Octave 3.8, jcobi/3

Percentage Accurate: 94.1% → 99.8%

Time: 13.6s

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: 94.1% 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

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 95.9%

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

3. Simplified 90.3%

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

   1. times-frac 97.0%

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

   2. +-commutative 97.0%

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

6. Applied egg-rr 97.0%

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

   1. +-commutative 97.0%

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

   2. +-commutative 97.0%

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

   3. +-commutative 97.0%

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

   4. +-commutative 97.0%

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

   5. associate-/r* 99.7%

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

   6. +-commutative 99.7%

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

   7. +-commutative 99.7%

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

   8. +-commutative 99.7%

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

   9. associate-+r+ 99.7%

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

   10. +-commutative 99.7%

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

   11. +-commutative 99.7%

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

8. Simplified 99.7%

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

   1. associate-*r/ 99.8%

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

   2. +-commutative 99.8%

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

   3. +-commutative 99.8%

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

   4. associate-+r+ 99.8%

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

   5. +-commutative 99.8%

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

   6. +-commutative 99.8%

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

10. Applied egg-rr 99.8%

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

11. Final simplification 99.8%

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

Alternative 2: 78.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + \left(\alpha + \beta\right)\\ \mathbf{if}\;\beta \leq 5.6 \cdot 10^{+17}:\\ \;\;\;\;\frac{1 + \beta}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \alpha\right) \cdot \frac{\frac{1 + \beta}{t\_0}}{t\_0}}{\alpha + \beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (+ alpha beta))))
   (if (<= beta 5.6e+17)
     (/ (+ 1.0 beta) (* (+ alpha (+ beta 2.0)) (* (+ beta 2.0) (+ beta 3.0))))
     (/ (* (+ 1.0 alpha) (/ (/ (+ 1.0 beta) t_0) t_0)) (+ alpha beta)))))

C

double code(double alpha, double beta) {
	double t_0 = 2.0 + (alpha + beta);
	double tmp;
	if (beta <= 5.6e+17) {
		tmp = (1.0 + beta) / ((alpha + (beta + 2.0)) * ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = ((1.0 + alpha) * (((1.0 + beta) / t_0) / t_0)) / (alpha + beta);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 + (alpha + beta)
    if (beta <= 5.6d+17) then
        tmp = (1.0d0 + beta) / ((alpha + (beta + 2.0d0)) * ((beta + 2.0d0) * (beta + 3.0d0)))
    else
        tmp = ((1.0d0 + alpha) * (((1.0d0 + beta) / t_0) / t_0)) / (alpha + beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = 2.0 + (alpha + beta);
	double tmp;
	if (beta <= 5.6e+17) {
		tmp = (1.0 + beta) / ((alpha + (beta + 2.0)) * ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = ((1.0 + alpha) * (((1.0 + beta) / t_0) / t_0)) / (alpha + beta);
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = 2.0 + (alpha + beta)
	tmp = 0
	if beta <= 5.6e+17:
		tmp = (1.0 + beta) / ((alpha + (beta + 2.0)) * ((beta + 2.0) * (beta + 3.0)))
	else:
		tmp = ((1.0 + alpha) * (((1.0 + beta) / t_0) / t_0)) / (alpha + beta)
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(alpha + beta))
	tmp = 0.0
	if (beta <= 5.6e+17)
		tmp = Float64(Float64(1.0 + beta) / Float64(Float64(alpha + Float64(beta + 2.0)) * Float64(Float64(beta + 2.0) * Float64(beta + 3.0))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) * Float64(Float64(Float64(1.0 + beta) / t_0) / t_0)) / Float64(alpha + beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = 2.0 + (alpha + beta);
	tmp = 0.0;
	if (beta <= 5.6e+17)
		tmp = (1.0 + beta) / ((alpha + (beta + 2.0)) * ((beta + 2.0) * (beta + 3.0)));
	else
		tmp = ((1.0 + alpha) * (((1.0 + beta) / t_0) / t_0)) / (alpha + beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 5.6e+17], N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] * N[(N[(N[(1.0 + beta), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(alpha + beta), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.6e17

   1. Initial program 99.7%

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

   3. Simplified 96.2%

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

   5. Taylor expanded in alpha around 0 85.3%

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

   6. Taylor expanded in alpha around 0 71.7%

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

      1. +-commutative 71.7%

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

      2. +-commutative 71.7%

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

   8. Simplified 71.7%

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

   if 5.6e17 < beta

   1. Initial program 87.3%

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

   3. Simplified 77.2%

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

      1. times-frac 90.9%

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

      2. +-commutative 90.9%

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

   6. Applied egg-rr 90.9%

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

      1. +-commutative 90.9%

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

      2. +-commutative 90.9%

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

      3. +-commutative 90.9%

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

      4. +-commutative 90.9%

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

      5. associate-/r* 99.7%

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

      6. +-commutative 99.7%

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

      7. +-commutative 99.7%

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

      8. +-commutative 99.7%

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

      9. associate-+r+ 99.7%

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

      10. +-commutative 99.7%

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

      11. +-commutative 99.7%

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

   8. Simplified 99.7%

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

      1. associate-*r/ 99.8%

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

      2. +-commutative 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+r+ 99.8%

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

      5. +-commutative 99.8%

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

      6. +-commutative 99.8%

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

   10. Applied egg-rr 99.8%

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

       1. associate-*l/ 99.8%

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

       2. associate-+r+ 99.8%

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

       3. associate-+r+ 99.8%

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

   12. Applied egg-rr 99.8%

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

       1. associate-/l* 99.8%

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

       2. +-commutative 99.8%

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

       3. +-commutative 99.8%

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

       4. +-commutative 99.8%

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

       5. +-commutative 99.8%

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

   14. Simplified 99.8%

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

   15. Taylor expanded in beta around inf 99.8%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.6 \cdot 10^{+17}:\\ \;\;\;\;\frac{1 + \beta}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \alpha\right) \cdot \frac{\frac{1 + \beta}{2 + \left(\alpha + \beta\right)}}{2 + \left(\alpha + \beta\right)}}{\alpha + \beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 300000000:\\ \;\;\;\;\frac{1 + \beta}{t\_0 \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{t\_0} \cdot \frac{1 - \frac{4 + \alpha \cdot 2}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 300000000.0)
     (/ (+ 1.0 beta) (* t_0 (* (+ beta 2.0) (+ beta 3.0))))
     (*
      (/ (+ 1.0 alpha) t_0)
      (/ (- 1.0 (/ (+ 4.0 (* alpha 2.0)) beta)) beta)))))

C

double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 300000000.0) {
		tmp = (1.0 + beta) / (t_0 * ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / t_0) * ((1.0 - ((4.0 + (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) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    if (beta <= 300000000.0d0) then
        tmp = (1.0d0 + beta) / (t_0 * ((beta + 2.0d0) * (beta + 3.0d0)))
    else
        tmp = ((1.0d0 + alpha) / t_0) * ((1.0d0 - ((4.0d0 + (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 tmp;
	if (beta <= 300000000.0) {
		tmp = (1.0 + beta) / (t_0 * ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / t_0) * ((1.0 - ((4.0 + (alpha * 2.0)) / beta)) / beta);
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 300000000.0:
		tmp = (1.0 + beta) / (t_0 * ((beta + 2.0) * (beta + 3.0)))
	else:
		tmp = ((1.0 + alpha) / t_0) * ((1.0 - ((4.0 + (alpha * 2.0)) / beta)) / beta)
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 300000000.0)
		tmp = Float64(Float64(1.0 + beta) / Float64(t_0 * Float64(Float64(beta + 2.0) * Float64(beta + 3.0))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / t_0) * Float64(Float64(1.0 - Float64(Float64(4.0 + Float64(alpha * 2.0)) / beta)) / beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 300000000.0)
		tmp = (1.0 + beta) / (t_0 * ((beta + 2.0) * (beta + 3.0)));
	else
		tmp = ((1.0 + alpha) / t_0) * ((1.0 - ((4.0 + (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]}, If[LessEqual[beta, 300000000.0], N[(N[(1.0 + beta), $MachinePrecision] / N[(t$95$0 * N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(N[(1.0 - N[(N[(4.0 + N[(alpha * 2.0), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 3e8

   1. Initial program 99.8%

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

   3. Simplified 96.2%

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

   5. Taylor expanded in alpha around 0 85.2%

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

   6. Taylor expanded in alpha around 0 71.5%

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

      1. +-commutative 71.5%

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

      2. +-commutative 71.5%

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

   8. Simplified 71.5%

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

   if 3e8 < beta

   1. Initial program 87.5%

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

   3. Simplified 77.4%

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

      1. times-frac 91.0%

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

      2. +-commutative 91.0%

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

   6. Applied egg-rr 91.0%

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

      1. +-commutative 91.0%

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

      2. +-commutative 91.0%

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

      3. +-commutative 91.0%

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

      4. +-commutative 91.0%

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

      5. associate-/r* 99.7%

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

      6. +-commutative 99.7%

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

      7. +-commutative 99.7%

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

      8. +-commutative 99.7%

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

      9. associate-+r+ 99.7%

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

      10. +-commutative 99.7%

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

      11. +-commutative 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in beta around inf 99.7%

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

   10. Taylor expanded in beta around inf 81.7%

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

       1. mul-1-neg 81.7%

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

       2. unsub-neg 81.7%

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

   12. Simplified 81.7%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 300000000:\\ \;\;\;\;\frac{1 + \beta}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 - \frac{4 + \alpha \cdot 2}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.9%

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

3. Simplified 90.3%

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

   1. times-frac 97.0%

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

   2. +-commutative 97.0%

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

6. Applied egg-rr 97.0%

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

   1. +-commutative 97.0%

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

   2. +-commutative 97.0%

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

   3. +-commutative 97.0%

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

   4. +-commutative 97.0%

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

   5. associate-/r* 99.7%

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

   6. +-commutative 99.7%

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

   7. +-commutative 99.7%

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

   8. +-commutative 99.7%

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

   9. associate-+r+ 99.7%

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

   10. +-commutative 99.7%

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

   11. +-commutative 99.7%

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

8. Simplified 99.7%

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

   1. associate-*r/ 99.8%

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

   2. +-commutative 99.8%

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

   3. +-commutative 99.8%

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

   4. associate-+r+ 99.8%

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

   5. +-commutative 99.8%

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

   6. +-commutative 99.8%

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

10. Applied egg-rr 99.8%

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

    1. associate-*l/ 99.7%

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

    2. associate-+r+ 99.7%

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

    3. associate-+r+ 99.7%

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

12. Applied egg-rr 99.7%

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

    1. associate-/l* 99.8%

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

    2. +-commutative 99.8%

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

    3. +-commutative 99.8%

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

    4. +-commutative 99.8%

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

    5. +-commutative 99.8%

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

14. Simplified 99.8%

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

15. Final simplification 99.8%

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

Alternative 5: 99.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

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 = ((1.0d0 + alpha) / t_0) * (((1.0d0 + beta) / t_0) / (3.0d0 + (alpha + beta)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.9%

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

3. Simplified 90.3%

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

   1. times-frac 97.0%

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

   2. +-commutative 97.0%

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

6. Applied egg-rr 97.0%

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

   1. +-commutative 97.0%

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

   2. +-commutative 97.0%

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

   3. +-commutative 97.0%

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

   4. +-commutative 97.0%

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

   5. associate-/r* 99.7%

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

   6. +-commutative 99.7%

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

   7. +-commutative 99.7%

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

   8. +-commutative 99.7%

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

   9. associate-+r+ 99.7%

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

   10. +-commutative 99.7%

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

   11. +-commutative 99.7%

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

8. Simplified 99.7%

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

9. Final simplification 99.7%

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

Alternative 6: 73.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.5 \cdot 10^{+17}:\\ \;\;\;\;\frac{1 + \beta}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \alpha\right) \cdot \frac{1}{2 + \left(\alpha + \beta\right)}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.5e+17)
   (/ (+ 1.0 beta) (* (+ alpha (+ beta 2.0)) (* (+ beta 2.0) (+ beta 3.0))))
   (/
    (* (+ 1.0 alpha) (/ 1.0 (+ 2.0 (+ alpha beta))))
    (+ alpha (+ beta 3.0)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.5e+17) {
		tmp = (1.0 + beta) / ((alpha + (beta + 2.0)) * ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = ((1.0 + alpha) * (1.0 / (2.0 + (alpha + 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 <= 6.5d+17) then
        tmp = (1.0d0 + beta) / ((alpha + (beta + 2.0d0)) * ((beta + 2.0d0) * (beta + 3.0d0)))
    else
        tmp = ((1.0d0 + alpha) * (1.0d0 / (2.0d0 + (alpha + beta)))) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.5e+17) {
		tmp = (1.0 + beta) / ((alpha + (beta + 2.0)) * ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = ((1.0 + alpha) * (1.0 / (2.0 + (alpha + beta)))) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 6.5e+17:
		tmp = (1.0 + beta) / ((alpha + (beta + 2.0)) * ((beta + 2.0) * (beta + 3.0)))
	else:
		tmp = ((1.0 + alpha) * (1.0 / (2.0 + (alpha + beta)))) / (alpha + (beta + 3.0))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.5e+17)
		tmp = Float64(Float64(1.0 + beta) / Float64(Float64(alpha + Float64(beta + 2.0)) * Float64(Float64(beta + 2.0) * Float64(beta + 3.0))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) * Float64(1.0 / Float64(2.0 + Float64(alpha + beta)))) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 6.5e+17)
		tmp = (1.0 + beta) / ((alpha + (beta + 2.0)) * ((beta + 2.0) * (beta + 3.0)));
	else
		tmp = ((1.0 + alpha) * (1.0 / (2.0 + (alpha + beta)))) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 6.5e+17], N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] * N[(1.0 / N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.5e17

   1. Initial program 99.7%

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

   3. Simplified 96.2%

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

   5. Taylor expanded in alpha around 0 85.3%

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

   6. Taylor expanded in alpha around 0 71.7%

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

      1. +-commutative 71.7%

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

      2. +-commutative 71.7%

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

   8. Simplified 71.7%

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

   if 6.5e17 < beta

   1. Initial program 87.3%

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

   3. Simplified 77.2%

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

      1. times-frac 90.9%

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

      2. +-commutative 90.9%

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

   6. Applied egg-rr 90.9%

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

      1. +-commutative 90.9%

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

      2. +-commutative 90.9%

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

      3. +-commutative 90.9%

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

      4. +-commutative 90.9%

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

      5. associate-/r* 99.7%

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

      6. +-commutative 99.7%

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

      7. +-commutative 99.7%

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

      8. +-commutative 99.7%

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

      9. associate-+r+ 99.7%

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

      10. +-commutative 99.7%

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

      11. +-commutative 99.7%

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

   8. Simplified 99.7%

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

      1. associate-*r/ 99.8%

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

      2. +-commutative 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+r+ 99.8%

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

      5. +-commutative 99.8%

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

      6. +-commutative 99.8%

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

   10. Applied egg-rr 99.8%

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

       1. associate-*l/ 99.8%

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

       2. associate-+r+ 99.8%

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

       3. associate-+r+ 99.8%

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

   12. Applied egg-rr 99.8%

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

       1. associate-/l* 99.8%

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

       2. +-commutative 99.8%

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

       3. +-commutative 99.8%

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

       4. +-commutative 99.8%

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

       5. +-commutative 99.8%

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

   14. Simplified 99.8%

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

   15. Taylor expanded in beta around inf 82.6%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.5 \cdot 10^{+17}:\\ \;\;\;\;\frac{1 + \beta}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \alpha\right) \cdot \frac{1}{2 + \left(\alpha + \beta\right)}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.2 \cdot 10^{+15}:\\ \;\;\;\;\frac{1}{\beta + 2} \cdot \frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \alpha\right) \cdot \frac{1}{2 + \left(\alpha + \beta\right)}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5.2e+15)
   (* (/ 1.0 (+ beta 2.0)) (/ (+ 1.0 beta) (* (+ beta 2.0) (+ beta 3.0))))
   (/
    (* (+ 1.0 alpha) (/ 1.0 (+ 2.0 (+ alpha beta))))
    (+ alpha (+ beta 3.0)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.2e+15) {
		tmp = (1.0 / (beta + 2.0)) * ((1.0 + beta) / ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = ((1.0 + alpha) * (1.0 / (2.0 + (alpha + 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 <= 5.2d+15) then
        tmp = (1.0d0 / (beta + 2.0d0)) * ((1.0d0 + beta) / ((beta + 2.0d0) * (beta + 3.0d0)))
    else
        tmp = ((1.0d0 + alpha) * (1.0d0 / (2.0d0 + (alpha + beta)))) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.2e+15) {
		tmp = (1.0 / (beta + 2.0)) * ((1.0 + beta) / ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = ((1.0 + alpha) * (1.0 / (2.0 + (alpha + beta)))) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 5.2e+15:
		tmp = (1.0 / (beta + 2.0)) * ((1.0 + beta) / ((beta + 2.0) * (beta + 3.0)))
	else:
		tmp = ((1.0 + alpha) * (1.0 / (2.0 + (alpha + beta)))) / (alpha + (beta + 3.0))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 5.2e+15)
		tmp = Float64(Float64(1.0 / Float64(beta + 2.0)) * Float64(Float64(1.0 + beta) / Float64(Float64(beta + 2.0) * Float64(beta + 3.0))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) * Float64(1.0 / Float64(2.0 + Float64(alpha + beta)))) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 5.2e+15)
		tmp = (1.0 / (beta + 2.0)) * ((1.0 + beta) / ((beta + 2.0) * (beta + 3.0)));
	else
		tmp = ((1.0 + alpha) * (1.0 / (2.0 + (alpha + beta)))) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 5.2e+15], N[(N[(1.0 / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] * N[(1.0 / N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.2e15

   1. Initial program 99.7%

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

   3. Simplified 96.2%

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

      1. times-frac 99.8%

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

      2. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. +-commutative 99.8%

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

      2. +-commutative 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

      5. associate-/r* 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. +-commutative 99.8%

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

      9. associate-+r+ 99.8%

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

      10. +-commutative 99.8%

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

      11. +-commutative 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in alpha around 0 86.9%

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

       1. +-commutative 86.9%

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

   11. Simplified 86.9%

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

   12. Taylor expanded in alpha around 0 70.8%

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

       1. +-commutative 70.8%

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

       2. +-commutative 70.8%

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

   14. Simplified 70.8%

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

   if 5.2e15 < beta

   1. Initial program 87.3%

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

   3. Simplified 77.2%

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

      1. times-frac 90.9%

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

      2. +-commutative 90.9%

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

   6. Applied egg-rr 90.9%

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

      1. +-commutative 90.9%

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

      2. +-commutative 90.9%

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

      3. +-commutative 90.9%

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

      4. +-commutative 90.9%

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

      5. associate-/r* 99.7%

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

      6. +-commutative 99.7%

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

      7. +-commutative 99.7%

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

      8. +-commutative 99.7%

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

      9. associate-+r+ 99.7%

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

      10. +-commutative 99.7%

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

      11. +-commutative 99.7%

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

   8. Simplified 99.7%

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

      1. associate-*r/ 99.8%

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

      2. +-commutative 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+r+ 99.8%

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

      5. +-commutative 99.8%

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

      6. +-commutative 99.8%

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

   10. Applied egg-rr 99.8%

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

       1. associate-*l/ 99.8%

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

       2. associate-+r+ 99.8%

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

       3. associate-+r+ 99.8%

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

   12. Applied egg-rr 99.8%

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

       1. associate-/l* 99.8%

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

       2. +-commutative 99.8%

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

       3. +-commutative 99.8%

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

       4. +-commutative 99.8%

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

       5. +-commutative 99.8%

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

   14. Simplified 99.8%

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

   15. Taylor expanded in beta around inf 82.6%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.2 \cdot 10^{+15}:\\ \;\;\;\;\frac{1}{\beta + 2} \cdot \frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \alpha\right) \cdot \frac{1}{2 + \left(\alpha + \beta\right)}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.4 \cdot 10^{+16}:\\ \;\;\;\;\frac{1}{\beta + 2} \cdot \frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.4e+16)
   (* (/ 1.0 (+ beta 2.0)) (/ (+ 1.0 beta) (* (+ beta 2.0) (+ beta 3.0))))
   (/ (/ (+ 1.0 alpha) beta) (+ 1.0 (+ 2.0 (+ alpha beta))))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.4e+16) {
		tmp = (1.0 / (beta + 2.0)) * ((1.0 + beta) / ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / 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 <= 2.4d+16) then
        tmp = (1.0d0 / (beta + 2.0d0)) * ((1.0d0 + beta) / ((beta + 2.0d0) * (beta + 3.0d0)))
    else
        tmp = ((1.0d0 + alpha) / 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 <= 2.4e+16) {
		tmp = (1.0 / (beta + 2.0)) * ((1.0 + beta) / ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / beta) / (1.0 + (2.0 + (alpha + beta)));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.4e+16:
		tmp = (1.0 / (beta + 2.0)) * ((1.0 + beta) / ((beta + 2.0) * (beta + 3.0)))
	else:
		tmp = ((1.0 + alpha) / beta) / (1.0 + (2.0 + (alpha + beta)))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.4e+16)
		tmp = Float64(Float64(1.0 / Float64(beta + 2.0)) * Float64(Float64(1.0 + beta) / Float64(Float64(beta + 2.0) * Float64(beta + 3.0))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / 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 <= 2.4e+16)
		tmp = (1.0 / (beta + 2.0)) * ((1.0 + beta) / ((beta + 2.0) * (beta + 3.0)));
	else
		tmp = ((1.0 + alpha) / beta) / (1.0 + (2.0 + (alpha + beta)));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.4e+16], N[(N[(1.0 / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(1.0 + N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.4e16

   1. Initial program 99.7%

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

   3. Simplified 96.2%

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

      1. times-frac 99.8%

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

      2. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. +-commutative 99.8%

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

      2. +-commutative 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

      5. associate-/r* 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. +-commutative 99.8%

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

      9. associate-+r+ 99.8%

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

      10. +-commutative 99.8%

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

      11. +-commutative 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in alpha around 0 86.9%

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

       1. +-commutative 86.9%

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

   11. Simplified 86.9%

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

   12. Taylor expanded in alpha around 0 70.8%

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

       1. +-commutative 70.8%

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

       2. +-commutative 70.8%

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

   14. Simplified 70.8%

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

   if 2.4e16 < beta

   1. Initial program 87.3%

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

   3. Taylor expanded in beta around inf 82.1%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.4 \cdot 10^{+16}:\\ \;\;\;\;\frac{1}{\beta + 2} \cdot \frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 82.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 7.2:\\ \;\;\;\;\frac{1}{\beta + 2} \cdot \frac{1}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 7.2)
   (* (/ 1.0 (+ beta 2.0)) (/ 1.0 (* (+ alpha 2.0) (+ alpha 3.0))))
   (/ (/ (+ 1.0 alpha) beta) (+ beta (+ alpha 3.0)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 7.2) {
		tmp = (1.0 / (beta + 2.0)) * (1.0 / ((alpha + 2.0) * (alpha + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 7.2d0) then
        tmp = (1.0d0 / (beta + 2.0d0)) * (1.0d0 / ((alpha + 2.0d0) * (alpha + 3.0d0)))
    else
        tmp = ((1.0d0 + alpha) / beta) / (beta + (alpha + 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 7.2) {
		tmp = (1.0 / (beta + 2.0)) * (1.0 / ((alpha + 2.0) * (alpha + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 3.0));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 7.2:
		tmp = (1.0 / (beta + 2.0)) * (1.0 / ((alpha + 2.0) * (alpha + 3.0)))
	else:
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 3.0))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 7.2)
		tmp = Float64(Float64(1.0 / Float64(beta + 2.0)) * Float64(1.0 / Float64(Float64(alpha + 2.0) * Float64(alpha + 3.0))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(beta + Float64(alpha + 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 7.2)
		tmp = (1.0 / (beta + 2.0)) * (1.0 / ((alpha + 2.0) * (alpha + 3.0)));
	else
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 7.2], N[(N[(1.0 / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[(alpha + 2.0), $MachinePrecision] * N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 7.20000000000000018

   1. Initial program 99.8%

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

   3. Simplified 96.1%

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

      1. times-frac 99.8%

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

      2. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. +-commutative 99.8%

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

      2. +-commutative 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

      5. associate-/r* 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. +-commutative 99.8%

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

      9. associate-+r+ 99.8%

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

      10. +-commutative 99.8%

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

      11. +-commutative 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in alpha around 0 87.1%

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

       1. +-commutative 87.1%

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

   11. Simplified 87.1%

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

   12. Taylor expanded in beta around 0 84.9%

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

   if 7.20000000000000018 < beta

   1. Initial program 88.0%

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

   3. Taylor expanded in beta around inf 78.9%

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

      1. metadata-eval 78.9%

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

      2. associate-+l+ 78.9%

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

      3. metadata-eval 78.9%

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

      4. +-commutative 78.9%

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

      5. associate-+r+ 78.9%

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

      6. *-un-lft-identity 78.9%

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

      7. fma-define 78.9%

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

   5. Applied egg-rr 78.9%

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

      1. fma-undefine 78.9%

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

      2. *-lft-identity 78.9%

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

      3. +-commutative 78.9%

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

   7. Simplified 78.9%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 7.2:\\ \;\;\;\;\frac{1}{\beta + 2} \cdot \frac{1}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 82.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 8:\\ \;\;\;\;\frac{0.5}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 8.0)
   (/ 0.5 (* (+ alpha 2.0) (+ alpha 3.0)))
   (/ (/ (+ 1.0 alpha) beta) (+ beta (+ alpha 3.0)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 8.0) {
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 8.0d0) then
        tmp = 0.5d0 / ((alpha + 2.0d0) * (alpha + 3.0d0))
    else
        tmp = ((1.0d0 + alpha) / beta) / (beta + (alpha + 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 8.0) {
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 3.0));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 8.0:
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0))
	else:
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 3.0))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 8.0)
		tmp = Float64(0.5 / Float64(Float64(alpha + 2.0) * Float64(alpha + 3.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(beta + Float64(alpha + 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 8.0)
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	else
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 8.0], N[(0.5 / N[(N[(alpha + 2.0), $MachinePrecision] * N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 8

   1. Initial program 99.8%

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

   3. Simplified 96.1%

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

      1. times-frac 99.8%

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

      2. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. +-commutative 99.8%

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

      2. +-commutative 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

      5. associate-/r* 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. +-commutative 99.8%

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

      9. associate-+r+ 99.8%

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

      10. +-commutative 99.8%

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

      11. +-commutative 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in alpha around 0 87.1%

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

       1. +-commutative 87.1%

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

   11. Simplified 87.1%

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

   12. Taylor expanded in beta around 0 84.8%

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

   if 8 < beta

   1. Initial program 88.0%

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

   3. Taylor expanded in beta around inf 78.9%

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

      1. metadata-eval 78.9%

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

      2. associate-+l+ 78.9%

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

      3. metadata-eval 78.9%

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

      4. +-commutative 78.9%

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

      5. associate-+r+ 78.9%

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

      6. *-un-lft-identity 78.9%

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

      7. fma-define 78.9%

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

   5. Applied egg-rr 78.9%

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

      1. fma-undefine 78.9%

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

      2. *-lft-identity 78.9%

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

      3. +-commutative 78.9%

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

   7. Simplified 78.9%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 8:\\ \;\;\;\;\frac{0.5}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 82.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.7:\\ \;\;\;\;\frac{0.5}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.7)
   (/ 0.5 (* (+ alpha 2.0) (+ alpha 3.0)))
   (/ (/ (+ 1.0 alpha) beta) (+ beta 3.0))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.7) {
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 3.7d0) then
        tmp = 0.5d0 / ((alpha + 2.0d0) * (alpha + 3.0d0))
    else
        tmp = ((1.0d0 + alpha) / beta) / (beta + 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.7) {
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 3.7:
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0))
	else:
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.7)
		tmp = Float64(0.5 / Float64(Float64(alpha + 2.0) * Float64(alpha + 3.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(beta + 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.7)
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	else
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 3.7], N[(0.5 / N[(N[(alpha + 2.0), $MachinePrecision] * N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.7000000000000002

   1. Initial program 99.8%

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

   3. Simplified 96.1%

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

      1. times-frac 99.8%

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

      2. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. +-commutative 99.8%

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

      2. +-commutative 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

      5. associate-/r* 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. +-commutative 99.8%

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

      9. associate-+r+ 99.8%

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

      10. +-commutative 99.8%

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

      11. +-commutative 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in alpha around 0 87.1%

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

       1. +-commutative 87.1%

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

   11. Simplified 87.1%

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

   12. Taylor expanded in beta around 0 84.8%

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

   if 3.7000000000000002 < beta

   1. Initial program 88.0%

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

   3. Taylor expanded in beta around inf 78.9%

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

   4. Taylor expanded in alpha around 0 78.6%

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

      1. +-commutative 78.6%

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

   6. Simplified 78.6%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.7:\\ \;\;\;\;\frac{0.5}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 82.7% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 4.5d0) then
        tmp = 0.5d0 / ((alpha + 2.0d0) * (alpha + 3.0d0))
    else
        tmp = ((1.0d0 + alpha) / beta) / beta
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if beta < 4.5

   1. Initial program 99.8%

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

   3. Simplified 96.1%

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

      1. times-frac 99.8%

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

      2. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. +-commutative 99.8%

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

      2. +-commutative 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

      5. associate-/r* 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. +-commutative 99.8%

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

      9. associate-+r+ 99.8%

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

      10. +-commutative 99.8%

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

      11. +-commutative 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in alpha around 0 87.1%

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

       1. +-commutative 87.1%

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

   11. Simplified 87.1%

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

   12. Taylor expanded in beta around 0 84.8%

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

   if 4.5 < beta

   1. Initial program 88.0%

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

   3. Taylor expanded in beta around inf 78.9%

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

   4. Taylor expanded in beta around -inf 78.9%

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

      1. associate-*r* 78.9%

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

      2. mul-1-neg 78.9%

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

      3. sub-neg 78.9%

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

      4. associate-*r/ 78.9%

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

      5. distribute-lft-in 78.9%

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

      6. metadata-eval 78.9%

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

      7. metadata-eval 78.9%

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

      8. mul-1-neg 78.9%

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

      9. unsub-neg 78.9%

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

      10. metadata-eval 78.9%

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

      11. metadata-eval 78.9%

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

   6. Simplified 78.9%

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

   7. Taylor expanded in beta around inf 78.6%

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

4. Final simplification 82.8%

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

Alternative 13: 28.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 4.5 \cdot 10^{-48}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 4.5e-48) (/ (/ 1.0 beta) (+ beta 3.0)) (/ (/ alpha beta) beta)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 4.5e-48) {
		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 (alpha <= 4.5d-48) 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 (alpha <= 4.5e-48) {
		tmp = (1.0 / beta) / (beta + 3.0);
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 4.5e-48:
		tmp = (1.0 / beta) / (beta + 3.0)
	else:
		tmp = (alpha / beta) / beta
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 4.5e-48)
		tmp = Float64(Float64(1.0 / 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 (alpha <= 4.5e-48)
		tmp = (1.0 / beta) / (beta + 3.0);
	else
		tmp = (alpha / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 4.5e-48], N[(N[(1.0 / beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 4.49999999999999988e-48

   1. Initial program 99.8%

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

   3. Taylor expanded in beta around inf 33.2%

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

   4. Taylor expanded in alpha around 0 32.7%

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

      1. +-commutative 32.7%

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

   6. Simplified 32.7%

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

      1. *-un-lft-identity 32.7%

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

      2. associate-/r* 32.7%

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

   8. Applied egg-rr 32.7%

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

      1. *-lft-identity 32.7%

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

   10. Simplified 32.7%

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

   if 4.49999999999999988e-48 < alpha

   1. Initial program 88.8%

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

   3. Taylor expanded in beta around inf 18.3%

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

   4. Taylor expanded in beta around -inf 17.8%

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

      1. associate-*r* 17.8%

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

      2. mul-1-neg 17.8%

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

      3. sub-neg 17.8%

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

      4. associate-*r/ 17.8%

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

      5. distribute-lft-in 17.8%

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

      6. metadata-eval 17.8%

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

      7. metadata-eval 17.8%

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

      8. mul-1-neg 17.8%

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

      9. unsub-neg 17.8%

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

      10. metadata-eval 17.8%

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

      11. metadata-eval 17.8%

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

   6. Simplified 17.8%

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

   7. Taylor expanded in alpha around inf 17.7%

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

   8. Taylor expanded in beta around inf 18.0%

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

Alternative 14: 28.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 6.7 \cdot 10^{-49}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 6.7e-49) (/ 1.0 (* beta (+ beta 3.0))) (/ (/ alpha beta) beta)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 6.7e-49) {
		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 (alpha <= 6.7d-49) 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 (alpha <= 6.7e-49) {
		tmp = 1.0 / (beta * (beta + 3.0));
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 6.7e-49:
		tmp = 1.0 / (beta * (beta + 3.0))
	else:
		tmp = (alpha / beta) / beta
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 6.7e-49)
		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 (alpha <= 6.7e-49)
		tmp = 1.0 / (beta * (beta + 3.0));
	else
		tmp = (alpha / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 6.7e-49], N[(1.0 / N[(beta * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 6.7e-49

   1. Initial program 99.8%

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

   3. Taylor expanded in beta around inf 33.2%

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

   4. Taylor expanded in alpha around 0 32.7%

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

      1. +-commutative 32.7%

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

   6. Simplified 32.7%

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

   if 6.7e-49 < alpha

   1. Initial program 88.8%

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

   3. Taylor expanded in beta around inf 18.3%

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

   4. Taylor expanded in beta around -inf 17.8%

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

      1. associate-*r* 17.8%

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

      2. mul-1-neg 17.8%

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

      3. sub-neg 17.8%

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

      4. associate-*r/ 17.8%

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

      5. distribute-lft-in 17.8%

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

      6. metadata-eval 17.8%

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

      7. metadata-eval 17.8%

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

      8. mul-1-neg 17.8%

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

      9. unsub-neg 17.8%

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

      10. metadata-eval 17.8%

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

      11. metadata-eval 17.8%

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

   6. Simplified 17.8%

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

   7. Taylor expanded in alpha around inf 17.7%

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

   8. Taylor expanded in beta around inf 18.0%

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

Alternative 15: 28.6% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 4.5 \cdot 10^{-48}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 4.5e-48) (/ 1.0 (* beta beta)) (/ (/ alpha beta) beta)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 4.5e-48) {
		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 (alpha <= 4.5d-48) 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 (alpha <= 4.5e-48) {
		tmp = 1.0 / (beta * beta);
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 4.5e-48:
		tmp = 1.0 / (beta * beta)
	else:
		tmp = (alpha / beta) / beta
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 4.5e-48)
		tmp = Float64(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 (alpha <= 4.5e-48)
		tmp = 1.0 / (beta * beta);
	else
		tmp = (alpha / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 4.5e-48], N[(1.0 / N[(beta * beta), $MachinePrecision]), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 4.49999999999999988e-48

   1. Initial program 99.8%

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

   3. Taylor expanded in beta around inf 33.2%

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

   4. Taylor expanded in alpha around 0 32.7%

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

      1. +-commutative 32.7%

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

   6. Simplified 32.7%

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

   7. Taylor expanded in beta around inf 33.3%

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

   if 4.49999999999999988e-48 < alpha

   1. Initial program 88.8%

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

   3. Taylor expanded in beta around inf 18.3%

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

   4. Taylor expanded in beta around -inf 17.8%

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

      1. associate-*r* 17.8%

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

      2. mul-1-neg 17.8%

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

      3. sub-neg 17.8%

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

      4. associate-*r/ 17.8%

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

      5. distribute-lft-in 17.8%

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

      6. metadata-eval 17.8%

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

      7. metadata-eval 17.8%

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

      8. mul-1-neg 17.8%

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

      9. unsub-neg 17.8%

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

      10. metadata-eval 17.8%

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

      11. metadata-eval 17.8%

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

   6. Simplified 17.8%

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

   7. Taylor expanded in alpha around inf 17.7%

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

   8. Taylor expanded in beta around inf 18.0%

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

Alternative 16: 29.9% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.9%

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

3. Taylor expanded in beta around inf 27.9%

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

4. Taylor expanded in beta around -inf 27.8%

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

   1. associate-*r* 27.8%

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

   2. mul-1-neg 27.8%

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

   3. sub-neg 27.8%

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

   4. associate-*r/ 27.8%

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

   5. distribute-lft-in 27.8%

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

   6. metadata-eval 27.8%

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

   7. metadata-eval 27.8%

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

   8. mul-1-neg 27.8%

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

   9. unsub-neg 27.8%

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

   10. metadata-eval 27.8%

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

   11. metadata-eval 27.8%

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

6. Simplified 27.8%

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

7. Taylor expanded in beta around inf 28.3%

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

Alternative 17: 27.4% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.9%

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

3. Taylor expanded in beta around inf 27.9%

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

4. Taylor expanded in alpha around 0 25.3%

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

   1. +-commutative 25.3%

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

6. Simplified 25.3%

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

7. Taylor expanded in beta around inf 25.7%

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

Alternative 18: 4.3% accurate, 11.7× speedup

Math

\[\begin{array}{l} \\ \frac{0.3333333333333333}{\beta} \end{array} \]

FPCore

(FPCore (alpha beta) :precision binary64 (/ 0.3333333333333333 beta))

C

double code(double alpha, double beta) {
	return 0.3333333333333333 / beta;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = 0.3333333333333333d0 / beta
end function

Java

public static double code(double alpha, double beta) {
	return 0.3333333333333333 / beta;
}

Python

def code(alpha, beta):
	return 0.3333333333333333 / beta

Julia

function code(alpha, beta)
	return Float64(0.3333333333333333 / beta)
end

MATLAB

function tmp = code(alpha, beta)
	tmp = 0.3333333333333333 / beta;
end

Wolfram

code[alpha_, beta_] := N[(0.3333333333333333 / beta), $MachinePrecision]

Derivation

1. Initial program 95.9%

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

3. Taylor expanded in beta around inf 27.9%

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

4. Taylor expanded in alpha around 0 25.3%

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

   1. +-commutative 25.3%

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

6. Simplified 25.3%

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

7. Taylor expanded in beta around 0 4.5%

   \[\leadsto \color{blue}{\frac{0.3333333333333333}{\beta}} \]
8. Add Preprocessing

Alternative 19: 3.6% accurate, 11.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.9%

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

3. Simplified 90.3%

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

   1. times-frac 97.0%

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

   2. +-commutative 97.0%

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

6. Applied egg-rr 97.0%

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

   1. +-commutative 97.0%

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

   2. +-commutative 97.0%

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

   3. +-commutative 97.0%

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

   4. +-commutative 97.0%

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

   5. associate-/r* 99.7%

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

   6. +-commutative 99.7%

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

   7. +-commutative 99.7%

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

   8. +-commutative 99.7%

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

   9. associate-+r+ 99.7%

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

   10. +-commutative 99.7%

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

   11. +-commutative 99.7%

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

8. Simplified 99.7%

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

   1. associate-*r/ 99.8%

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

   2. +-commutative 99.8%

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

   3. +-commutative 99.8%

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

   4. associate-+r+ 99.8%

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

   5. +-commutative 99.8%

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

   6. +-commutative 99.8%

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

10. Applied egg-rr 99.8%

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

11. Taylor expanded in beta around -inf 27.9%

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

    1. +-commutative 27.9%

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

    2. mul-1-neg 27.9%

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

    3. unsub-neg 27.9%

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

13. Simplified 27.9%

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

14. Taylor expanded in alpha around inf 3.6%

    \[\leadsto \color{blue}{\frac{-1}{\beta}} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024130 
(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)))