Octave 3.8, jcobi/3

Percentage Accurate: 94.6% → 99.8%

Time: 18.9s

Alternatives: 10

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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.8%

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

3. Simplified 84.4%

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

   1. times-frac 96.0%

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

   2. +-commutative 96.0%

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

6. Applied egg-rr 96.0%

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

   1. associate-*r/ 96.0%

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

   2. +-commutative 96.0%

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

   3. +-commutative 96.0%

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

   4. +-commutative 96.0%

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

   5. +-commutative 96.0%

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

   6. +-commutative 96.0%

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

   7. +-commutative 96.0%

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

   8. +-commutative 96.0%

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

   9. +-commutative 96.0%

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

   10. +-commutative 96.0%

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

   11. +-commutative 96.0%

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

8. Simplified 96.0%

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

   1. associate-*l/ 92.7%

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

   2. +-commutative 92.7%

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

10. Applied egg-rr 92.7%

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

    1. associate-/l* 96.0%

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

    2. +-commutative 96.0%

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

    3. *-commutative 96.0%

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

    4. times-frac 99.8%

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

    5. associate-+l+ 99.8%

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

    6. associate-+l+ 99.8%

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

    7. associate-+l+ 99.8%

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

12. Applied egg-rr 99.8%

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

13. Final simplification 99.8%

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

Alternative 2: 90.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ t_1 := \beta + \left(\alpha + 3\right)\\ \mathbf{if}\;\beta \leq 5.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{\alpha + \left(1 + \beta\right)}{t\_1 \cdot \left(t\_0 \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{t\_1} \cdot \frac{1 - \frac{3 + \alpha \cdot 2}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))) (t_1 (+ beta (+ alpha 3.0))))
   (if (<= beta 5.5e+15)
     (/ (+ alpha (+ 1.0 beta)) (* t_1 (* t_0 t_0)))
     (*
      (/ (+ 1.0 alpha) t_1)
      (/ (- 1.0 (/ (+ 3.0 (* alpha 2.0)) beta)) beta)))))

C

double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double t_1 = beta + (alpha + 3.0);
	double tmp;
	if (beta <= 5.5e+15) {
		tmp = (alpha + (1.0 + beta)) / (t_1 * (t_0 * t_0));
	} else {
		tmp = ((1.0 + alpha) / t_1) * ((1.0 - ((3.0 + (alpha * 2.0)) / beta)) / beta);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    t_1 = beta + (alpha + 3.0d0)
    if (beta <= 5.5d+15) then
        tmp = (alpha + (1.0d0 + beta)) / (t_1 * (t_0 * t_0))
    else
        tmp = ((1.0d0 + alpha) / t_1) * ((1.0d0 - ((3.0d0 + (alpha * 2.0d0)) / beta)) / beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double t_1 = beta + (alpha + 3.0);
	double tmp;
	if (beta <= 5.5e+15) {
		tmp = (alpha + (1.0 + beta)) / (t_1 * (t_0 * t_0));
	} else {
		tmp = ((1.0 + alpha) / t_1) * ((1.0 - ((3.0 + (alpha * 2.0)) / beta)) / beta);
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	t_1 = beta + (alpha + 3.0)
	tmp = 0
	if beta <= 5.5e+15:
		tmp = (alpha + (1.0 + beta)) / (t_1 * (t_0 * t_0))
	else:
		tmp = ((1.0 + alpha) / t_1) * ((1.0 - ((3.0 + (alpha * 2.0)) / beta)) / beta)
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	t_1 = Float64(beta + Float64(alpha + 3.0))
	tmp = 0.0
	if (beta <= 5.5e+15)
		tmp = Float64(Float64(alpha + Float64(1.0 + beta)) / Float64(t_1 * Float64(t_0 * t_0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / t_1) * Float64(Float64(1.0 - Float64(Float64(3.0 + Float64(alpha * 2.0)) / beta)) / beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	t_1 = beta + (alpha + 3.0);
	tmp = 0.0;
	if (beta <= 5.5e+15)
		tmp = (alpha + (1.0 + beta)) / (t_1 * (t_0 * t_0));
	else
		tmp = ((1.0 + alpha) / t_1) * ((1.0 - ((3.0 + (alpha * 2.0)) / beta)) / beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 5.5e+15], N[(N[(alpha + N[(1.0 + beta), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$1), $MachinePrecision] * N[(N[(1.0 - N[(N[(3.0 + N[(alpha * 2.0), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.5e15

   1. Initial program 99.8%

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

   3. Simplified 92.9%

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

   5. Taylor expanded in alpha around 0 92.2%

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

   if 5.5e15 < beta

   1. Initial program 83.8%

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

   3. Simplified 65.7%

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

      1. times-frac 87.6%

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

      2. +-commutative 87.6%

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

   6. Applied egg-rr 87.6%

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

      1. associate-*r/ 87.6%

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

      2. +-commutative 87.6%

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

      3. +-commutative 87.6%

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

      4. +-commutative 87.6%

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

      5. +-commutative 87.6%

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

      6. +-commutative 87.6%

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

      7. +-commutative 87.6%

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

      8. +-commutative 87.6%

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

      9. +-commutative 87.6%

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

      10. +-commutative 87.6%

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

      11. +-commutative 87.6%

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

   8. Simplified 87.6%

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

      1. associate-*l/ 77.3%

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

      2. +-commutative 77.3%

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

   10. Applied egg-rr 77.3%

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

       1. associate-/l* 87.6%

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

       2. +-commutative 87.6%

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

       3. *-commutative 87.6%

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

       4. times-frac 99.7%

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

       5. associate-+l+ 99.7%

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

       6. associate-+l+ 99.7%

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

       7. associate-+l+ 99.7%

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

   12. Applied egg-rr 99.7%

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

   13. Taylor expanded in beta around inf 81.4%

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

       1. mul-1-neg 81.4%

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

       2. unsub-neg 81.4%

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

   15. Simplified 81.4%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{\alpha + \left(1 + \beta\right)}{\left(\beta + \left(\alpha + 3\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta + \left(\alpha + 3\right)} \cdot \frac{1 - \frac{3 + \alpha \cdot 2}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4600000000000:\\ \;\;\;\;\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(6 + \beta \cdot \left(\beta + 5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta + \left(\alpha + 3\right)} \cdot \frac{1 - \frac{3 + \alpha \cdot 2}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4600000000000.0)
   (/ (+ 1.0 beta) (* (+ beta 2.0) (+ 6.0 (* beta (+ beta 5.0)))))
   (*
    (/ (+ 1.0 alpha) (+ beta (+ alpha 3.0)))
    (/ (- 1.0 (/ (+ 3.0 (* alpha 2.0)) beta)) beta))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4600000000000.0) {
		tmp = (1.0 + beta) / ((beta + 2.0) * (6.0 + (beta * (beta + 5.0))));
	} else {
		tmp = ((1.0 + alpha) / (beta + (alpha + 3.0))) * ((1.0 - ((3.0 + (alpha * 2.0)) / beta)) / beta);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 4600000000000.0d0) then
        tmp = (1.0d0 + beta) / ((beta + 2.0d0) * (6.0d0 + (beta * (beta + 5.0d0))))
    else
        tmp = ((1.0d0 + alpha) / (beta + (alpha + 3.0d0))) * ((1.0d0 - ((3.0d0 + (alpha * 2.0d0)) / beta)) / beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4600000000000.0) {
		tmp = (1.0 + beta) / ((beta + 2.0) * (6.0 + (beta * (beta + 5.0))));
	} else {
		tmp = ((1.0 + alpha) / (beta + (alpha + 3.0))) * ((1.0 - ((3.0 + (alpha * 2.0)) / beta)) / beta);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 4600000000000.0:
		tmp = (1.0 + beta) / ((beta + 2.0) * (6.0 + (beta * (beta + 5.0))))
	else:
		tmp = ((1.0 + alpha) / (beta + (alpha + 3.0))) * ((1.0 - ((3.0 + (alpha * 2.0)) / beta)) / beta)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4600000000000.0)
		tmp = Float64(Float64(1.0 + beta) / Float64(Float64(beta + 2.0) * Float64(6.0 + Float64(beta * Float64(beta + 5.0)))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(beta + Float64(alpha + 3.0))) * Float64(Float64(1.0 - Float64(Float64(3.0 + Float64(alpha * 2.0)) / beta)) / beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4600000000000.0)
		tmp = (1.0 + beta) / ((beta + 2.0) * (6.0 + (beta * (beta + 5.0))));
	else
		tmp = ((1.0 + alpha) / (beta + (alpha + 3.0))) * ((1.0 - ((3.0 + (alpha * 2.0)) / beta)) / beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 4600000000000.0], N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(6.0 + N[(beta * N[(beta + 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[(N[(3.0 + N[(alpha * 2.0), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.6e12

   1. Initial program 99.8%

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

   3. Simplified 92.9%

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

   5. Taylor expanded in beta around 0 92.9%

      \[\leadsto \frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \color{blue}{\left(\beta \cdot \left(5 + \left(\beta + 2 \cdot \alpha\right)\right) + \left(2 + \alpha\right) \cdot \left(3 + \alpha\right)\right)}} \]

   6. Taylor expanded in alpha around 0 69.1%

      \[\leadsto \color{blue}{\frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(6 + \beta \cdot \left(5 + \beta\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative 69.1%

         \[\leadsto \frac{1 + \beta}{\color{blue}{\left(\beta + 2\right)} \cdot \left(6 + \beta \cdot \left(5 + \beta\right)\right)} \]

      2. +-commutative 69.1%

         \[\leadsto \frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(6 + \beta \cdot \color{blue}{\left(\beta + 5\right)}\right)} \]

   8. Simplified 69.1%

      \[\leadsto \color{blue}{\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(6 + \beta \cdot \left(\beta + 5\right)\right)}} \]

   if 4.6e12 < beta

   1. Initial program 84.0%

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

   3. Simplified 66.1%

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

      1. times-frac 87.8%

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

      2. +-commutative 87.8%

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

   6. Applied egg-rr 87.8%

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

      1. associate-*r/ 87.8%

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

      2. +-commutative 87.8%

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

      3. +-commutative 87.8%

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

      4. +-commutative 87.8%

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

      5. +-commutative 87.8%

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

      6. +-commutative 87.8%

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

      7. +-commutative 87.8%

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

      8. +-commutative 87.8%

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

      9. +-commutative 87.8%

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

      10. +-commutative 87.8%

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

      11. +-commutative 87.8%

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

   8. Simplified 87.8%

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

      1. associate-*l/ 77.6%

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

      2. +-commutative 77.6%

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

   10. Applied egg-rr 77.6%

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

       1. associate-/l* 87.8%

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

       2. +-commutative 87.8%

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

       3. *-commutative 87.8%

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

       4. times-frac 99.7%

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

       5. associate-+l+ 99.7%

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

       6. associate-+l+ 99.7%

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

       7. associate-+l+ 99.7%

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

   12. Applied egg-rr 99.7%

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

   13. Taylor expanded in beta around inf 81.6%

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

       1. mul-1-neg 81.6%

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

       2. unsub-neg 81.6%

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

   15. Simplified 81.6%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4600000000000:\\ \;\;\;\;\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(6 + \beta \cdot \left(\beta + 5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta + \left(\alpha + 3\right)} \cdot \frac{1 - \frac{3 + \alpha \cdot 2}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.25 \cdot 10^{+15}:\\ \;\;\;\;\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(6 + \beta \cdot \left(\beta + 5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta + \left(\alpha + 3\right)} \cdot \frac{1}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.25e+15)
   (/ (+ 1.0 beta) (* (+ beta 2.0) (+ 6.0 (* beta (+ beta 5.0)))))
   (* (/ (+ 1.0 alpha) (+ beta (+ alpha 3.0))) (/ 1.0 beta))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.25e+15) {
		tmp = (1.0 + beta) / ((beta + 2.0) * (6.0 + (beta * (beta + 5.0))));
	} else {
		tmp = ((1.0 + alpha) / (beta + (alpha + 3.0))) * (1.0 / beta);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 3.25d+15) then
        tmp = (1.0d0 + beta) / ((beta + 2.0d0) * (6.0d0 + (beta * (beta + 5.0d0))))
    else
        tmp = ((1.0d0 + alpha) / (beta + (alpha + 3.0d0))) * (1.0d0 / beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.25e+15) {
		tmp = (1.0 + beta) / ((beta + 2.0) * (6.0 + (beta * (beta + 5.0))));
	} else {
		tmp = ((1.0 + alpha) / (beta + (alpha + 3.0))) * (1.0 / beta);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 3.25e+15:
		tmp = (1.0 + beta) / ((beta + 2.0) * (6.0 + (beta * (beta + 5.0))))
	else:
		tmp = ((1.0 + alpha) / (beta + (alpha + 3.0))) * (1.0 / beta)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.25e+15)
		tmp = Float64(Float64(1.0 + beta) / Float64(Float64(beta + 2.0) * Float64(6.0 + Float64(beta * Float64(beta + 5.0)))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(beta + Float64(alpha + 3.0))) * Float64(1.0 / beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.25e+15)
		tmp = (1.0 + beta) / ((beta + 2.0) * (6.0 + (beta * (beta + 5.0))));
	else
		tmp = ((1.0 + alpha) / (beta + (alpha + 3.0))) * (1.0 / beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 3.25e+15], N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(6.0 + N[(beta * N[(beta + 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.25e15

   1. Initial program 99.8%

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

   3. Simplified 92.9%

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

   5. Taylor expanded in beta around 0 92.9%

      \[\leadsto \frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \color{blue}{\left(\beta \cdot \left(5 + \left(\beta + 2 \cdot \alpha\right)\right) + \left(2 + \alpha\right) \cdot \left(3 + \alpha\right)\right)}} \]

   6. Taylor expanded in alpha around 0 69.2%

      \[\leadsto \color{blue}{\frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(6 + \beta \cdot \left(5 + \beta\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative 69.2%

         \[\leadsto \frac{1 + \beta}{\color{blue}{\left(\beta + 2\right)} \cdot \left(6 + \beta \cdot \left(5 + \beta\right)\right)} \]

      2. +-commutative 69.2%

         \[\leadsto \frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(6 + \beta \cdot \color{blue}{\left(\beta + 5\right)}\right)} \]

   8. Simplified 69.2%

      \[\leadsto \color{blue}{\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(6 + \beta \cdot \left(\beta + 5\right)\right)}} \]

   if 3.25e15 < beta

   1. Initial program 83.8%

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

   3. Simplified 65.7%

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

      1. times-frac 87.6%

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

      2. +-commutative 87.6%

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

   6. Applied egg-rr 87.6%

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

      1. associate-*r/ 87.6%

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

      2. +-commutative 87.6%

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

      3. +-commutative 87.6%

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

      4. +-commutative 87.6%

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

      5. +-commutative 87.6%

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

      6. +-commutative 87.6%

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

      7. +-commutative 87.6%

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

      8. +-commutative 87.6%

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

      9. +-commutative 87.6%

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

      10. +-commutative 87.6%

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

      11. +-commutative 87.6%

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

   8. Simplified 87.6%

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

      1. associate-*l/ 77.3%

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

      2. +-commutative 77.3%

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

   10. Applied egg-rr 77.3%

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

       1. associate-/l* 87.6%

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

       2. +-commutative 87.6%

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

       3. *-commutative 87.6%

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

       4. times-frac 99.7%

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

       5. associate-+l+ 99.7%

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

       6. associate-+l+ 99.7%

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

       7. associate-+l+ 99.7%

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

   12. Applied egg-rr 99.7%

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

   13. Taylor expanded in beta around inf 81.0%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.25 \cdot 10^{+15}:\\ \;\;\;\;\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(6 + \beta \cdot \left(\beta + 5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta + \left(\alpha + 3\right)} \cdot \frac{1}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.75 \cdot 10^{+15}:\\ \;\;\;\;\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(6 + \beta \cdot \left(\beta + 5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.75e+15)
   (/ (+ 1.0 beta) (* (+ beta 2.0) (+ 6.0 (* beta (+ beta 5.0)))))
   (/ (/ (+ 1.0 alpha) beta) (+ 1.0 (+ 2.0 (+ alpha beta))))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.75e+15) {
		tmp = (1.0 + beta) / ((beta + 2.0) * (6.0 + (beta * (beta + 5.0))));
	} else {
		tmp = ((1.0 + alpha) / beta) / (1.0 + (2.0 + (alpha + beta)));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.75d+15) then
        tmp = (1.0d0 + beta) / ((beta + 2.0d0) * (6.0d0 + (beta * (beta + 5.0d0))))
    else
        tmp = ((1.0d0 + alpha) / beta) / (1.0d0 + (2.0d0 + (alpha + beta)))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.75e+15) {
		tmp = (1.0 + beta) / ((beta + 2.0) * (6.0 + (beta * (beta + 5.0))));
	} else {
		tmp = ((1.0 + alpha) / beta) / (1.0 + (2.0 + (alpha + beta)));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 1.75e+15:
		tmp = (1.0 + beta) / ((beta + 2.0) * (6.0 + (beta * (beta + 5.0))))
	else:
		tmp = ((1.0 + alpha) / beta) / (1.0 + (2.0 + (alpha + beta)))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.75e+15)
		tmp = Float64(Float64(1.0 + beta) / Float64(Float64(beta + 2.0) * Float64(6.0 + Float64(beta * Float64(beta + 5.0)))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(1.0 + Float64(2.0 + Float64(alpha + beta))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.75e+15)
		tmp = (1.0 + beta) / ((beta + 2.0) * (6.0 + (beta * (beta + 5.0))));
	else
		tmp = ((1.0 + alpha) / beta) / (1.0 + (2.0 + (alpha + beta)));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 1.75e+15], N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(6.0 + N[(beta * N[(beta + 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(1.0 + N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.75e15

   1. Initial program 99.8%

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

   3. Simplified 92.9%

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

   5. Taylor expanded in beta around 0 92.9%

      \[\leadsto \frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \color{blue}{\left(\beta \cdot \left(5 + \left(\beta + 2 \cdot \alpha\right)\right) + \left(2 + \alpha\right) \cdot \left(3 + \alpha\right)\right)}} \]

   6. Taylor expanded in alpha around 0 69.2%

      \[\leadsto \color{blue}{\frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(6 + \beta \cdot \left(5 + \beta\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative 69.2%

         \[\leadsto \frac{1 + \beta}{\color{blue}{\left(\beta + 2\right)} \cdot \left(6 + \beta \cdot \left(5 + \beta\right)\right)} \]

      2. +-commutative 69.2%

         \[\leadsto \frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(6 + \beta \cdot \color{blue}{\left(\beta + 5\right)}\right)} \]

   8. Simplified 69.2%

      \[\leadsto \color{blue}{\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(6 + \beta \cdot \left(\beta + 5\right)\right)}} \]

   if 1.75e15 < beta

   1. Initial program 83.8%

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

   3. Taylor expanded in beta around inf 81.1%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.75 \cdot 10^{+15}:\\ \;\;\;\;\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(6 + \beta \cdot \left(\beta + 5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.6:\\ \;\;\;\;\frac{1 + \beta}{12 + \beta \cdot 16}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta + \left(\alpha + 3\right)} \cdot \frac{1}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.6)
   (/ (+ 1.0 beta) (+ 12.0 (* beta 16.0)))
   (* (/ (+ 1.0 alpha) (+ beta (+ alpha 3.0))) (/ 1.0 beta))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.6) {
		tmp = (1.0 + beta) / (12.0 + (beta * 16.0));
	} else {
		tmp = ((1.0 + alpha) / (beta + (alpha + 3.0))) * (1.0 / beta);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.6d0) then
        tmp = (1.0d0 + beta) / (12.0d0 + (beta * 16.0d0))
    else
        tmp = ((1.0d0 + alpha) / (beta + (alpha + 3.0d0))) * (1.0d0 / beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.6) {
		tmp = (1.0 + beta) / (12.0 + (beta * 16.0));
	} else {
		tmp = ((1.0 + alpha) / (beta + (alpha + 3.0))) * (1.0 / beta);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.6:
		tmp = (1.0 + beta) / (12.0 + (beta * 16.0))
	else:
		tmp = ((1.0 + alpha) / (beta + (alpha + 3.0))) * (1.0 / beta)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.6)
		tmp = Float64(Float64(1.0 + beta) / Float64(12.0 + Float64(beta * 16.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(beta + Float64(alpha + 3.0))) * Float64(1.0 / beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.6)
		tmp = (1.0 + beta) / (12.0 + (beta * 16.0));
	else
		tmp = ((1.0 + alpha) / (beta + (alpha + 3.0))) * (1.0 / beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.6], N[(N[(1.0 + beta), $MachinePrecision] / N[(12.0 + N[(beta * 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.60000000000000009

   1. Initial program 99.8%

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

   3. Simplified 93.3%

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

   5. Taylor expanded in alpha around 0 69.7%

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

      1. +-commutative 69.7%

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

      2. +-commutative 69.7%

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

   7. Simplified 69.7%

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

   8. Taylor expanded in beta around 0 68.6%

      \[\leadsto \frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\color{blue}{6 \cdot \left(2 + \alpha\right) + \beta \cdot \left(2 \cdot \left(5 + \alpha\right) + 3 \cdot \left(2 + \alpha\right)\right)}} \]

   9. Taylor expanded in alpha around 0 68.7%

      \[\leadsto \color{blue}{\frac{1 + \beta}{12 + 16 \cdot \beta}} \]
   10. Step-by-step derivation

       1. *-commutative 68.7%

          \[\leadsto \frac{1 + \beta}{12 + \color{blue}{\beta \cdot 16}} \]

   11. Simplified 68.7%

       \[\leadsto \color{blue}{\frac{1 + \beta}{12 + \beta \cdot 16}} \]

   if 2.60000000000000009 < beta

   1. Initial program 84.4%

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

   3. Simplified 65.8%

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

      1. times-frac 88.1%

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

      2. +-commutative 88.1%

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

   6. Applied egg-rr 88.1%

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

      1. associate-*r/ 88.1%

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

      2. +-commutative 88.1%

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

      3. +-commutative 88.1%

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

      4. +-commutative 88.1%

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

      5. +-commutative 88.1%

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

      6. +-commutative 88.1%

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

      7. +-commutative 88.1%

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

      8. +-commutative 88.1%

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

      9. +-commutative 88.1%

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

      10. +-commutative 88.1%

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

      11. +-commutative 88.1%

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

   8. Simplified 88.1%

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

      1. associate-*l/ 78.1%

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

      2. +-commutative 78.1%

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

   10. Applied egg-rr 78.1%

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

       1. associate-/l* 88.1%

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

       2. +-commutative 88.1%

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

       3. *-commutative 88.1%

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

       4. times-frac 99.7%

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

       5. associate-+l+ 99.7%

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

       6. associate-+l+ 99.7%

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

       7. associate-+l+ 99.7%

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

   12. Applied egg-rr 99.7%

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

   13. Taylor expanded in beta around inf 79.3%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.6:\\ \;\;\;\;\frac{1 + \beta}{12 + \beta \cdot 16}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta + \left(\alpha + 3\right)} \cdot \frac{1}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.0) 0.08333333333333333 (/ 1.0 (* (+ beta 2.0) (+ beta 3.0)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.0) {
		tmp = 0.08333333333333333;
	} else {
		tmp = 1.0 / ((beta + 2.0) * (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 <= 1.0d0) then
        tmp = 0.08333333333333333d0
    else
        tmp = 1.0d0 / ((beta + 2.0d0) * (beta + 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.0) {
		tmp = 0.08333333333333333;
	} else {
		tmp = 1.0 / ((beta + 2.0) * (beta + 3.0));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 1.0:
		tmp = 0.08333333333333333
	else:
		tmp = 1.0 / ((beta + 2.0) * (beta + 3.0))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.0)
		tmp = 0.08333333333333333;
	else
		tmp = Float64(1.0 / Float64(Float64(beta + 2.0) * Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.0)
		tmp = 0.08333333333333333;
	else
		tmp = 1.0 / ((beta + 2.0) * (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 1.0], 0.08333333333333333, N[(1.0 / N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1

   1. Initial program 99.8%

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

   3. Simplified 93.3%

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

   5. Taylor expanded in alpha around 0 69.7%

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

      1. +-commutative 69.7%

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

      2. +-commutative 69.7%

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

   7. Simplified 69.7%

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

   8. Taylor expanded in beta around 0 67.4%

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

      1. distribute-lft-in 67.4%

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

      2. metadata-eval 67.4%

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\color{blue}{12} + 6 \cdot \alpha} \]

   10. Simplified 67.4%

       \[\leadsto \frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\color{blue}{12 + 6 \cdot \alpha}} \]

   11. Taylor expanded in alpha around 0 67.4%

       \[\leadsto \color{blue}{0.08333333333333333 \cdot \left(1 + \beta\right)} \]

   12. Taylor expanded in beta around 0 67.6%

       \[\leadsto \color{blue}{0.08333333333333333} \]

   if 1 < beta

   1. Initial program 84.4%

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

      1. associate-/l/ 78.1%

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

      2. +-commutative 78.1%

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

      3. associate-+l+ 78.1%

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

      4. *-commutative 78.1%

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

      5. metadata-eval 78.1%

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

      6. associate-+l+ 78.1%

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

      7. metadata-eval 78.1%

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

      8. associate-+l+ 78.1%

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

      9. metadata-eval 78.1%

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

      10. metadata-eval 78.1%

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

      11. associate-+l+ 78.1%

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

   3. Simplified 78.1%

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

   5. Taylor expanded in beta around inf 83.9%

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

   6. Taylor expanded in alpha around 0 69.5%

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

      1. +-commutative 69.5%

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

      2. +-commutative 69.5%

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

   8. Simplified 69.5%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.35:\\ \;\;\;\;\frac{1 + \beta}{12 + \beta \cdot 16}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.35)
   (/ (+ 1.0 beta) (+ 12.0 (* beta 16.0)))
   (/ 1.0 (* (+ beta 2.0) (+ beta 3.0)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.35) {
		tmp = (1.0 + beta) / (12.0 + (beta * 16.0));
	} else {
		tmp = 1.0 / ((beta + 2.0) * (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 <= 1.35d0) then
        tmp = (1.0d0 + beta) / (12.0d0 + (beta * 16.0d0))
    else
        tmp = 1.0d0 / ((beta + 2.0d0) * (beta + 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.35) {
		tmp = (1.0 + beta) / (12.0 + (beta * 16.0));
	} else {
		tmp = 1.0 / ((beta + 2.0) * (beta + 3.0));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 1.35:
		tmp = (1.0 + beta) / (12.0 + (beta * 16.0))
	else:
		tmp = 1.0 / ((beta + 2.0) * (beta + 3.0))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.35)
		tmp = Float64(Float64(1.0 + beta) / Float64(12.0 + Float64(beta * 16.0)));
	else
		tmp = Float64(1.0 / Float64(Float64(beta + 2.0) * Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.35)
		tmp = (1.0 + beta) / (12.0 + (beta * 16.0));
	else
		tmp = 1.0 / ((beta + 2.0) * (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 1.35], N[(N[(1.0 + beta), $MachinePrecision] / N[(12.0 + N[(beta * 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.3500000000000001

   1. Initial program 99.8%

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

   3. Simplified 93.3%

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

   5. Taylor expanded in alpha around 0 69.7%

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

      1. +-commutative 69.7%

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

      2. +-commutative 69.7%

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

   7. Simplified 69.7%

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

   8. Taylor expanded in beta around 0 68.6%

      \[\leadsto \frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\color{blue}{6 \cdot \left(2 + \alpha\right) + \beta \cdot \left(2 \cdot \left(5 + \alpha\right) + 3 \cdot \left(2 + \alpha\right)\right)}} \]

   9. Taylor expanded in alpha around 0 68.7%

      \[\leadsto \color{blue}{\frac{1 + \beta}{12 + 16 \cdot \beta}} \]
   10. Step-by-step derivation

       1. *-commutative 68.7%

          \[\leadsto \frac{1 + \beta}{12 + \color{blue}{\beta \cdot 16}} \]

   11. Simplified 68.7%

       \[\leadsto \color{blue}{\frac{1 + \beta}{12 + \beta \cdot 16}} \]

   if 1.3500000000000001 < beta

   1. Initial program 84.4%

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

      1. associate-/l/ 78.1%

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

      2. +-commutative 78.1%

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

      3. associate-+l+ 78.1%

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

      4. *-commutative 78.1%

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

      5. metadata-eval 78.1%

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

      6. associate-+l+ 78.1%

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

      7. metadata-eval 78.1%

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

      8. associate-+l+ 78.1%

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

      9. metadata-eval 78.1%

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

      10. metadata-eval 78.1%

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

      11. associate-+l+ 78.1%

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

   3. Simplified 78.1%

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

   5. Taylor expanded in beta around inf 83.9%

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

   6. Taylor expanded in alpha around 0 69.5%

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

      1. +-commutative 69.5%

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

      2. +-commutative 69.5%

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

   8. Simplified 69.5%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.35:\\ \;\;\;\;\frac{1 + \beta}{12 + \beta \cdot 16}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 45.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{4}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.0) 0.08333333333333333 (/ (- 1.0 (/ 4.0 beta)) beta)))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.0) {
		tmp = 0.08333333333333333;
	} else {
		tmp = (1.0 - (4.0 / 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 <= 4.0d0) then
        tmp = 0.08333333333333333d0
    else
        tmp = (1.0d0 - (4.0d0 / beta)) / beta
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.0) {
		tmp = 0.08333333333333333;
	} else {
		tmp = (1.0 - (4.0 / beta)) / beta;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 4.0:
		tmp = 0.08333333333333333
	else:
		tmp = (1.0 - (4.0 / beta)) / beta
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.0)
		tmp = 0.08333333333333333;
	else
		tmp = Float64(Float64(1.0 - Float64(4.0 / beta)) / beta);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4.0)
		tmp = 0.08333333333333333;
	else
		tmp = (1.0 - (4.0 / beta)) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 4.0], 0.08333333333333333, N[(N[(1.0 - N[(4.0 / beta), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4

   1. Initial program 99.8%

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

   3. Simplified 93.3%

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

   5. Taylor expanded in alpha around 0 69.7%

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

      1. +-commutative 69.7%

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

      2. +-commutative 69.7%

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

   7. Simplified 69.7%

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

   8. Taylor expanded in beta around 0 67.4%

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

      1. distribute-lft-in 67.4%

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

      2. metadata-eval 67.4%

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\color{blue}{12} + 6 \cdot \alpha} \]

   10. Simplified 67.4%

       \[\leadsto \frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\color{blue}{12 + 6 \cdot \alpha}} \]

   11. Taylor expanded in alpha around 0 67.4%

       \[\leadsto \color{blue}{0.08333333333333333 \cdot \left(1 + \beta\right)} \]

   12. Taylor expanded in beta around 0 67.6%

       \[\leadsto \color{blue}{0.08333333333333333} \]

   if 4 < beta

   1. Initial program 84.4%

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

   3. Simplified 65.8%

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

   5. Taylor expanded in alpha around 0 61.5%

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

      1. +-commutative 61.5%

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

      2. +-commutative 61.5%

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

   7. Simplified 61.5%

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

   8. Taylor expanded in alpha around inf 44.0%

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

      1. associate-/r* 6.8%

         \[\leadsto \color{blue}{\frac{\frac{1 + \beta}{2 + \beta}}{3 + \beta}} \]

      2. +-commutative 6.8%

         \[\leadsto \frac{\frac{1 + \beta}{\color{blue}{\beta + 2}}}{3 + \beta} \]

      3. +-commutative 6.8%

         \[\leadsto \frac{\frac{1 + \beta}{\beta + 2}}{\color{blue}{\beta + 3}} \]

      4. associate-/r* 44.0%

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

   10. Simplified 44.0%

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

   11. Taylor expanded in beta around inf 6.8%

       \[\leadsto \color{blue}{\frac{1 - 4 \cdot \frac{1}{\beta}}{\beta}} \]
   12. Step-by-step derivation

       1. associate-*r/ 6.8%

          \[\leadsto \frac{1 - \color{blue}{\frac{4 \cdot 1}{\beta}}}{\beta} \]

       2. metadata-eval 6.8%

          \[\leadsto \frac{1 - \frac{\color{blue}{4}}{\beta}}{\beta} \]

   13. Simplified 6.8%

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

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{4}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 45.0% accurate, 35.0× speedup

Math

\[\begin{array}{l} \\ 0.08333333333333333 \end{array} \]

FPCore

(FPCore (alpha beta) :precision binary64 0.08333333333333333)

C

double code(double alpha, double beta) {
	return 0.08333333333333333;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = 0.08333333333333333d0
end function

Java

public static double code(double alpha, double beta) {
	return 0.08333333333333333;
}

Python

def code(alpha, beta):
	return 0.08333333333333333

Julia

function code(alpha, beta)
	return 0.08333333333333333
end

MATLAB

function tmp = code(alpha, beta)
	tmp = 0.08333333333333333;
end

Wolfram

code[alpha_, beta_] := 0.08333333333333333

Derivation

1. Initial program 94.8%

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

3. Simplified 84.4%

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

5. Taylor expanded in alpha around 0 67.1%

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

   1. +-commutative 67.1%

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

   2. +-commutative 67.1%

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

7. Simplified 67.1%

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

8. Taylor expanded in beta around 0 46.4%

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

   1. distribute-lft-in 46.4%

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

   2. metadata-eval 46.4%

      \[\leadsto \frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\color{blue}{12} + 6 \cdot \alpha} \]

10. Simplified 46.4%

    \[\leadsto \frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\color{blue}{12 + 6 \cdot \alpha}} \]

11. Taylor expanded in alpha around 0 46.5%

    \[\leadsto \color{blue}{0.08333333333333333 \cdot \left(1 + \beta\right)} \]

12. Taylor expanded in beta around 0 46.9%

    \[\leadsto \color{blue}{0.08333333333333333} \]

13. Final simplification 46.9%

    \[\leadsto 0.08333333333333333 \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024053 
(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)))