Octave 3.8, jcobi/3

Percentage Accurate: 94.0% → 97.7%

Time: 1.8s

Alternatives: 7

Speedup: 0.5×

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: 94.0% 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: 97.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot 1\\ t_1 := \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{t\_0}}{t\_0}}{t\_0 + 1}\\ \mathbf{if}\;t\_1 \leq 0.1:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 \cdot 1}{t\_0}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 1.0)))
        (t_1
         (/
          (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) t_0) t_0)
          (+ t_0 1.0))))
   (if (<= t_1 0.1) t_1 (/ (/ (* 2.0 1.0) t_0) t_0))))

C

double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + (2.0 * 1.0);
	double t_1 = (((((alpha + beta) + (beta * alpha)) + 1.0) / t_0) / t_0) / (t_0 + 1.0);
	double tmp;
	if (t_1 <= 0.1) {
		tmp = t_1;
	} else {
		tmp = ((2.0 * 1.0) / t_0) / t_0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (alpha + beta) + (2.0d0 * 1.0d0)
    t_1 = (((((alpha + beta) + (beta * alpha)) + 1.0d0) / t_0) / t_0) / (t_0 + 1.0d0)
    if (t_1 <= 0.1d0) then
        tmp = t_1
    else
        tmp = ((2.0d0 * 1.0d0) / t_0) / t_0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + (2.0 * 1.0);
	double t_1 = (((((alpha + beta) + (beta * alpha)) + 1.0) / t_0) / t_0) / (t_0 + 1.0);
	double tmp;
	if (t_1 <= 0.1) {
		tmp = t_1;
	} else {
		tmp = ((2.0 * 1.0) / t_0) / t_0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (alpha + beta) + (2.0 * 1.0)
	t_1 = (((((alpha + beta) + (beta * alpha)) + 1.0) / t_0) / t_0) / (t_0 + 1.0)
	tmp = 0
	if t_1 <= 0.1:
		tmp = t_1
	else:
		tmp = ((2.0 * 1.0) / t_0) / t_0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * 1.0))
	t_1 = Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) + Float64(beta * alpha)) + 1.0) / t_0) / t_0) / Float64(t_0 + 1.0))
	tmp = 0.0
	if (t_1 <= 0.1)
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(2.0 * 1.0) / t_0) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (alpha + beta) + (2.0 * 1.0);
	t_1 = (((((alpha + beta) + (beta * alpha)) + 1.0) / t_0) / t_0) / (t_0 + 1.0);
	tmp = 0.0;
	if (t_1 <= 0.1)
		tmp = t_1;
	else
		tmp = ((2.0 * 1.0) / t_0) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, 0.1], t$95$1, N[(N[(N[(2.0 * 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (/.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 beta alpha)) #s(literal 1 binary64)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) #s(literal 1 binary64)))) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) #s(literal 1 binary64)))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) #s(literal 1 binary64))) #s(literal 1 binary64))) < 0.10000000000000001

   1. Initial program 99.8%

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

   if 0.10000000000000001 < (/.f64 (/.f64 (/.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 beta alpha)) #s(literal 1 binary64)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) #s(literal 1 binary64)))) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) #s(literal 1 binary64)))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) #s(literal 1 binary64))) #s(literal 1 binary64)))

   1. Initial program 1.6%

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

   3. Taylor expanded in alpha around 0

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

   5. Applied rewrites 1.6%

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

   6. Taylor expanded in alpha around 0

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

   8. Applied rewrites 1.6%

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

   9. Taylor expanded in alpha around 0

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

   11. Applied rewrites 68.1%

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

   12. Taylor expanded in alpha around 0

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

   14. Applied rewrites 68.1%

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

Alternative 2: 61.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot 1\\ t_1 := \frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{t\_0}}{t\_0}\\ \mathbf{if}\;\frac{t\_1}{t\_0 + 1} \leq 0.1:\\ \;\;\;\;\frac{t\_1}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 \cdot 1}{t\_0}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 1.0)))
        (t_1 (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) t_0) t_0)))
   (if (<= (/ t_1 (+ t_0 1.0)) 0.1) (/ t_1 t_0) (/ (/ (* 2.0 1.0) t_0) t_0))))

C

double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + (2.0 * 1.0);
	double t_1 = ((((alpha + beta) + (beta * alpha)) + 1.0) / t_0) / t_0;
	double tmp;
	if ((t_1 / (t_0 + 1.0)) <= 0.1) {
		tmp = t_1 / t_0;
	} else {
		tmp = ((2.0 * 1.0) / t_0) / t_0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (alpha + beta) + (2.0d0 * 1.0d0)
    t_1 = ((((alpha + beta) + (beta * alpha)) + 1.0d0) / t_0) / t_0
    if ((t_1 / (t_0 + 1.0d0)) <= 0.1d0) then
        tmp = t_1 / t_0
    else
        tmp = ((2.0d0 * 1.0d0) / t_0) / t_0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + (2.0 * 1.0);
	double t_1 = ((((alpha + beta) + (beta * alpha)) + 1.0) / t_0) / t_0;
	double tmp;
	if ((t_1 / (t_0 + 1.0)) <= 0.1) {
		tmp = t_1 / t_0;
	} else {
		tmp = ((2.0 * 1.0) / t_0) / t_0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (alpha + beta) + (2.0 * 1.0)
	t_1 = ((((alpha + beta) + (beta * alpha)) + 1.0) / t_0) / t_0
	tmp = 0
	if (t_1 / (t_0 + 1.0)) <= 0.1:
		tmp = t_1 / t_0
	else:
		tmp = ((2.0 * 1.0) / t_0) / t_0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * 1.0))
	t_1 = Float64(Float64(Float64(Float64(Float64(alpha + beta) + Float64(beta * alpha)) + 1.0) / t_0) / t_0)
	tmp = 0.0
	if (Float64(t_1 / Float64(t_0 + 1.0)) <= 0.1)
		tmp = Float64(t_1 / t_0);
	else
		tmp = Float64(Float64(Float64(2.0 * 1.0) / t_0) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (alpha + beta) + (2.0 * 1.0);
	t_1 = ((((alpha + beta) + (beta * alpha)) + 1.0) / t_0) / t_0;
	tmp = 0.0;
	if ((t_1 / (t_0 + 1.0)) <= 0.1)
		tmp = t_1 / t_0;
	else
		tmp = ((2.0 * 1.0) / t_0) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] + N[(beta * alpha), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[N[(t$95$1 / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision], 0.1], N[(t$95$1 / t$95$0), $MachinePrecision], N[(N[(N[(2.0 * 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (/.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 beta alpha)) #s(literal 1 binary64)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) #s(literal 1 binary64)))) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) #s(literal 1 binary64)))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) #s(literal 1 binary64))) #s(literal 1 binary64))) < 0.10000000000000001

   1. Initial program 99.8%

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

   3. Taylor expanded in alpha around 0

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

   5. Applied rewrites 11.6%

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

   6. Taylor expanded in alpha around 0

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

   8. Applied rewrites 63.5%

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

   if 0.10000000000000001 < (/.f64 (/.f64 (/.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 beta alpha)) #s(literal 1 binary64)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) #s(literal 1 binary64)))) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) #s(literal 1 binary64)))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) #s(literal 1 binary64))) #s(literal 1 binary64)))

   1. Initial program 1.6%

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

   3. Taylor expanded in alpha around 0

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

   5. Applied rewrites 1.6%

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

   6. Taylor expanded in alpha around 0

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

   8. Applied rewrites 1.6%

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

   9. Taylor expanded in alpha around 0

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

   11. Applied rewrites 68.1%

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

   12. Taylor expanded in alpha around 0

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

   14. Applied rewrites 68.1%

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

Alternative 3: 56.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + (2.0 * 1.0);
	return (((t_0 + 1.0) / t_0) / t_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 * 1.0d0)
    code = (((t_0 + 1.0d0) / t_0) / t_0) / t_0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 93.3%

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

3. Taylor expanded in alpha around 0

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

5. Applied rewrites 10.9%

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

6. Taylor expanded in alpha around 0

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

8. Applied rewrites 59.4%

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

9. Taylor expanded in alpha around 0

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

11. Applied rewrites 58.7%

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

Alternative 4: 39.3% accurate, 1.7× speedup

Math

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

FPCore

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

C

double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + (2.0 * 1.0);
	return ((2.0 * 1.0) / t_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 * 1.0d0)
    code = ((2.0d0 * 1.0d0) / t_0) / t_0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 93.3%

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

3. Taylor expanded in alpha around 0

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

5. Applied rewrites 10.9%

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

6. Taylor expanded in alpha around 0

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

8. Applied rewrites 59.4%

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

9. Taylor expanded in alpha around 0

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

11. Applied rewrites 58.7%

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

12. Taylor expanded in alpha around 0

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

14. Applied rewrites 39.8%

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

Alternative 5: 10.9% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.3%

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

3. Taylor expanded in alpha around 0

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

5. Applied rewrites 10.9%

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

6. Taylor expanded in alpha around 0

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

8. Applied rewrites 10.8%

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

Alternative 6: 9.0% accurate, 14.0× speedup

Math

\[\begin{array}{l} \\ 2 \cdot 1 \end{array} \]

FPCore

(FPCore (alpha beta) :precision binary64 (* 2.0 1.0))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[alpha_, beta_] := N[(2.0 * 1.0), $MachinePrecision]

Derivation

1. Initial program 93.3%

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

3. Taylor expanded in alpha around 0

   \[\leadsto \color{blue}{\alpha \cdot \left(\left(-1 \cdot \left(\alpha \cdot \left(\frac{\left(1 + \beta\right) \cdot \left(7 + \left(\beta + 2 \cdot \beta\right)\right)}{{\left(2 + \beta\right)}^{4} \cdot {\left(3 + \beta\right)}^{2}} + \frac{\left(\left(3 + \beta\right) \cdot \left(4 + 2 \cdot \beta\right) + {\left(2 + \beta\right)}^{2}\right) \cdot \left(\left(\frac{1}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)} + \frac{\beta}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)}\right) - \frac{\left(1 + \beta\right) \cdot \left(\left(3 + \beta\right) \cdot \left(4 + 2 \cdot \beta\right) + {\left(2 + \beta\right)}^{2}\right)}{{\left(2 + \beta\right)}^{4} \cdot {\left(3 + \beta\right)}^{2}}\right)}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)}\right)\right) + \left(\frac{1}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)} + \frac{\beta}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)}\right)\right) - \frac{\left(1 + \beta\right) \cdot \left(\left(3 + \beta\right) \cdot \left(4 + 2 \cdot \beta\right) + {\left(2 + \beta\right)}^{2}\right)}{{\left(2 + \beta\right)}^{4} \cdot {\left(3 + \beta\right)}^{2}}\right) + \left(\frac{1}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)} + \frac{\beta}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)}\right)} \]

5. Applied rewrites 9.2%

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

6. Taylor expanded in alpha around 0

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

8. Applied rewrites 8.8%

   \[\leadsto 2 \cdot \color{blue}{1} \]
9. Add Preprocessing

Alternative 7: 3.6% accurate, 21.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.3%

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

3. Taylor expanded in alpha around 0

   \[\leadsto \color{blue}{\alpha \cdot \left(\left(\alpha \cdot \left(-1 \cdot \left(\alpha \cdot \left(-1 \cdot \frac{\left(\left(3 + \beta\right) \cdot \left(4 + 2 \cdot \beta\right) + {\left(2 + \beta\right)}^{2}\right) \cdot \left(\frac{\left(1 + \beta\right) \cdot \left(7 + \left(\beta + 2 \cdot \beta\right)\right)}{{\left(2 + \beta\right)}^{4} \cdot {\left(3 + \beta\right)}^{2}} + \frac{\left(\left(3 + \beta\right) \cdot \left(4 + 2 \cdot \beta\right) + {\left(2 + \beta\right)}^{2}\right) \cdot \left(\left(\frac{1}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)} + \frac{\beta}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)}\right) - \frac{\left(1 + \beta\right) \cdot \left(\left(3 + \beta\right) \cdot \left(4 + 2 \cdot \beta\right) + {\left(2 + \beta\right)}^{2}\right)}{{\left(2 + \beta\right)}^{4} \cdot {\left(3 + \beta\right)}^{2}}\right)}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)}\right)}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)} + \left(\frac{1}{{\left(2 + \beta\right)}^{4} \cdot {\left(3 + \beta\right)}^{2}} + \left(\frac{\beta}{{\left(2 + \beta\right)}^{4} \cdot {\left(3 + \beta\right)}^{2}} + \frac{\left(7 + \left(\beta + 2 \cdot \beta\right)\right) \cdot \left(\left(\frac{1}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)} + \frac{\beta}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)}\right) - \frac{\left(1 + \beta\right) \cdot \left(\left(3 + \beta\right) \cdot \left(4 + 2 \cdot \beta\right) + {\left(2 + \beta\right)}^{2}\right)}{{\left(2 + \beta\right)}^{4} \cdot {\left(3 + \beta\right)}^{2}}\right)}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)}\right)\right)\right)\right) - \left(\frac{\left(1 + \beta\right) \cdot \left(7 + \left(\beta + 2 \cdot \beta\right)\right)}{{\left(2 + \beta\right)}^{4} \cdot {\left(3 + \beta\right)}^{2}} + \frac{\left(\left(3 + \beta\right) \cdot \left(4 + 2 \cdot \beta\right) + {\left(2 + \beta\right)}^{2}\right) \cdot \left(\left(\frac{1}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)} + \frac{\beta}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)}\right) - \frac{\left(1 + \beta\right) \cdot \left(\left(3 + \beta\right) \cdot \left(4 + 2 \cdot \beta\right) + {\left(2 + \beta\right)}^{2}\right)}{{\left(2 + \beta\right)}^{4} \cdot {\left(3 + \beta\right)}^{2}}\right)}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)}\right)\right) + \left(\frac{1}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)} + \frac{\beta}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)}\right)\right) - \frac{\left(1 + \beta\right) \cdot \left(\left(3 + \beta\right) \cdot \left(4 + 2 \cdot \beta\right) + {\left(2 + \beta\right)}^{2}\right)}{{\left(2 + \beta\right)}^{4} \cdot {\left(3 + \beta\right)}^{2}}\right) + \left(\frac{1}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)} + \frac{\beta}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)}\right)} \]

5. Applied rewrites 8.3%

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

6. Taylor expanded in alpha around 0

   \[\leadsto \color{blue}{\alpha \cdot \left(\left(\alpha \cdot \left(-1 \cdot \left(\alpha \cdot \left(-1 \cdot \frac{\left(\left(3 + \beta\right) \cdot \left(4 + 2 \cdot \beta\right) + {\left(2 + \beta\right)}^{2}\right) \cdot \left(\frac{\left(1 + \beta\right) \cdot \left(7 + \left(\beta + 2 \cdot \beta\right)\right)}{{\left(2 + \beta\right)}^{4} \cdot {\left(3 + \beta\right)}^{2}} + \frac{\left(\left(3 + \beta\right) \cdot \left(4 + 2 \cdot \beta\right) + {\left(2 + \beta\right)}^{2}\right) \cdot \left(\left(\frac{1}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)} + \frac{\beta}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)}\right) - \frac{\left(1 + \beta\right) \cdot \left(\left(3 + \beta\right) \cdot \left(4 + 2 \cdot \beta\right) + {\left(2 + \beta\right)}^{2}\right)}{{\left(2 + \beta\right)}^{4} \cdot {\left(3 + \beta\right)}^{2}}\right)}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)}\right)}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)} + \left(\frac{1}{{\left(2 + \beta\right)}^{4} \cdot {\left(3 + \beta\right)}^{2}} + \left(\frac{\beta}{{\left(2 + \beta\right)}^{4} \cdot {\left(3 + \beta\right)}^{2}} + \frac{\left(7 + \left(\beta + 2 \cdot \beta\right)\right) \cdot \left(\left(\frac{1}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)} + \frac{\beta}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)}\right) - \frac{\left(1 + \beta\right) \cdot \left(\left(3 + \beta\right) \cdot \left(4 + 2 \cdot \beta\right) + {\left(2 + \beta\right)}^{2}\right)}{{\left(2 + \beta\right)}^{4} \cdot {\left(3 + \beta\right)}^{2}}\right)}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)}\right)\right)\right)\right) - \left(\frac{\left(1 + \beta\right) \cdot \left(7 + \left(\beta + 2 \cdot \beta\right)\right)}{{\left(2 + \beta\right)}^{4} \cdot {\left(3 + \beta\right)}^{2}} + \frac{\left(\left(3 + \beta\right) \cdot \left(4 + 2 \cdot \beta\right) + {\left(2 + \beta\right)}^{2}\right) \cdot \left(\left(\frac{1}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)} + \frac{\beta}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)}\right) - \frac{\left(1 + \beta\right) \cdot \left(\left(3 + \beta\right) \cdot \left(4 + 2 \cdot \beta\right) + {\left(2 + \beta\right)}^{2}\right)}{{\left(2 + \beta\right)}^{4} \cdot {\left(3 + \beta\right)}^{2}}\right)}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)}\right)\right) + \left(\frac{1}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)} + \frac{\beta}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)}\right)\right) - \frac{\left(1 + \beta\right) \cdot \left(\left(3 + \beta\right) \cdot \left(4 + 2 \cdot \beta\right) + {\left(2 + \beta\right)}^{2}\right)}{{\left(2 + \beta\right)}^{4} \cdot {\left(3 + \beta\right)}^{2}}\right) + \left(\frac{1}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)} + \frac{\beta}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)}\right)} \]

8. Applied rewrites 3.7%

   \[\leadsto \color{blue}{\alpha + \beta} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024321 
(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)))