+154)
		tmp = (alpha + 1.0) / (beta * beta);
	else
		tmp = (alpha / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.85], N[(0.08333333333333333 + N[(beta * -0.027777777777777776), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 4e+154], N[(N[(alpha + 1.0), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / beta), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if beta < 2.85000000000000009

   1. Initial program 99.9%

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

      1. associate-/l/ 99.4%

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

      2. +-commutative 99.4%

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

      3. +-commutative 99.4%

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

      4. associate-+r+ 99.4%

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

      5. associate-+r+ 99.4%

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

      6. associate-+r+ 99.4%

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

      7. distribute-rgt1-in 99.5%

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

      8. +-commutative 99.5%

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

      9. *-commutative 99.5%

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

      10. distribute-rgt1-in 99.5%

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

      11. +-commutative 99.5%

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

      12. metadata-eval 99.5%

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

      13. associate-+l+ 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in alpha around 0 65.6%

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

      1. +-commutative 65.6%

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

      2. unpow2 65.6%

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

      3. +-commutative 65.6%

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

   7. Simplified 65.6%

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

   8. Taylor expanded in beta around 0 64.9%

      \[\leadsto \color{blue}{0.08333333333333333 + -0.027777777777777776 \cdot \beta} \]
   9. Step-by-step derivation

      1. *-commutative 64.9%

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

   10. Simplified 64.9%

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

   if 2.85000000000000009 < beta < 4.00000000000000015e154

   1. Initial program 90.1%

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

      1. associate-/l/ 88.1%

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

      2. +-commutative 88.1%

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

      3. +-commutative 88.1%

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

      4. associate-+r+ 88.1%

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

      5. associate-+r+ 88.1%

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

      6. associate-+r+ 88.1%

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

      7. distribute-rgt1-in 88.1%

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

      8. +-commutative 88.1%

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

      9. *-commutative 88.1%

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

      10. distribute-rgt1-in 88.1%

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

      11. +-commutative 88.1%

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

      12. metadata-eval 88.1%

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

      13. associate-+l+ 88.1%

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

   3. Simplified 88.1%

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

   5. Taylor expanded in beta around inf 69.9%

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

      1. unpow2 69.9%

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

   7. Simplified 69.9%

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

   if 4.00000000000000015e154 < beta

   1. Initial program 71.2%

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

      1. associate-/l/ 67.6%

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

      2. +-commutative 67.6%

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

      3. +-commutative 67.6%

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

      4. associate-+r+ 67.6%

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

      5. associate-+r+ 67.6%

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

      6. associate-+r+ 67.6%

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

      7. distribute-rgt1-in 67.6%

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

      8. +-commutative 67.6%

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

      9. *-commutative 67.6%

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

      10. distribute-rgt1-in 67.6%

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

      11. +-commutative 67.6%

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

      12. metadata-eval 67.6%

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

      13. associate-+l+ 67.6%

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

   3. Simplified 67.6%

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

   5. Taylor expanded in beta around inf 82.5%

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

      1. unpow2 82.5%

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

   7. Simplified 82.5%

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

      1. associate-/r* 94.7%

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

      2. +-commutative 94.7%

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

   9. Applied egg-rr 94.7%

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

   10. Taylor expanded in alpha around inf 93.3%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.85:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{elif}\;\beta \leq 4 \cdot 10^{+154}:\\ \;\;\;\;\frac{\alpha + 1}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 69.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.5:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{elif}\;\beta \leq 4.2 \cdot 10^{+151}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.5)
   (+ 0.08333333333333333 (* beta -0.027777777777777776))
   (if (<= beta 4.2e+151)
     (/ 1.0 (* beta (+ beta 3.0)))
     (/ (/ alpha beta) beta))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.5) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else if (beta <= 4.2e+151) {
		tmp = 1.0 / (beta * (beta + 3.0));
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.5d0) then
        tmp = 0.08333333333333333d0 + (beta * (-0.027777777777777776d0))
    else if (beta <= 4.2d+151) then
        tmp = 1.0d0 / (beta * (beta + 3.0d0))
    else
        tmp = (alpha / beta) / beta
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.5) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else if (beta <= 4.2e+151) {
		tmp = 1.0 / (beta * (beta + 3.0));
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.5:
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776)
	elif beta <= 4.2e+151:
		tmp = 1.0 / (beta * (beta + 3.0))
	else:
		tmp = (alpha / beta) / beta
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.5)
		tmp = Float64(0.08333333333333333 + Float64(beta * -0.027777777777777776));
	elseif (beta <= 4.2e+151)
		tmp = Float64(1.0 / Float64(beta * Float64(beta + 3.0)));
	else
		tmp = Float64(Float64(alpha / beta) / beta);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.5)
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	elseif (beta <= 4.2e+151)
		tmp = 1.0 / (beta * (beta + 3.0));
	else
		tmp = (alpha / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.5], N[(0.08333333333333333 + N[(beta * -0.027777777777777776), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 4.2e+151], N[(1.0 / N[(beta * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / beta), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if beta < 2.5

   1. Initial program 99.9%

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

      1. associate-/l/ 99.4%

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

      2. +-commutative 99.4%

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

      3. +-commutative 99.4%

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

      4. associate-+r+ 99.4%

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

      5. associate-+r+ 99.4%

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

      6. associate-+r+ 99.4%

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

      7. distribute-rgt1-in 99.5%

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

      8. +-commutative 99.5%

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

      9. *-commutative 99.5%

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

      10. distribute-rgt1-in 99.5%

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

      11. +-commutative 99.5%

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

      12. metadata-eval 99.5%

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

      13. associate-+l+ 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in alpha around 0 65.6%

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

      1. +-commutative 65.6%

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

      2. unpow2 65.6%

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

      3. +-commutative 65.6%

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

   7. Simplified 65.6%

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

   8. Taylor expanded in beta around 0 64.9%

      \[\leadsto \color{blue}{0.08333333333333333 + -0.027777777777777776 \cdot \beta} \]
   9. Step-by-step derivation

      1. *-commutative 64.9%

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

   10. Simplified 64.9%

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

   if 2.5 < beta < 4.2000000000000001e151

   1. Initial program 89.9%

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

   3. Taylor expanded in beta around inf 69.8%

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

   4. Taylor expanded in alpha around 0 57.6%

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

   if 4.2000000000000001e151 < beta

   1. Initial program 71.9%

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

      1. associate-/l/ 68.3%

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

      2. +-commutative 68.3%

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

      3. +-commutative 68.3%

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

      4. associate-+r+ 68.3%

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

      5. associate-+r+ 68.3%

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

      6. associate-+r+ 68.3%

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

      7. distribute-rgt1-in 68.3%

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

      8. +-commutative 68.3%

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

      9. *-commutative 68.3%

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

      10. distribute-rgt1-in 68.3%

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

      11. +-commutative 68.3%

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

      12. metadata-eval 68.3%

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

      13. associate-+l+ 68.3%

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

   3. Simplified 68.3%

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

   5. Taylor expanded in beta around inf 82.9%

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

      1. unpow2 82.9%

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

   7. Simplified 82.9%

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

      1. associate-/r* 94.8%

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

      2. +-commutative 94.8%

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

   9. Applied egg-rr 94.8%

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

   10. Taylor expanded in alpha around inf 93.5%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.5:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{elif}\;\beta \leq 4.2 \cdot 10^{+151}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 71.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.5:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.5)
   (+ 0.08333333333333333 (* beta -0.027777777777777776))
   (/ (/ (+ alpha 1.0) beta) (+ alpha (+ beta 3.0)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.5) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.5d0) then
        tmp = 0.08333333333333333d0 + (beta * (-0.027777777777777776d0))
    else
        tmp = ((alpha + 1.0d0) / beta) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.5) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.5:
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776)
	else:
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 3.0))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.5)
		tmp = Float64(0.08333333333333333 + Float64(beta * -0.027777777777777776));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.5)
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	else
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.5], N[(0.08333333333333333 + N[(beta * -0.027777777777777776), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.5

   1. Initial program 99.9%

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

      1. associate-/l/ 99.4%

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

      2. +-commutative 99.4%

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

      3. +-commutative 99.4%

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

      4. associate-+r+ 99.4%

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

      5. associate-+r+ 99.4%

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

      6. associate-+r+ 99.4%

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

      7. distribute-rgt1-in 99.5%

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

      8. +-commutative 99.5%

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

      9. *-commutative 99.5%

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

      10. distribute-rgt1-in 99.5%

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

      11. +-commutative 99.5%

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

      12. metadata-eval 99.5%

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

      13. associate-+l+ 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in alpha around 0 65.6%

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

      1. +-commutative 65.6%

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

      2. unpow2 65.6%

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

      3. +-commutative 65.6%

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

   7. Simplified 65.6%

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

   8. Taylor expanded in beta around 0 64.9%

      \[\leadsto \color{blue}{0.08333333333333333 + -0.027777777777777776 \cdot \beta} \]
   9. Step-by-step derivation

      1. *-commutative 64.9%

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

   10. Simplified 64.9%

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

   if 2.5 < beta

   1. Initial program 81.8%

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

   3. Taylor expanded in beta around inf 81.1%

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

      1. +-commutative 81.1%

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

      2. metadata-eval 81.1%

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

      3. associate-+l+ 81.1%

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

      4. metadata-eval 81.1%

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

      5. associate-+r+ 81.1%

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

   5. Applied egg-rr 81.1%

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

Alternative 12: 69.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.8:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{elif}\;\beta \leq 4.2 \cdot 10^{+151}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.8)
   (+ 0.08333333333333333 (* beta -0.027777777777777776))
   (if (<= beta 4.2e+151) (/ (/ 1.0 beta) beta) (/ (/ alpha beta) beta))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.8) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else if (beta <= 4.2e+151) {
		tmp = (1.0 / beta) / beta;
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.8d0) then
        tmp = 0.08333333333333333d0 + (beta * (-0.027777777777777776d0))
    else if (beta <= 4.2d+151) then
        tmp = (1.0d0 / beta) / beta
    else
        tmp = (alpha / beta) / beta
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.8) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else if (beta <= 4.2e+151) {
		tmp = (1.0 / beta) / beta;
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.8:
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776)
	elif beta <= 4.2e+151:
		tmp = (1.0 / beta) / beta
	else:
		tmp = (alpha / beta) / beta
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.8)
		tmp = Float64(0.08333333333333333 + Float64(beta * -0.027777777777777776));
	elseif (beta <= 4.2e+151)
		tmp = Float64(Float64(1.0 / beta) / beta);
	else
		tmp = Float64(Float64(alpha / beta) / beta);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.8)
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	elseif (beta <= 4.2e+151)
		tmp = (1.0 / beta) / beta;
	else
		tmp = (alpha / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.8], N[(0.08333333333333333 + N[(beta * -0.027777777777777776), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 4.2e+151], N[(N[(1.0 / beta), $MachinePrecision] / beta), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / beta), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if beta < 2.7999999999999998

   1. Initial program 99.9%

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

      1. associate-/l/ 99.4%

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

      2. +-commutative 99.4%

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

      3. +-commutative 99.4%

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

      4. associate-+r+ 99.4%

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

      5. associate-+r+ 99.4%

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

      6. associate-+r+ 99.4%

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

      7. distribute-rgt1-in 99.5%

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

      8. +-commutative 99.5%

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

      9. *-commutative 99.5%

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

      10. distribute-rgt1-in 99.5%

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

      11. +-commutative 99.5%

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

      12. metadata-eval 99.5%

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

      13. associate-+l+ 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in alpha around 0 65.6%

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

      1. +-commutative 65.6%

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

      2. unpow2 65.6%

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

      3. +-commutative 65.6%

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

   7. Simplified 65.6%

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

   8. Taylor expanded in beta around 0 64.9%

      \[\leadsto \color{blue}{0.08333333333333333 + -0.027777777777777776 \cdot \beta} \]
   9. Step-by-step derivation

      1. *-commutative 64.9%

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

   10. Simplified 64.9%

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

   if 2.7999999999999998 < beta < 4.2000000000000001e151

   1. Initial program 89.9%

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

      1. associate-/l/ 87.9%

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

      2. +-commutative 87.9%

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

      3. +-commutative 87.9%

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

      4. associate-+r+ 87.9%

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

      5. associate-+r+ 87.9%

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

      6. associate-+r+ 87.9%

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

      7. distribute-rgt1-in 87.9%

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

      8. +-commutative 87.9%

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

      9. *-commutative 87.9%

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

      10. distribute-rgt1-in 87.9%

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

      11. +-commutative 87.9%

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

      12. metadata-eval 87.9%

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

      13. associate-+l+ 87.9%

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

   3. Simplified 87.9%

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

   5. Taylor expanded in beta around inf 69.3%

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

      1. unpow2 69.3%

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

   7. Simplified 69.3%

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

      1. associate-/r* 69.4%

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

      2. +-commutative 69.4%

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

   9. Applied egg-rr 69.4%

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

   10. Taylor expanded in alpha around 0 57.6%

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

   if 4.2000000000000001e151 < beta

   1. Initial program 71.9%

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

      1. associate-/l/ 68.3%

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

      2. +-commutative 68.3%

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

      3. +-commutative 68.3%

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

      4. associate-+r+ 68.3%

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

      5. associate-+r+ 68.3%

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

      6. associate-+r+ 68.3%

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

      7. distribute-rgt1-in 68.3%

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

      8. +-commutative 68.3%

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

      9. *-commutative 68.3%

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

      10. distribute-rgt1-in 68.3%

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

      11. +-commutative 68.3%

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

      12. metadata-eval 68.3%

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

      13. associate-+l+ 68.3%

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

   3. Simplified 68.3%

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

   5. Taylor expanded in beta around inf 82.9%

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

      1. unpow2 82.9%

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

   7. Simplified 82.9%

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

      1. associate-/r* 94.8%

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

      2. +-commutative 94.8%

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

   9. Applied egg-rr 94.8%

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

   10. Taylor expanded in alpha around inf 93.5%

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

Alternative 13: 71.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.85:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta} + \frac{1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.85)
   (+ 0.08333333333333333 (* beta -0.027777777777777776))
   (/ (+ (/ alpha beta) (/ 1.0 beta)) beta)))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.85) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = ((alpha / beta) + (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 <= 2.85d0) then
        tmp = 0.08333333333333333d0 + (beta * (-0.027777777777777776d0))
    else
        tmp = ((alpha / beta) + (1.0d0 / beta)) / beta
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.85) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = ((alpha / beta) + (1.0 / beta)) / beta;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.85:
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776)
	else:
		tmp = ((alpha / beta) + (1.0 / beta)) / beta
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.85)
		tmp = Float64(0.08333333333333333 + Float64(beta * -0.027777777777777776));
	else
		tmp = Float64(Float64(Float64(alpha / beta) + Float64(1.0 / beta)) / beta);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.85)
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	else
		tmp = ((alpha / beta) + (1.0 / beta)) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.85], N[(0.08333333333333333 + N[(beta * -0.027777777777777776), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha / beta), $MachinePrecision] + N[(1.0 / beta), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.85000000000000009

   1. Initial program 99.9%

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

      1. associate-/l/ 99.4%

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

      2. +-commutative 99.4%

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

      3. +-commutative 99.4%

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

      4. associate-+r+ 99.4%

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

      5. associate-+r+ 99.4%

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

      6. associate-+r+ 99.4%

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

      7. distribute-rgt1-in 99.5%

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

      8. +-commutative 99.5%

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

      9. *-commutative 99.5%

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

      10. distribute-rgt1-in 99.5%

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

      11. +-commutative 99.5%

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

      12. metadata-eval 99.5%

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

      13. associate-+l+ 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in alpha around 0 65.6%

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

      1. +-commutative 65.6%

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

      2. unpow2 65.6%

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

      3. +-commutative 65.6%

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

   7. Simplified 65.6%

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

   8. Taylor expanded in beta around 0 64.9%

      \[\leadsto \color{blue}{0.08333333333333333 + -0.027777777777777776 \cdot \beta} \]
   9. Step-by-step derivation

      1. *-commutative 64.9%

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

   10. Simplified 64.9%

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

   if 2.85000000000000009 < beta

   1. Initial program 81.8%

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

      1. associate-/l/ 79.1%

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

      2. +-commutative 79.1%

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

      3. +-commutative 79.1%

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

      4. associate-+r+ 79.1%

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

      5. associate-+r+ 79.1%

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

      6. associate-+r+ 79.1%

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

      7. distribute-rgt1-in 79.1%

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

      8. +-commutative 79.1%

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

      9. *-commutative 79.1%

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

      10. distribute-rgt1-in 79.1%

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

      11. +-commutative 79.1%

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

      12. metadata-eval 79.1%

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

      13. associate-+l+ 79.1%

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

   3. Simplified 79.1%

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

   5. Taylor expanded in beta around inf 75.5%

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

      1. unpow2 75.5%

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

   7. Simplified 75.5%

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

      1. associate-/r* 80.9%

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

      2. +-commutative 80.9%

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

   9. Applied egg-rr 80.9%

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

   10. Taylor expanded in alpha around 0 80.9%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.85:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta} + \frac{1}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 71.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.8:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.8)
   (+ 0.08333333333333333 (* beta -0.027777777777777776))
   (/ (/ (+ alpha 1.0) beta) beta)))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.8) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = ((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 <= 2.8d0) then
        tmp = 0.08333333333333333d0 + (beta * (-0.027777777777777776d0))
    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 <= 2.8) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.8:
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776)
	else:
		tmp = ((alpha + 1.0) / beta) / beta
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.8)
		tmp = Float64(0.08333333333333333 + Float64(beta * -0.027777777777777776));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / beta);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.8)
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	else
		tmp = ((alpha + 1.0) / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.8], N[(0.08333333333333333 + N[(beta * -0.027777777777777776), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.7999999999999998

   1. Initial program 99.9%

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

      1. associate-/l/ 99.4%

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

      2. +-commutative 99.4%

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

      3. +-commutative 99.4%

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

      4. associate-+r+ 99.4%

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

      5. associate-+r+ 99.4%

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

      6. associate-+r+ 99.4%

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

      7. distribute-rgt1-in 99.5%

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

      8. +-commutative 99.5%

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

      9. *-commutative 99.5%

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

      10. distribute-rgt1-in 99.5%

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

      11. +-commutative 99.5%

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

      12. metadata-eval 99.5%

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

      13. associate-+l+ 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in alpha around 0 65.6%

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

      1. +-commutative 65.6%

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

      2. unpow2 65.6%

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

      3. +-commutative 65.6%

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

   7. Simplified 65.6%

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

   8. Taylor expanded in beta around 0 64.9%

      \[\leadsto \color{blue}{0.08333333333333333 + -0.027777777777777776 \cdot \beta} \]
   9. Step-by-step derivation

      1. *-commutative 64.9%

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

   10. Simplified 64.9%

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

   if 2.7999999999999998 < beta

   1. Initial program 81.8%

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

      1. associate-/l/ 79.1%

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

      2. +-commutative 79.1%

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

      3. +-commutative 79.1%

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

      4. associate-+r+ 79.1%

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

      5. associate-+r+ 79.1%

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

      6. associate-+r+ 79.1%

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

      7. distribute-rgt1-in 79.1%

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

      8. +-commutative 79.1%

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

      9. *-commutative 79.1%

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

      10. distribute-rgt1-in 79.1%

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

      11. +-commutative 79.1%

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

      12. metadata-eval 79.1%

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

      13. associate-+l+ 79.1%

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

   3. Simplified 79.1%

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

   5. Taylor expanded in beta around inf 75.5%

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

      1. unpow2 75.5%

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

   7. Simplified 75.5%

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

      1. associate-/r* 80.9%

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

      2. +-commutative 80.9%

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

   9. Applied egg-rr 80.9%

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

Alternative 15: 69.3% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.8:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.8)
   (+ 0.08333333333333333 (* beta -0.027777777777777776))
   (/ (/ 1.0 beta) beta)))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.8) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = (1.0 / beta) / beta;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.8d0) then
        tmp = 0.08333333333333333d0 + (beta * (-0.027777777777777776d0))
    else
        tmp = (1.0d0 / beta) / beta
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.8) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = (1.0 / beta) / beta;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.8:
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776)
	else:
		tmp = (1.0 / beta) / beta
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.8)
		tmp = Float64(0.08333333333333333 + Float64(beta * -0.027777777777777776));
	else
		tmp = Float64(Float64(1.0 / beta) / beta);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.8)
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	else
		tmp = (1.0 / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.8], N[(0.08333333333333333 + N[(beta * -0.027777777777777776), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.7999999999999998

   1. Initial program 99.9%

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

      1. associate-/l/ 99.4%

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

      2. +-commutative 99.4%

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

      3. +-commutative 99.4%

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

      4. associate-+r+ 99.4%

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

      5. associate-+r+ 99.4%

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

      6. associate-+r+ 99.4%

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

      7. distribute-rgt1-in 99.5%

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

      8. +-commutative 99.5%

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

      9. *-commutative 99.5%

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

      10. distribute-rgt1-in 99.5%

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

      11. +-commutative 99.5%

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

      12. metadata-eval 99.5%

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

      13. associate-+l+ 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in alpha around 0 65.6%

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

      1. +-commutative 65.6%

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

      2. unpow2 65.6%

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

      3. +-commutative 65.6%

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

   7. Simplified 65.6%

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

   8. Taylor expanded in beta around 0 64.9%

      \[\leadsto \color{blue}{0.08333333333333333 + -0.027777777777777776 \cdot \beta} \]
   9. Step-by-step derivation

      1. *-commutative 64.9%

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

   10. Simplified 64.9%

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

   if 2.7999999999999998 < beta

   1. Initial program 81.8%

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

      1. associate-/l/ 79.1%

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

      2. +-commutative 79.1%

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

      3. +-commutative 79.1%

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

      4. associate-+r+ 79.1%

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

      5. associate-+r+ 79.1%

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

      6. associate-+r+ 79.1%

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

      7. distribute-rgt1-in 79.1%

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

      8. +-commutative 79.1%

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

      9. *-commutative 79.1%

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

      10. distribute-rgt1-in 79.1%

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

      11. +-commutative 79.1%

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

      12. metadata-eval 79.1%

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

      13. associate-+l+ 79.1%

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

   3. Simplified 79.1%

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

   5. Taylor expanded in beta around inf 75.5%

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

      1. unpow2 75.5%

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

   7. Simplified 75.5%

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

      1. associate-/r* 80.9%

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

      2. +-commutative 80.9%

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

   9. Applied egg-rr 80.9%

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

   10. Taylor expanded in alpha around 0 68.2%

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

Alternative 16: 69.1% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.8:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.8)
   (+ 0.08333333333333333 (* beta -0.027777777777777776))
   (/ 1.0 (* beta beta))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.8) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = 1.0 / (beta * beta);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.8d0) then
        tmp = 0.08333333333333333d0 + (beta * (-0.027777777777777776d0))
    else
        tmp = 1.0d0 / (beta * beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.8) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = 1.0 / (beta * beta);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.8:
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776)
	else:
		tmp = 1.0 / (beta * beta)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.8)
		tmp = Float64(0.08333333333333333 + Float64(beta * -0.027777777777777776));
	else
		tmp = Float64(1.0 / Float64(beta * beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.8)
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	else
		tmp = 1.0 / (beta * beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.8], N[(0.08333333333333333 + N[(beta * -0.027777777777777776), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.7999999999999998

   1. Initial program 99.9%

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

      1. associate-/l/ 99.4%

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

      2. +-commutative 99.4%

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

      3. +-commutative 99.4%

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

      4. associate-+r+ 99.4%

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

      5. associate-+r+ 99.4%

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

      6. associate-+r+ 99.4%

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

      7. distribute-rgt1-in 99.5%

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

      8. +-commutative 99.5%

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

      9. *-commutative 99.5%

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

      10. distribute-rgt1-in 99.5%

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

      11. +-commutative 99.5%

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

      12. metadata-eval 99.5%

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

      13. associate-+l+ 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in alpha around 0 65.6%

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

      1. +-commutative 65.6%

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

      2. unpow2 65.6%

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

      3. +-commutative 65.6%

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

   7. Simplified 65.6%

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

   8. Taylor expanded in beta around 0 64.9%

      \[\leadsto \color{blue}{0.08333333333333333 + -0.027777777777777776 \cdot \beta} \]
   9. Step-by-step derivation

      1. *-commutative 64.9%

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

   10. Simplified 64.9%

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

   if 2.7999999999999998 < beta

   1. Initial program 81.8%

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

      1. associate-/l/ 79.1%

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

      2. +-commutative 79.1%

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

      3. +-commutative 79.1%

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

      4. associate-+r+ 79.1%

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

      5. associate-+r+ 79.1%

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

      6. associate-+r+ 79.1%

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

      7. distribute-rgt1-in 79.1%

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

      8. +-commutative 79.1%

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

      9. *-commutative 79.1%

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

      10. distribute-rgt1-in 79.1%

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

      11. +-commutative 79.1%

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

      12. metadata-eval 79.1%

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

      13. associate-+l+ 79.1%

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

   3. Simplified 79.1%

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

   5. Taylor expanded in beta around inf 75.5%

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

      1. unpow2 75.5%

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

   7. Simplified 75.5%

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

   8. Taylor expanded in alpha around 0 68.0%

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

      1. unpow2 68.0%

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

   10. Simplified 68.0%

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

Alternative 17: 46.2% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.9:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.9)
   (+ 0.08333333333333333 (* beta -0.027777777777777776))
   (/ 1.0 beta)))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.9) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = 1.0 / beta;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.9d0) then
        tmp = 0.08333333333333333d0 + (beta * (-0.027777777777777776d0))
    else
        tmp = 1.0d0 / beta
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.9) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = 1.0 / beta;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.9:
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776)
	else:
		tmp = 1.0 / beta
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.9)
		tmp = Float64(0.08333333333333333 + Float64(beta * -0.027777777777777776));
	else
		tmp = Float64(1.0 / beta);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.9)
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	else
		tmp = 1.0 / beta;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.9], N[(0.08333333333333333 + N[(beta * -0.027777777777777776), $MachinePrecision]), $MachinePrecision], N[(1.0 / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.89999999999999991

   1. Initial program 99.9%

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

      1. associate-/l/ 99.4%

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

      2. +-commutative 99.4%

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

      3. +-commutative 99.4%

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

      4. associate-+r+ 99.4%

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

      5. associate-+r+ 99.4%

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

      6. associate-+r+ 99.4%

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

      7. distribute-rgt1-in 99.5%

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

      8. +-commutative 99.5%

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

      9. *-commutative 99.5%

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

      10. distribute-rgt1-in 99.5%

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

      11. +-commutative 99.5%

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

      12. metadata-eval 99.5%

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

      13. associate-+l+ 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in alpha around 0 65.6%

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

      1. +-commutative 65.6%

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

      2. unpow2 65.6%

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

      3. +-commutative 65.6%

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

   7. Simplified 65.6%

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

   8. Taylor expanded in beta around 0 64.9%

      \[\leadsto \color{blue}{0.08333333333333333 + -0.027777777777777776 \cdot \beta} \]
   9. Step-by-step derivation

      1. *-commutative 64.9%

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

   10. Simplified 64.9%

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

   if 2.89999999999999991 < beta

   1. Initial program 81.8%

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

   3. Taylor expanded in beta around inf 81.1%

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

   4. Taylor expanded in alpha around inf 7.0%

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

Alternative 18: 45.7% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 12:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 12.0) 0.08333333333333333 (/ 1.0 beta)))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 12.0) {
		tmp = 0.08333333333333333;
	} else {
		tmp = 1.0 / beta;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 12.0d0) then
        tmp = 0.08333333333333333d0
    else
        tmp = 1.0d0 / beta
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 12.0) {
		tmp = 0.08333333333333333;
	} else {
		tmp = 1.0 / beta;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 12.0:
		tmp = 0.08333333333333333
	else:
		tmp = 1.0 / beta
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 12.0)
		tmp = 0.08333333333333333;
	else
		tmp = Float64(1.0 / beta);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 12.0)
		tmp = 0.08333333333333333;
	else
		tmp = 1.0 / beta;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 12.0], 0.08333333333333333, N[(1.0 / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 12

   1. Initial program 99.9%

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

      1. associate-/l/ 99.4%

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

      2. +-commutative 99.4%

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

      3. +-commutative 99.4%

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

      4. associate-+r+ 99.4%

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

      5. associate-+r+ 99.4%

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

      6. associate-+r+ 99.4%

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

      7. distribute-rgt1-in 99.5%

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

      8. +-commutative 99.5%

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

      9. *-commutative 99.5%

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

      10. distribute-rgt1-in 99.5%

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

      11. +-commutative 99.5%

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

      12. metadata-eval 99.5%

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

      13. associate-+l+ 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in alpha around 0 65.6%

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

      1. +-commutative 65.6%

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

      2. unpow2 65.6%

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

      3. +-commutative 65.6%

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

   7. Simplified 65.6%

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

   8. Taylor expanded in beta around 0 64.7%

      \[\leadsto \color{blue}{0.08333333333333333} \]

   if 12 < beta

   1. Initial program 81.8%

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

   3. Taylor expanded in beta around inf 81.1%

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

   4. Taylor expanded in alpha around inf 7.0%

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

Alternative 19: 44.8% accurate, 35.0× speedup

Math

\[\begin{array}{l} \\ 0.08333333333333333 \end{array} \]

FPCore

(FPCore (alpha beta) :precision binary64 0.08333333333333333)

C

double code(double alpha, double beta) {
	return 0.08333333333333333;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = 0.08333333333333333d0
end function

Java

public static double code(double alpha, double beta) {
	return 0.08333333333333333;
}

Python

def code(alpha, beta):
	return 0.08333333333333333

Julia

function code(alpha, beta)
	return 0.08333333333333333
end

MATLAB

function tmp = code(alpha, beta)
	tmp = 0.08333333333333333;
end

Wolfram

code[alpha_, beta_] := 0.08333333333333333

Derivation

1. Initial program 93.3%

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

   1. associate-/l/ 92.0%

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

   2. +-commutative 92.0%

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

   3. +-commutative 92.0%

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

   4. associate-+r+ 92.0%

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

   5. associate-+r+ 92.0%

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

   6. associate-+r+ 92.0%

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

   7. distribute-rgt1-in 92.0%

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

   8. +-commutative 92.0%

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

   9. *-commutative 92.0%

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

   10. distribute-rgt1-in 92.0%

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

   11. +-commutative 92.0%

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

   12. metadata-eval 92.0%

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

   13. associate-+l+ 92.0%

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

3. Simplified 92.0%

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

5. Taylor expanded in alpha around 0 63.0%

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

   1. +-commutative 63.0%

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

   2. unpow2 63.0%

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

   3. +-commutative 63.0%

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

7. Simplified 63.0%

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

8. Taylor expanded in beta around 0 42.7%

   \[\leadsto \color{blue}{0.08333333333333333} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024100 
(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)))