Octave 3.8, jcobi/3

Percentage Accurate: 94.6% → 99.8%

Time: 12.8s

Alternatives: 15

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: 94.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.1%

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

3. Simplified 83.4%

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

   1. times-frac 96.5%

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

   2. +-commutative 96.5%

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

6. Applied egg-rr 96.5%

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

   1. associate-*l/ 96.5%

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

   2. associate-+r+ 96.5%

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

   3. associate-/r* 99.9%

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

   4. associate-+r+ 99.9%

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

8. Applied egg-rr 99.9%

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

   1. associate-+r+ 99.9%

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

   2. associate-*r/ 99.9%

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

   3. associate-+r+ 99.9%

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

   4. +-commutative 99.9%

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

   5. associate-+r+ 99.9%

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

10. Applied egg-rr 99.9%

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

Alternative 2: 99.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.1%

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

3. Simplified 83.4%

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

   1. times-frac 96.5%

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

   2. +-commutative 96.5%

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

6. Applied egg-rr 96.5%

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

   1. associate-*l/ 96.5%

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

   2. associate-+r+ 96.5%

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

   3. associate-/r* 99.9%

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

   4. associate-+r+ 99.9%

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

8. Applied egg-rr 99.9%

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

Alternative 3: 73.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.6 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 2}}{\left(\beta + 2\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\alpha + 1\right) \cdot \frac{1 - 2 \cdot \frac{\alpha}{\beta}}{\beta}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.6e+16)
   (/ (/ (+ 1.0 beta) (+ beta 2.0)) (* (+ beta 2.0) (+ (+ alpha beta) 3.0)))
   (/
    (* (+ alpha 1.0) (/ (- 1.0 (* 2.0 (/ alpha beta))) beta))
    (+ alpha (+ beta 2.0)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.6e+16) {
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * ((alpha + beta) + 3.0));
	} else {
		tmp = ((alpha + 1.0) * ((1.0 - (2.0 * (alpha / beta))) / beta)) / (alpha + (beta + 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 3.6d+16) then
        tmp = ((1.0d0 + beta) / (beta + 2.0d0)) / ((beta + 2.0d0) * ((alpha + beta) + 3.0d0))
    else
        tmp = ((alpha + 1.0d0) * ((1.0d0 - (2.0d0 * (alpha / beta))) / beta)) / (alpha + (beta + 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.6e+16) {
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * ((alpha + beta) + 3.0));
	} else {
		tmp = ((alpha + 1.0) * ((1.0 - (2.0 * (alpha / beta))) / beta)) / (alpha + (beta + 2.0));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 3.6e+16:
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * ((alpha + beta) + 3.0))
	else:
		tmp = ((alpha + 1.0) * ((1.0 - (2.0 * (alpha / beta))) / beta)) / (alpha + (beta + 2.0))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.6e+16)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(beta + 2.0)) / Float64(Float64(beta + 2.0) * Float64(Float64(alpha + beta) + 3.0)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) * Float64(Float64(1.0 - Float64(2.0 * Float64(alpha / beta))) / beta)) / Float64(alpha + Float64(beta + 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.6e+16)
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * ((alpha + beta) + 3.0));
	else
		tmp = ((alpha + 1.0) * ((1.0 - (2.0 * (alpha / beta))) / beta)) / (alpha + (beta + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 3.6e+16], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] * N[(N[(1.0 - N[(2.0 * N[(alpha / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.6e16

   1. Initial program 99.9%

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

      1. associate-/l/ 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+l+ 99.9%

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

      4. *-commutative 99.9%

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

      5. metadata-eval 99.9%

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

      6. associate-+l+ 99.9%

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

      7. metadata-eval 99.9%

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

      8. +-commutative 99.9%

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

      9. +-commutative 99.9%

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

      10. +-commutative 99.9%

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

      11. metadata-eval 99.9%

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

      12. metadata-eval 99.9%

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

      13. associate-+l+ 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in alpha around 0 83.3%

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

      1. +-commutative 83.3%

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

   7. Simplified 83.3%

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

   8. Taylor expanded in alpha around 0 67.0%

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

      1. +-commutative 67.0%

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

   10. Simplified 67.0%

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

   if 3.6e16 < beta

   1. Initial program 79.3%

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

   3. Simplified 59.3%

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

      1. times-frac 88.0%

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

      2. +-commutative 88.0%

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

   6. Applied egg-rr 88.0%

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

      1. associate-*l/ 88.1%

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

      2. associate-+r+ 88.1%

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

      3. associate-/r* 99.8%

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

      4. associate-+r+ 99.8%

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in beta around inf 83.6%

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

       1. mul-1-neg 83.6%

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

       2. unsub-neg 83.6%

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

   11. Simplified 83.6%

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

   12. Taylor expanded in alpha around inf 83.6%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.6 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 2}}{\left(\beta + 2\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\alpha + 1\right) \cdot \frac{1 - 2 \cdot \frac{\alpha}{\beta}}{\beta}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 8.7 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 2}}{\left(\beta + 2\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\alpha + \left(\beta + 3\right)}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 8.7e+15)
   (/ (/ (+ 1.0 beta) (+ beta 2.0)) (* (+ beta 2.0) (+ (+ alpha beta) 3.0)))
   (/ (/ (+ alpha 1.0) (+ alpha (+ beta 3.0))) (+ alpha (+ beta 2.0)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 8.7e+15) {
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * ((alpha + beta) + 3.0));
	} else {
		tmp = ((alpha + 1.0) / (alpha + (beta + 3.0))) / (alpha + (beta + 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 8.7d+15) then
        tmp = ((1.0d0 + beta) / (beta + 2.0d0)) / ((beta + 2.0d0) * ((alpha + beta) + 3.0d0))
    else
        tmp = ((alpha + 1.0d0) / (alpha + (beta + 3.0d0))) / (alpha + (beta + 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 8.7e+15) {
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * ((alpha + beta) + 3.0));
	} else {
		tmp = ((alpha + 1.0) / (alpha + (beta + 3.0))) / (alpha + (beta + 2.0));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 8.7e+15:
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * ((alpha + beta) + 3.0))
	else:
		tmp = ((alpha + 1.0) / (alpha + (beta + 3.0))) / (alpha + (beta + 2.0))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 8.7e+15)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(beta + 2.0)) / Float64(Float64(beta + 2.0) * Float64(Float64(alpha + beta) + 3.0)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(alpha + Float64(beta + 3.0))) / Float64(alpha + Float64(beta + 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 8.7e+15)
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * ((alpha + beta) + 3.0));
	else
		tmp = ((alpha + 1.0) / (alpha + (beta + 3.0))) / (alpha + (beta + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 8.7e+15], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 8.7e15

   1. Initial program 99.9%

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

      1. associate-/l/ 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+l+ 99.9%

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

      4. *-commutative 99.9%

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

      5. metadata-eval 99.9%

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

      6. associate-+l+ 99.9%

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

      7. metadata-eval 99.9%

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

      8. +-commutative 99.9%

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

      9. +-commutative 99.9%

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

      10. +-commutative 99.9%

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

      11. metadata-eval 99.9%

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

      12. metadata-eval 99.9%

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

      13. associate-+l+ 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in alpha around 0 83.3%

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

      1. +-commutative 83.3%

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

   7. Simplified 83.3%

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

   8. Taylor expanded in alpha around 0 67.0%

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

      1. +-commutative 67.0%

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

   10. Simplified 67.0%

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

   if 8.7e15 < beta

   1. Initial program 79.3%

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

   3. Simplified 59.3%

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

      1. times-frac 88.0%

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

      2. +-commutative 88.0%

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

   6. Applied egg-rr 88.0%

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

      1. associate-*l/ 88.1%

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

      2. associate-+r+ 88.1%

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

      3. associate-/r* 99.8%

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

      4. associate-+r+ 99.8%

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

   8. Applied egg-rr 99.8%

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

      1. associate-+r+ 99.8%

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

      2. associate-*r/ 99.9%

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

      3. associate-+r+ 99.9%

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

      4. +-commutative 99.9%

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

      5. associate-+r+ 99.9%

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

   10. Applied egg-rr 99.9%

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

   11. Taylor expanded in beta around inf 84.0%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 8.7 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 2}}{\left(\beta + 2\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\alpha + \left(\beta + 3\right)}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.35 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{1}{\frac{\beta + 2}{1 + \beta}}}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\alpha + \left(\beta + 3\right)}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.35e+15)
   (/ (/ 1.0 (/ (+ beta 2.0) (+ 1.0 beta))) (* (+ beta 3.0) (+ beta 2.0)))
   (/ (/ (+ alpha 1.0) (+ alpha (+ beta 3.0))) (+ alpha (+ beta 2.0)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.35e+15) {
		tmp = (1.0 / ((beta + 2.0) / (1.0 + beta))) / ((beta + 3.0) * (beta + 2.0));
	} else {
		tmp = ((alpha + 1.0) / (alpha + (beta + 3.0))) / (alpha + (beta + 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.35d+15) then
        tmp = (1.0d0 / ((beta + 2.0d0) / (1.0d0 + beta))) / ((beta + 3.0d0) * (beta + 2.0d0))
    else
        tmp = ((alpha + 1.0d0) / (alpha + (beta + 3.0d0))) / (alpha + (beta + 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.35e+15) {
		tmp = (1.0 / ((beta + 2.0) / (1.0 + beta))) / ((beta + 3.0) * (beta + 2.0));
	} else {
		tmp = ((alpha + 1.0) / (alpha + (beta + 3.0))) / (alpha + (beta + 2.0));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 1.35e+15:
		tmp = (1.0 / ((beta + 2.0) / (1.0 + beta))) / ((beta + 3.0) * (beta + 2.0))
	else:
		tmp = ((alpha + 1.0) / (alpha + (beta + 3.0))) / (alpha + (beta + 2.0))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.35e+15)
		tmp = Float64(Float64(1.0 / Float64(Float64(beta + 2.0) / Float64(1.0 + beta))) / Float64(Float64(beta + 3.0) * Float64(beta + 2.0)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(alpha + Float64(beta + 3.0))) / Float64(alpha + Float64(beta + 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.35e+15)
		tmp = (1.0 / ((beta + 2.0) / (1.0 + beta))) / ((beta + 3.0) * (beta + 2.0));
	else
		tmp = ((alpha + 1.0) / (alpha + (beta + 3.0))) / (alpha + (beta + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 1.35e+15], N[(N[(1.0 / N[(N[(beta + 2.0), $MachinePrecision] / N[(1.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + 3.0), $MachinePrecision] * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.35e15

   1. Initial program 99.9%

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

      1. associate-/l/ 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+l+ 99.9%

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

      4. *-commutative 99.9%

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

      5. metadata-eval 99.9%

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

      6. associate-+l+ 99.9%

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

      7. metadata-eval 99.9%

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

      8. +-commutative 99.9%

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

      9. +-commutative 99.9%

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

      10. +-commutative 99.9%

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

      11. metadata-eval 99.9%

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

      12. metadata-eval 99.9%

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

      13. associate-+l+ 99.9%

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

   3. Simplified 99.9%

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

      1. clear-num 99.9%

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

      2. inv-pow 99.9%

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

      3. +-commutative 99.9%

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

      4. distribute-rgt1-in 99.9%

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

      5. fma-define 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. unpow-1 99.9%

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

      2. +-commutative 99.9%

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

      3. +-commutative 99.9%

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

      4. +-commutative 99.9%

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

      5. +-commutative 99.9%

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

      6. fma-undefine 99.9%

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

      7. +-commutative 99.9%

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

      8. *-commutative 99.9%

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

      9. +-commutative 99.9%

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

      10. associate-+r+ 99.9%

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

      11. distribute-lft1-in 99.9%

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

      12. +-commutative 99.9%

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

      13. +-commutative 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in alpha around 0 83.2%

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

       1. +-commutative 83.2%

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

   11. Simplified 83.2%

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

   12. Taylor expanded in alpha around 0 66.2%

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

       1. +-commutative 66.2%

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

       2. +-commutative 66.2%

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

   14. Simplified 66.2%

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

   if 1.35e15 < beta

   1. Initial program 79.6%

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

   3. Simplified 59.9%

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

      1. times-frac 88.2%

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

      2. +-commutative 88.2%

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

   6. Applied egg-rr 88.2%

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

      1. associate-*l/ 88.2%

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

      2. associate-+r+ 88.2%

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

      3. associate-/r* 99.8%

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

      4. associate-+r+ 99.8%

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

   8. Applied egg-rr 99.8%

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

      1. associate-+r+ 99.8%

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

      2. associate-*r/ 99.9%

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

      3. associate-+r+ 99.9%

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

      4. +-commutative 99.9%

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

      5. associate-+r+ 99.9%

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

   10. Applied egg-rr 99.9%

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

   11. Taylor expanded in beta around inf 82.9%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.35 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{1}{\frac{\beta + 2}{1 + \beta}}}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\alpha + \left(\beta + 3\right)}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 2}}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\alpha + \left(\beta + 3\right)}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.5e+15)
   (/ (/ (+ 1.0 beta) (+ beta 2.0)) (* (+ beta 3.0) (+ beta 2.0)))
   (/ (/ (+ alpha 1.0) (+ alpha (+ beta 3.0))) (+ alpha (+ beta 2.0)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.5e+15) {
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 3.0) * (beta + 2.0));
	} else {
		tmp = ((alpha + 1.0) / (alpha + (beta + 3.0))) / (alpha + (beta + 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.5d+15) then
        tmp = ((1.0d0 + beta) / (beta + 2.0d0)) / ((beta + 3.0d0) * (beta + 2.0d0))
    else
        tmp = ((alpha + 1.0d0) / (alpha + (beta + 3.0d0))) / (alpha + (beta + 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.5e+15) {
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 3.0) * (beta + 2.0));
	} else {
		tmp = ((alpha + 1.0) / (alpha + (beta + 3.0))) / (alpha + (beta + 2.0));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 1.5e+15:
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 3.0) * (beta + 2.0))
	else:
		tmp = ((alpha + 1.0) / (alpha + (beta + 3.0))) / (alpha + (beta + 2.0))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.5e+15)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(beta + 2.0)) / Float64(Float64(beta + 3.0) * Float64(beta + 2.0)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(alpha + Float64(beta + 3.0))) / Float64(alpha + Float64(beta + 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.5e+15)
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 3.0) * (beta + 2.0));
	else
		tmp = ((alpha + 1.0) / (alpha + (beta + 3.0))) / (alpha + (beta + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 1.5e+15], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + 3.0), $MachinePrecision] * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.5e15

   1. Initial program 99.9%

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

      1. associate-/l/ 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+l+ 99.9%

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

      4. *-commutative 99.9%

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

      5. metadata-eval 99.9%

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

      6. associate-+l+ 99.9%

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

      7. metadata-eval 99.9%

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

      8. +-commutative 99.9%

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

      9. +-commutative 99.9%

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

      10. +-commutative 99.9%

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

      11. metadata-eval 99.9%

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

      12. metadata-eval 99.9%

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

      13. associate-+l+ 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in alpha around 0 83.2%

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

      1. +-commutative 83.2%

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

   7. Simplified 83.2%

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

   8. Taylor expanded in alpha around 0 66.2%

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

      1. +-commutative 66.2%

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

      2. +-commutative 66.2%

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

   10. Simplified 66.2%

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

   if 1.5e15 < beta

   1. Initial program 79.6%

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

   3. Simplified 59.9%

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

      1. times-frac 88.2%

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

      2. +-commutative 88.2%

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

   6. Applied egg-rr 88.2%

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

      1. associate-*l/ 88.2%

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

      2. associate-+r+ 88.2%

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

      3. associate-/r* 99.8%

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

      4. associate-+r+ 99.8%

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

   8. Applied egg-rr 99.8%

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

      1. associate-+r+ 99.8%

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

      2. associate-*r/ 99.9%

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

      3. associate-+r+ 99.9%

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

      4. +-commutative 99.9%

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

      5. associate-+r+ 99.9%

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

   10. Applied egg-rr 99.9%

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

   11. Taylor expanded in beta around inf 82.9%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 2}}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\alpha + \left(\beta + 3\right)}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.2:\\ \;\;\;\;\frac{0.5}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\alpha + \left(\beta + 3\right)}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.2)
   (/ 0.5 (* (+ alpha 2.0) (+ alpha 3.0)))
   (/ (/ (+ alpha 1.0) (+ alpha (+ beta 3.0))) (+ alpha (+ beta 2.0)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.2) {
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	} else {
		tmp = ((alpha + 1.0) / (alpha + (beta + 3.0))) / (alpha + (beta + 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 4.2d0) then
        tmp = 0.5d0 / ((alpha + 2.0d0) * (alpha + 3.0d0))
    else
        tmp = ((alpha + 1.0d0) / (alpha + (beta + 3.0d0))) / (alpha + (beta + 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.2) {
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	} else {
		tmp = ((alpha + 1.0) / (alpha + (beta + 3.0))) / (alpha + (beta + 2.0));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 4.2:
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0))
	else:
		tmp = ((alpha + 1.0) / (alpha + (beta + 3.0))) / (alpha + (beta + 2.0))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.2)
		tmp = Float64(0.5 / Float64(Float64(alpha + 2.0) * Float64(alpha + 3.0)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(alpha + Float64(beta + 3.0))) / Float64(alpha + Float64(beta + 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4.2)
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	else
		tmp = ((alpha + 1.0) / (alpha + (beta + 3.0))) / (alpha + (beta + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 4.2], N[(0.5 / N[(N[(alpha + 2.0), $MachinePrecision] * N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.20000000000000018

   1. Initial program 99.9%

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

      1. associate-/l/ 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+l+ 99.9%

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

      4. *-commutative 99.9%

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

      5. metadata-eval 99.9%

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

      6. associate-+l+ 99.9%

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

      7. metadata-eval 99.9%

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

      8. +-commutative 99.9%

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

      9. +-commutative 99.9%

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

      10. +-commutative 99.9%

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

      11. metadata-eval 99.9%

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

      12. metadata-eval 99.9%

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

      13. associate-+l+ 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in alpha around 0 83.1%

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

      1. +-commutative 83.1%

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

   7. Simplified 83.1%

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

   8. Taylor expanded in beta around 0 82.3%

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

   if 4.20000000000000018 < beta

   1. Initial program 79.9%

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

   3. Simplified 60.4%

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

      1. times-frac 88.3%

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

      2. +-commutative 88.3%

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

   6. Applied egg-rr 88.3%

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

      1. associate-*l/ 88.4%

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

      2. associate-+r+ 88.4%

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

      3. associate-/r* 99.8%

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

      4. associate-+r+ 99.8%

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

   8. Applied egg-rr 99.8%

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

      1. associate-+r+ 99.8%

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

      2. associate-*r/ 99.9%

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

      3. associate-+r+ 99.9%

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

      4. +-commutative 99.9%

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

      5. associate-+r+ 99.9%

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

   10. Applied egg-rr 99.9%

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

   11. Taylor expanded in beta around inf 82.4%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.2:\\ \;\;\;\;\frac{0.5}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\alpha + \left(\beta + 3\right)}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 82.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 7.5:\\ \;\;\;\;\frac{0.5}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 7.5)
   (/ 0.5 (* (+ alpha 2.0) (+ alpha 3.0)))
   (/ (/ (+ alpha 1.0) beta) (+ alpha (+ beta 3.0)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 7.5) {
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	} else {
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 7.5d0) then
        tmp = 0.5d0 / ((alpha + 2.0d0) * (alpha + 3.0d0))
    else
        tmp = ((alpha + 1.0d0) / beta) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 7.5) {
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	} else {
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 7.5:
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0))
	else:
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 3.0))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 7.5)
		tmp = Float64(0.5 / Float64(Float64(alpha + 2.0) * Float64(alpha + 3.0)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 7.5)
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	else
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 7.5], N[(0.5 / N[(N[(alpha + 2.0), $MachinePrecision] * N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 7.5

   1. Initial program 99.9%

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

      1. associate-/l/ 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+l+ 99.9%

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

      4. *-commutative 99.9%

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

      5. metadata-eval 99.9%

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

      6. associate-+l+ 99.9%

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

      7. metadata-eval 99.9%

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

      8. +-commutative 99.9%

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

      9. +-commutative 99.9%

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

      10. +-commutative 99.9%

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

      11. metadata-eval 99.9%

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

      12. metadata-eval 99.9%

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

      13. associate-+l+ 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in alpha around 0 83.1%

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

      1. +-commutative 83.1%

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

   7. Simplified 83.1%

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

   8. Taylor expanded in beta around 0 82.3%

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

   if 7.5 < beta

   1. Initial program 79.9%

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

   3. Taylor expanded in beta around inf 81.8%

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

      1. div-inv 81.7%

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

      2. +-commutative 81.7%

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

      3. metadata-eval 81.7%

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

      4. associate-+l+ 81.7%

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

      5. metadata-eval 81.7%

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

      6. associate-+r+ 81.7%

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

   5. Applied egg-rr 81.7%

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

      1. associate-*r/ 81.8%

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

      2. *-commutative 81.8%

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

      3. *-lft-identity 81.8%

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

      4. +-commutative 81.8%

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

      5. +-commutative 81.8%

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

      6. +-commutative 81.8%

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

      7. +-commutative 81.8%

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

   7. Simplified 81.8%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 7.5:\\ \;\;\;\;\frac{0.5}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 82.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 9.5:\\ \;\;\;\;\frac{0.5}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta + 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 9.5)
   (/ 0.5 (* (+ alpha 2.0) (+ alpha 3.0)))
   (/ (/ (+ alpha 1.0) beta) (+ beta 3.0))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 9.5) {
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	} else {
		tmp = ((alpha + 1.0) / beta) / (beta + 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 9.5d0) then
        tmp = 0.5d0 / ((alpha + 2.0d0) * (alpha + 3.0d0))
    else
        tmp = ((alpha + 1.0d0) / beta) / (beta + 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 9.5) {
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	} else {
		tmp = ((alpha + 1.0) / beta) / (beta + 3.0);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 9.5:
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0))
	else:
		tmp = ((alpha + 1.0) / beta) / (beta + 3.0)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 9.5)
		tmp = Float64(0.5 / Float64(Float64(alpha + 2.0) * Float64(alpha + 3.0)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / Float64(beta + 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 9.5)
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	else
		tmp = ((alpha + 1.0) / beta) / (beta + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 9.5], N[(0.5 / N[(N[(alpha + 2.0), $MachinePrecision] * N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 9.5

   1. Initial program 99.9%

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

      1. associate-/l/ 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+l+ 99.9%

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

      4. *-commutative 99.9%

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

      5. metadata-eval 99.9%

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

      6. associate-+l+ 99.9%

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

      7. metadata-eval 99.9%

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

      8. +-commutative 99.9%

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

      9. +-commutative 99.9%

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

      10. +-commutative 99.9%

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

      11. metadata-eval 99.9%

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

      12. metadata-eval 99.9%

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

      13. associate-+l+ 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in alpha around 0 83.1%

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

      1. +-commutative 83.1%

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

   7. Simplified 83.1%

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

   8. Taylor expanded in beta around 0 82.3%

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

   if 9.5 < beta

   1. Initial program 79.9%

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

   3. Taylor expanded in beta around inf 81.8%

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

   4. Taylor expanded in alpha around 0 81.6%

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

      1. +-commutative 81.6%

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

   6. Simplified 81.6%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 9.5:\\ \;\;\;\;\frac{0.5}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 82.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 42:\\ \;\;\;\;\frac{0.5}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 42.0)
   (/ 0.5 (* (+ alpha 2.0) (+ alpha 3.0)))
   (/ (/ (+ alpha 1.0) beta) beta)))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 42.0) {
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 42.0d0) then
        tmp = 0.5d0 / ((alpha + 2.0d0) * (alpha + 3.0d0))
    else
        tmp = ((alpha + 1.0d0) / beta) / beta
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 42.0) {
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 42.0:
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0))
	else:
		tmp = ((alpha + 1.0) / beta) / beta
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 42.0)
		tmp = Float64(0.5 / Float64(Float64(alpha + 2.0) * Float64(alpha + 3.0)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / beta);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 42.0)
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	else
		tmp = ((alpha + 1.0) / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 42.0], N[(0.5 / N[(N[(alpha + 2.0), $MachinePrecision] * N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 42

   1. Initial program 99.9%

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

      1. associate-/l/ 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+l+ 99.9%

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

      4. *-commutative 99.9%

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

      5. metadata-eval 99.9%

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

      6. associate-+l+ 99.9%

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

      7. metadata-eval 99.9%

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

      8. +-commutative 99.9%

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

      9. +-commutative 99.9%

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

      10. +-commutative 99.9%

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

      11. metadata-eval 99.9%

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

      12. metadata-eval 99.9%

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

      13. associate-+l+ 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in alpha around 0 83.1%

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

      1. +-commutative 83.1%

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

   7. Simplified 83.1%

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

   8. Taylor expanded in beta around 0 82.3%

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

   if 42 < beta

   1. Initial program 79.9%

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

   3. Taylor expanded in beta around inf 81.8%

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

      1. div-inv 81.7%

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

      2. +-commutative 81.7%

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

      3. metadata-eval 81.7%

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

      4. associate-+l+ 81.7%

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

      5. metadata-eval 81.7%

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

      6. associate-+r+ 81.7%

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

   5. Applied egg-rr 81.7%

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

      1. associate-*r/ 81.8%

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

      2. *-commutative 81.8%

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

      3. *-lft-identity 81.8%

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

      4. +-commutative 81.8%

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

      5. +-commutative 81.8%

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

      6. +-commutative 81.8%

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

      7. +-commutative 81.8%

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

   7. Simplified 81.8%

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

   8. Taylor expanded in beta around inf 81.6%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 42:\\ \;\;\;\;\frac{0.5}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 28.8% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.1%

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

3. Taylor expanded in beta around inf 25.7%

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

   1. div-inv 25.6%

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

   2. +-commutative 25.6%

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

   3. metadata-eval 25.6%

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

   4. associate-+l+ 25.6%

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

   5. metadata-eval 25.6%

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

   6. associate-+r+ 25.6%

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

5. Applied egg-rr 25.6%

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

   1. associate-*r/ 25.7%

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

   2. *-commutative 25.7%

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

   3. *-lft-identity 25.7%

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

   4. +-commutative 25.7%

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

   5. +-commutative 25.7%

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

   6. +-commutative 25.7%

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

   7. +-commutative 25.7%

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

7. Simplified 25.7%

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

8. Taylor expanded in beta around inf 26.1%

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

9. Final simplification 26.1%

   \[\leadsto \frac{\frac{\alpha + 1}{\beta}}{\beta} \]
10. Add Preprocessing

Alternative 12: 26.6% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.1%

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

3. Taylor expanded in beta around inf 25.7%

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

4. Taylor expanded in alpha around 0 21.8%

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

   1. associate-/r* 21.9%

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

   2. +-commutative 21.9%

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

6. Simplified 21.9%

   \[\leadsto \color{blue}{\frac{\frac{1}{\beta}}{\beta + 3}} \]
7. Add Preprocessing

Alternative 13: 26.5% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.1%

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

3. Taylor expanded in beta around inf 25.7%

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

4. Taylor expanded in alpha around 0 21.8%

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

5. Final simplification 21.8%

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

Alternative 14: 26.9% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.1%

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

3. Taylor expanded in beta around inf 25.7%

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

4. Taylor expanded in alpha around 0 21.8%

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

5. Taylor expanded in beta around inf 22.2%

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

Alternative 15: 4.2% 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 94.1%

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

3. Taylor expanded in beta around inf 25.7%

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

4. Taylor expanded in alpha around 0 21.8%

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

5. Taylor expanded in beta around 0 4.0%

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

Reproduce

herbie shell --seed 2024180 
(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)))