Octave 3.8, jcobi/3

Percentage Accurate: 94.1% → 99.8%

Time: 18.5s

Alternatives: 20

Speedup: 1.4×

Specification

\[\alpha > -1 \land \beta > -1\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.9%

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

3. Simplified 84.6%

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

   1. times-frac 96.5%

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

   2. +-commutative 96.5%

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

6. Applied egg-rr 96.5%

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

   1. associate-*l/ 96.5%

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

   2. +-commutative 96.5%

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

   3. +-commutative 96.5%

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

   4. associate-+l+ 96.5%

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

   5. +-commutative 96.5%

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

   6. +-commutative 96.5%

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

   7. associate-+l+ 96.5%

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

   8. +-commutative 96.5%

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

8. Applied egg-rr 96.5%

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

   1. associate-*r/ 92.2%

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

   2. times-frac 99.8%

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

   3. +-commutative 99.8%

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

   4. +-commutative 99.8%

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

   5. associate-+r+ 99.8%

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

   6. +-commutative 99.8%

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

   7. +-commutative 99.8%

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

10. Simplified 99.8%

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

11. Final simplification 99.8%

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

Alternative 2: 91.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + 2\right) + \beta\\ \mathbf{if}\;\beta \leq 4.4 \cdot 10^{+38}:\\ \;\;\;\;\left(1 + \alpha\right) \cdot \frac{\frac{\frac{1 + \beta}{t\_0}}{\left(\alpha + \beta\right) + 3}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{t\_0} \cdot \left(1 - \frac{\alpha + 2}{\beta}\right)}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ alpha 2.0) beta)))
   (if (<= beta 4.4e+38)
     (* (+ 1.0 alpha) (/ (/ (/ (+ 1.0 beta) t_0) (+ (+ alpha beta) 3.0)) t_0))
     (/ (* (/ (+ 1.0 alpha) t_0) (- 1.0 (/ (+ alpha 2.0) beta))) t_0))))

C

double code(double alpha, double beta) {
	double t_0 = (alpha + 2.0) + beta;
	double tmp;
	if (beta <= 4.4e+38) {
		tmp = (1.0 + alpha) * ((((1.0 + beta) / t_0) / ((alpha + beta) + 3.0)) / t_0);
	} else {
		tmp = (((1.0 + alpha) / t_0) * (1.0 - ((alpha + 2.0) / beta))) / t_0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (alpha + 2.0d0) + beta
    if (beta <= 4.4d+38) then
        tmp = (1.0d0 + alpha) * ((((1.0d0 + beta) / t_0) / ((alpha + beta) + 3.0d0)) / t_0)
    else
        tmp = (((1.0d0 + alpha) / t_0) * (1.0d0 - ((alpha + 2.0d0) / beta))) / t_0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (alpha + 2.0) + beta;
	double tmp;
	if (beta <= 4.4e+38) {
		tmp = (1.0 + alpha) * ((((1.0 + beta) / t_0) / ((alpha + beta) + 3.0)) / t_0);
	} else {
		tmp = (((1.0 + alpha) / t_0) * (1.0 - ((alpha + 2.0) / beta))) / t_0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (alpha + 2.0) + beta
	tmp = 0
	if beta <= 4.4e+38:
		tmp = (1.0 + alpha) * ((((1.0 + beta) / t_0) / ((alpha + beta) + 3.0)) / t_0)
	else:
		tmp = (((1.0 + alpha) / t_0) * (1.0 - ((alpha + 2.0) / beta))) / t_0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(alpha + 2.0) + beta)
	tmp = 0.0
	if (beta <= 4.4e+38)
		tmp = Float64(Float64(1.0 + alpha) * Float64(Float64(Float64(Float64(1.0 + beta) / t_0) / Float64(Float64(alpha + beta) + 3.0)) / t_0));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + alpha) / t_0) * Float64(1.0 - Float64(Float64(alpha + 2.0) / beta))) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (alpha + 2.0) + beta;
	tmp = 0.0;
	if (beta <= 4.4e+38)
		tmp = (1.0 + alpha) * ((((1.0 + beta) / t_0) / ((alpha + beta) + 3.0)) / t_0);
	else
		tmp = (((1.0 + alpha) / t_0) * (1.0 - ((alpha + 2.0) / beta))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(alpha + 2.0), $MachinePrecision] + beta), $MachinePrecision]}, If[LessEqual[beta, 4.4e+38], N[(N[(1.0 + alpha), $MachinePrecision] * N[(N[(N[(N[(1.0 + beta), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(1.0 - N[(N[(alpha + 2.0), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.40000000000000013e38

   1. Initial program 99.9%

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

   3. Simplified 96.3%

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

      1. times-frac 99.6%

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

      2. +-commutative 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. associate-*l/ 99.6%

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

      2. +-commutative 99.6%

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

      3. +-commutative 99.6%

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

      4. associate-+l+ 99.6%

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

      5. +-commutative 99.6%

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

      6. +-commutative 99.6%

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

      7. associate-+l+ 99.6%

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

      8. +-commutative 99.6%

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

   8. Applied egg-rr 99.6%

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

      1. associate-/l* 96.4%

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

      2. associate-/r* 96.3%

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

      3. +-commutative 96.3%

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

      4. +-commutative 96.3%

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

      5. associate-+r+ 96.3%

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

      6. +-commutative 96.3%

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

      7. +-commutative 96.3%

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

   10. Simplified 96.3%

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

   if 4.40000000000000013e38 < beta

   1. Initial program 78.4%

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

   3. Simplified 60.3%

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

      1. times-frac 90.2%

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

      2. +-commutative 90.2%

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

   6. Applied egg-rr 90.2%

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

      1. associate-*l/ 90.2%

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

      2. +-commutative 90.2%

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

      3. +-commutative 90.2%

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

      4. associate-+l+ 90.2%

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

      5. +-commutative 90.2%

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

      6. +-commutative 90.2%

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

      7. associate-+l+ 90.2%

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

      8. +-commutative 90.2%

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

   8. Applied egg-rr 90.2%

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

      1. associate-*r/ 76.9%

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

      2. times-frac 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

      5. associate-+r+ 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in beta around inf 83.0%

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

       1. associate-*r/ 83.0%

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

       2. mul-1-neg 83.0%

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

   13. Simplified 83.0%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.4 \cdot 10^{+38}:\\ \;\;\;\;\left(1 + \alpha\right) \cdot \frac{\frac{\frac{1 + \beta}{\left(\alpha + 2\right) + \beta}}{\left(\alpha + \beta\right) + 3}}{\left(\alpha + 2\right) + \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\left(\alpha + 2\right) + \beta} \cdot \left(1 - \frac{\alpha + 2}{\beta}\right)}{\left(\alpha + 2\right) + \beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + 2\right) + \beta\\ t_1 := \alpha + \left(2 + \beta\right)\\ \mathbf{if}\;\beta \leq 1650000000:\\ \;\;\;\;\frac{1 + \alpha}{t\_1} \cdot \frac{1 + \beta}{t\_1 \cdot \left(\alpha + \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{t\_0} \cdot \left(1 - \frac{\alpha + 2}{\beta}\right)}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ alpha 2.0) beta)) (t_1 (+ alpha (+ 2.0 beta))))
   (if (<= beta 1650000000.0)
     (* (/ (+ 1.0 alpha) t_1) (/ (+ 1.0 beta) (* t_1 (+ alpha (+ beta 3.0)))))
     (/ (* (/ (+ 1.0 alpha) t_0) (- 1.0 (/ (+ alpha 2.0) beta))) t_0))))

C

double code(double alpha, double beta) {
	double t_0 = (alpha + 2.0) + beta;
	double t_1 = alpha + (2.0 + beta);
	double tmp;
	if (beta <= 1650000000.0) {
		tmp = ((1.0 + alpha) / t_1) * ((1.0 + beta) / (t_1 * (alpha + (beta + 3.0))));
	} else {
		tmp = (((1.0 + alpha) / t_0) * (1.0 - ((alpha + 2.0) / beta))) / t_0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (alpha + 2.0d0) + beta
    t_1 = alpha + (2.0d0 + beta)
    if (beta <= 1650000000.0d0) then
        tmp = ((1.0d0 + alpha) / t_1) * ((1.0d0 + beta) / (t_1 * (alpha + (beta + 3.0d0))))
    else
        tmp = (((1.0d0 + alpha) / t_0) * (1.0d0 - ((alpha + 2.0d0) / beta))) / t_0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (alpha + 2.0) + beta;
	double t_1 = alpha + (2.0 + beta);
	double tmp;
	if (beta <= 1650000000.0) {
		tmp = ((1.0 + alpha) / t_1) * ((1.0 + beta) / (t_1 * (alpha + (beta + 3.0))));
	} else {
		tmp = (((1.0 + alpha) / t_0) * (1.0 - ((alpha + 2.0) / beta))) / t_0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (alpha + 2.0) + beta
	t_1 = alpha + (2.0 + beta)
	tmp = 0
	if beta <= 1650000000.0:
		tmp = ((1.0 + alpha) / t_1) * ((1.0 + beta) / (t_1 * (alpha + (beta + 3.0))))
	else:
		tmp = (((1.0 + alpha) / t_0) * (1.0 - ((alpha + 2.0) / beta))) / t_0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(alpha + 2.0) + beta)
	t_1 = Float64(alpha + Float64(2.0 + beta))
	tmp = 0.0
	if (beta <= 1650000000.0)
		tmp = Float64(Float64(Float64(1.0 + alpha) / t_1) * Float64(Float64(1.0 + beta) / Float64(t_1 * Float64(alpha + Float64(beta + 3.0)))));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + alpha) / t_0) * Float64(1.0 - Float64(Float64(alpha + 2.0) / beta))) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (alpha + 2.0) + beta;
	t_1 = alpha + (2.0 + beta);
	tmp = 0.0;
	if (beta <= 1650000000.0)
		tmp = ((1.0 + alpha) / t_1) * ((1.0 + beta) / (t_1 * (alpha + (beta + 3.0))));
	else
		tmp = (((1.0 + alpha) / t_0) * (1.0 - ((alpha + 2.0) / beta))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(alpha + 2.0), $MachinePrecision] + beta), $MachinePrecision]}, Block[{t$95$1 = N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1650000000.0], N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$1), $MachinePrecision] * N[(N[(1.0 + beta), $MachinePrecision] / N[(t$95$1 * N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(1.0 - N[(N[(alpha + 2.0), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.65e9

   1. Initial program 99.9%

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

   3. Simplified 96.2%

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

      1. times-frac 99.6%

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

      2. +-commutative 99.6%

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

   6. Applied egg-rr 99.6%

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

   if 1.65e9 < beta

   1. Initial program 80.0%

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

   3. Simplified 63.3%

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

      1. times-frac 90.9%

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

      2. +-commutative 90.9%

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

   6. Applied egg-rr 90.9%

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

      1. associate-*l/ 90.9%

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

      2. +-commutative 90.9%

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

      3. +-commutative 90.9%

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

      4. associate-+l+ 90.9%

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

      5. +-commutative 90.9%

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

      6. +-commutative 90.9%

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

      7. associate-+l+ 90.9%

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

      8. +-commutative 90.9%

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

   8. Applied egg-rr 90.9%

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

      1. associate-*r/ 78.6%

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

      2. times-frac 99.7%

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

      3. +-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. associate-+r+ 99.7%

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

      6. +-commutative 99.7%

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

      7. +-commutative 99.7%

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

   10. Simplified 99.7%

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

   11. Taylor expanded in beta around inf 82.1%

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

       1. associate-*r/ 82.1%

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

       2. mul-1-neg 82.1%

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

   13. Simplified 82.1%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1650000000:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 + \beta}{\left(\alpha + \left(2 + \beta\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\left(\alpha + 2\right) + \beta} \cdot \left(1 - \frac{\alpha + 2}{\beta}\right)}{\left(\alpha + 2\right) + \beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + 2\right) + \beta\\ \mathbf{if}\;\beta \leq 9 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{\frac{1 + \beta}{2 + \beta}}{\beta + 3}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{t\_0} \cdot \left(1 - \frac{\alpha + 2}{\beta}\right)}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ alpha 2.0) beta)))
   (if (<= beta 9e+15)
     (/ (/ (/ (+ 1.0 beta) (+ 2.0 beta)) (+ beta 3.0)) t_0)
     (/ (* (/ (+ 1.0 alpha) t_0) (- 1.0 (/ (+ alpha 2.0) beta))) t_0))))

C

double code(double alpha, double beta) {
	double t_0 = (alpha + 2.0) + beta;
	double tmp;
	if (beta <= 9e+15) {
		tmp = (((1.0 + beta) / (2.0 + beta)) / (beta + 3.0)) / t_0;
	} else {
		tmp = (((1.0 + alpha) / t_0) * (1.0 - ((alpha + 2.0) / beta))) / t_0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (alpha + 2.0d0) + beta
    if (beta <= 9d+15) then
        tmp = (((1.0d0 + beta) / (2.0d0 + beta)) / (beta + 3.0d0)) / t_0
    else
        tmp = (((1.0d0 + alpha) / t_0) * (1.0d0 - ((alpha + 2.0d0) / beta))) / t_0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (alpha + 2.0) + beta;
	double tmp;
	if (beta <= 9e+15) {
		tmp = (((1.0 + beta) / (2.0 + beta)) / (beta + 3.0)) / t_0;
	} else {
		tmp = (((1.0 + alpha) / t_0) * (1.0 - ((alpha + 2.0) / beta))) / t_0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (alpha + 2.0) + beta
	tmp = 0
	if beta <= 9e+15:
		tmp = (((1.0 + beta) / (2.0 + beta)) / (beta + 3.0)) / t_0
	else:
		tmp = (((1.0 + alpha) / t_0) * (1.0 - ((alpha + 2.0) / beta))) / t_0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(alpha + 2.0) + beta)
	tmp = 0.0
	if (beta <= 9e+15)
		tmp = Float64(Float64(Float64(Float64(1.0 + beta) / Float64(2.0 + beta)) / Float64(beta + 3.0)) / t_0);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + alpha) / t_0) * Float64(1.0 - Float64(Float64(alpha + 2.0) / beta))) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (alpha + 2.0) + beta;
	tmp = 0.0;
	if (beta <= 9e+15)
		tmp = (((1.0 + beta) / (2.0 + beta)) / (beta + 3.0)) / t_0;
	else
		tmp = (((1.0 + alpha) / t_0) * (1.0 - ((alpha + 2.0) / beta))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(alpha + 2.0), $MachinePrecision] + beta), $MachinePrecision]}, If[LessEqual[beta, 9e+15], N[(N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(1.0 - N[(N[(alpha + 2.0), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 9e15

   1. Initial program 99.9%

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

   3. Simplified 96.2%

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

      1. times-frac 99.6%

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

      2. +-commutative 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. associate-*l/ 99.6%

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

      2. +-commutative 99.6%

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

      3. +-commutative 99.6%

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

      4. associate-+l+ 99.6%

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

      5. +-commutative 99.6%

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

      6. +-commutative 99.6%

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

      7. associate-+l+ 99.6%

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

      8. +-commutative 99.6%

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

   8. Applied egg-rr 99.6%

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

      1. associate-*r/ 99.5%

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

      2. times-frac 99.9%

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

      3. +-commutative 99.9%

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

      4. +-commutative 99.9%

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

      5. associate-+r+ 99.9%

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

      6. +-commutative 99.9%

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

      7. +-commutative 99.9%

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

   10. Simplified 99.9%

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

   11. Taylor expanded in alpha around 0 65.5%

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

       1. associate-/r* 65.5%

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

       2. +-commutative 65.5%

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

       3. +-commutative 65.5%

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

   13. Simplified 65.5%

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

   if 9e15 < beta

   1. Initial program 79.8%

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

   3. Simplified 62.9%

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

      1. times-frac 90.8%

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

      2. +-commutative 90.8%

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

   6. Applied egg-rr 90.8%

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

      1. associate-*l/ 90.8%

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

      2. +-commutative 90.8%

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

      3. +-commutative 90.8%

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

      4. associate-+l+ 90.8%

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

      5. +-commutative 90.8%

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

      6. +-commutative 90.8%

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

      7. associate-+l+ 90.8%

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

      8. +-commutative 90.8%

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

   8. Applied egg-rr 90.8%

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

      1. associate-*r/ 78.4%

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

      2. times-frac 99.7%

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

      3. +-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. associate-+r+ 99.7%

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

      6. +-commutative 99.7%

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

      7. +-commutative 99.7%

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

   10. Simplified 99.7%

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

   11. Taylor expanded in beta around inf 83.0%

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

       1. associate-*r/ 83.0%

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

       2. mul-1-neg 83.0%

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

   13. Simplified 83.0%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 9 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{\frac{1 + \beta}{2 + \beta}}{\beta + 3}}{\left(\alpha + 2\right) + \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\left(\alpha + 2\right) + \beta} \cdot \left(1 - \frac{\alpha + 2}{\beta}\right)}{\left(\alpha + 2\right) + \beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5 \cdot 10^{+33}:\\ \;\;\;\;\frac{\frac{\frac{1 + \beta}{2 + \beta}}{\beta + 3}}{\left(\alpha + 2\right) + \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 - \frac{2 \cdot \left(\alpha + 2\right)}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5e+33)
   (/ (/ (/ (+ 1.0 beta) (+ 2.0 beta)) (+ beta 3.0)) (+ (+ alpha 2.0) beta))
   (*
    (/ (+ 1.0 alpha) (+ alpha (+ 2.0 beta)))
    (/ (- 1.0 (/ (* 2.0 (+ alpha 2.0)) beta)) beta))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5e+33) {
		tmp = (((1.0 + beta) / (2.0 + beta)) / (beta + 3.0)) / ((alpha + 2.0) + beta);
	} else {
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) * ((1.0 - ((2.0 * (alpha + 2.0)) / beta)) / beta);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 5d+33) then
        tmp = (((1.0d0 + beta) / (2.0d0 + beta)) / (beta + 3.0d0)) / ((alpha + 2.0d0) + beta)
    else
        tmp = ((1.0d0 + alpha) / (alpha + (2.0d0 + beta))) * ((1.0d0 - ((2.0d0 * (alpha + 2.0d0)) / beta)) / beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5e+33) {
		tmp = (((1.0 + beta) / (2.0 + beta)) / (beta + 3.0)) / ((alpha + 2.0) + beta);
	} else {
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) * ((1.0 - ((2.0 * (alpha + 2.0)) / beta)) / beta);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 5e+33:
		tmp = (((1.0 + beta) / (2.0 + beta)) / (beta + 3.0)) / ((alpha + 2.0) + beta)
	else:
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) * ((1.0 - ((2.0 * (alpha + 2.0)) / beta)) / beta)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 5e+33)
		tmp = Float64(Float64(Float64(Float64(1.0 + beta) / Float64(2.0 + beta)) / Float64(beta + 3.0)) / Float64(Float64(alpha + 2.0) + beta));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(2.0 + beta))) * Float64(Float64(1.0 - Float64(Float64(2.0 * Float64(alpha + 2.0)) / beta)) / beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 5e+33)
		tmp = (((1.0 + beta) / (2.0 + beta)) / (beta + 3.0)) / ((alpha + 2.0) + beta);
	else
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) * ((1.0 - ((2.0 * (alpha + 2.0)) / beta)) / beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 5e+33], N[(N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + 2.0), $MachinePrecision] + beta), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[(N[(2.0 * N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.99999999999999973e33

   1. Initial program 99.9%

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

   3. Simplified 96.3%

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

      1. times-frac 99.6%

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

      2. +-commutative 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. associate-*l/ 99.6%

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

      2. +-commutative 99.6%

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

      3. +-commutative 99.6%

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

      4. associate-+l+ 99.6%

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

      5. +-commutative 99.6%

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

      6. +-commutative 99.6%

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

      7. associate-+l+ 99.6%

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

      8. +-commutative 99.6%

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

   8. Applied egg-rr 99.6%

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

      1. associate-*r/ 99.5%

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

      2. times-frac 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

      5. associate-+r+ 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in alpha around 0 66.5%

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

       1. associate-/r* 66.5%

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

       2. +-commutative 66.5%

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

       3. +-commutative 66.5%

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

   13. Simplified 66.5%

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

   if 4.99999999999999973e33 < beta

   1. Initial program 78.7%

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

   3. Simplified 60.7%

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

      1. times-frac 90.3%

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

      2. +-commutative 90.3%

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

   6. Applied egg-rr 90.3%

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

   7. Taylor expanded in beta around inf 81.8%

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

      1. mul-1-neg 81.8%

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

      2. metadata-eval 81.8%

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

      3. distribute-lft-in 81.8%

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

   9. Simplified 81.8%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5 \cdot 10^{+33}:\\ \;\;\;\;\frac{\frac{\frac{1 + \beta}{2 + \beta}}{\beta + 3}}{\left(\alpha + 2\right) + \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 - \frac{2 \cdot \left(\alpha + 2\right)}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.9%

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

3. Simplified 84.6%

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

   1. times-frac 96.5%

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

   2. +-commutative 96.5%

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

6. Applied egg-rr 96.5%

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

   1. *-un-lft-identity 96.5%

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

   2. +-commutative 96.5%

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

   3. associate-+l+ 96.5%

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

   4. +-commutative 96.5%

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

8. Applied egg-rr 96.5%

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

   1. *-lft-identity 96.5%

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

   2. associate-/r* 99.8%

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

   3. +-commutative 99.8%

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

   4. +-commutative 99.8%

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

   5. associate-+r+ 99.8%

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

10. Simplified 99.8%

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

11. Final simplification 99.8%

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

Alternative 7: 72.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 8.4 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{\frac{1 + \beta}{2 + \beta}}{\beta + 3}}{\left(\alpha + 2\right) + \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 8.4e+15)
   (/ (/ (/ (+ 1.0 beta) (+ 2.0 beta)) (+ beta 3.0)) (+ (+ alpha 2.0) beta))
   (/ (/ (+ 1.0 alpha) beta) (+ 1.0 (+ 2.0 (+ alpha beta))))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 8.4e+15) {
		tmp = (((1.0 + beta) / (2.0 + beta)) / (beta + 3.0)) / ((alpha + 2.0) + beta);
	} 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 <= 8.4d+15) then
        tmp = (((1.0d0 + beta) / (2.0d0 + beta)) / (beta + 3.0d0)) / ((alpha + 2.0d0) + beta)
    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 <= 8.4e+15) {
		tmp = (((1.0 + beta) / (2.0 + beta)) / (beta + 3.0)) / ((alpha + 2.0) + beta);
	} else {
		tmp = ((1.0 + alpha) / beta) / (1.0 + (2.0 + (alpha + beta)));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 8.4e+15:
		tmp = (((1.0 + beta) / (2.0 + beta)) / (beta + 3.0)) / ((alpha + 2.0) + beta)
	else:
		tmp = ((1.0 + alpha) / beta) / (1.0 + (2.0 + (alpha + beta)))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 8.4e+15)
		tmp = Float64(Float64(Float64(Float64(1.0 + beta) / Float64(2.0 + beta)) / Float64(beta + 3.0)) / Float64(Float64(alpha + 2.0) + beta));
	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 <= 8.4e+15)
		tmp = (((1.0 + beta) / (2.0 + beta)) / (beta + 3.0)) / ((alpha + 2.0) + beta);
	else
		tmp = ((1.0 + alpha) / beta) / (1.0 + (2.0 + (alpha + beta)));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 8.4e+15], N[(N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + 2.0), $MachinePrecision] + beta), $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 < 8.4e15

   1. Initial program 99.9%

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

   3. Simplified 96.2%

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

      1. times-frac 99.6%

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

      2. +-commutative 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. associate-*l/ 99.6%

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

      2. +-commutative 99.6%

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

      3. +-commutative 99.6%

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

      4. associate-+l+ 99.6%

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

      5. +-commutative 99.6%

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

      6. +-commutative 99.6%

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

      7. associate-+l+ 99.6%

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

      8. +-commutative 99.6%

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

   8. Applied egg-rr 99.6%

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

      1. associate-*r/ 99.5%

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

      2. times-frac 99.9%

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

      3. +-commutative 99.9%

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

      4. +-commutative 99.9%

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

      5. associate-+r+ 99.9%

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

      6. +-commutative 99.9%

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

      7. +-commutative 99.9%

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

   10. Simplified 99.9%

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

   11. Taylor expanded in alpha around 0 65.5%

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

       1. associate-/r* 65.5%

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

       2. +-commutative 65.5%

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

       3. +-commutative 65.5%

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

   13. Simplified 65.5%

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

   if 8.4e15 < beta

   1. Initial program 79.8%

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

   3. Taylor expanded in beta around inf 82.9%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 8.4 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{\frac{1 + \beta}{2 + \beta}}{\beta + 3}}{\left(\alpha + 2\right) + \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.6:\\ \;\;\;\;\frac{0.25}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \mathbf{elif}\;\beta \leq 5 \cdot 10^{+152}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \left(\left(\alpha + \beta\right) + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.6)
   (/ 0.25 (+ 1.0 (+ 2.0 (+ alpha beta))))
   (if (<= beta 5e+152)
     (/ (+ 1.0 alpha) (* beta (+ (+ alpha beta) 3.0)))
     (/ (/ alpha beta) (+ alpha (+ beta 3.0))))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.6) {
		tmp = 0.25 / (1.0 + (2.0 + (alpha + beta)));
	} else if (beta <= 5e+152) {
		tmp = (1.0 + alpha) / (beta * ((alpha + beta) + 3.0));
	} else {
		tmp = (alpha / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 4.6d0) then
        tmp = 0.25d0 / (1.0d0 + (2.0d0 + (alpha + beta)))
    else if (beta <= 5d+152) then
        tmp = (1.0d0 + alpha) / (beta * ((alpha + beta) + 3.0d0))
    else
        tmp = (alpha / beta) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.6) {
		tmp = 0.25 / (1.0 + (2.0 + (alpha + beta)));
	} else if (beta <= 5e+152) {
		tmp = (1.0 + alpha) / (beta * ((alpha + beta) + 3.0));
	} else {
		tmp = (alpha / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 4.6:
		tmp = 0.25 / (1.0 + (2.0 + (alpha + beta)))
	elif beta <= 5e+152:
		tmp = (1.0 + alpha) / (beta * ((alpha + beta) + 3.0))
	else:
		tmp = (alpha / beta) / (alpha + (beta + 3.0))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.6)
		tmp = Float64(0.25 / Float64(1.0 + Float64(2.0 + Float64(alpha + beta))));
	elseif (beta <= 5e+152)
		tmp = Float64(Float64(1.0 + alpha) / Float64(beta * Float64(Float64(alpha + beta) + 3.0)));
	else
		tmp = Float64(Float64(alpha / beta) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4.6)
		tmp = 0.25 / (1.0 + (2.0 + (alpha + beta)));
	elseif (beta <= 5e+152)
		tmp = (1.0 + alpha) / (beta * ((alpha + beta) + 3.0));
	else
		tmp = (alpha / beta) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 4.6], N[(0.25 / N[(1.0 + N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 5e+152], N[(N[(1.0 + alpha), $MachinePrecision] / N[(beta * N[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if beta < 4.5999999999999996

   1. Initial program 99.9%

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

   3. Taylor expanded in beta around 0 98.6%

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

   4. Taylor expanded in alpha around 0 65.8%

      \[\leadsto \frac{\color{blue}{0.25}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   if 4.5999999999999996 < beta < 5e152

   1. Initial program 92.9%

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

   3. Taylor expanded in beta around inf 78.6%

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

      1. *-un-lft-identity 78.6%

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

      2. associate-/l/ 84.8%

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

      3. metadata-eval 84.8%

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

      4. associate-+l+ 84.8%

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

      5. metadata-eval 84.8%

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

      6. associate-+r+ 84.8%

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

   5. Applied egg-rr 84.8%

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

      1. *-lft-identity 84.8%

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

      2. *-commutative 84.8%

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

      3. associate-+r+ 84.8%

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

   7. Simplified 84.8%

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

   if 5e152 < beta

   1. Initial program 67.9%

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

   3. Taylor expanded in beta around inf 83.6%

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

   4. Taylor expanded in alpha around inf 83.6%

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

      1. *-un-lft-identity 83.6%

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

      2. metadata-eval 83.6%

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

      3. associate-+l+ 83.6%

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

      4. metadata-eval 83.6%

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

      5. associate-+l+ 83.6%

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

   6. Applied egg-rr 83.6%

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

      1. *-lft-identity 83.6%

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

      2. +-commutative 83.6%

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

      3. +-commutative 83.6%

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

      4. +-commutative 83.6%

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

   8. Simplified 83.6%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.6:\\ \;\;\;\;\frac{0.25}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \mathbf{elif}\;\beta \leq 5 \cdot 10^{+152}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \left(\left(\alpha + \beta\right) + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 71.8% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3e+15)
   (/ (+ 1.0 beta) (* (+ 2.0 beta) (* (+ 2.0 beta) (+ beta 3.0))))
   (* (/ (+ 1.0 alpha) beta) (/ 1.0 (+ alpha (+ beta 3.0))))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3e+15) {
		tmp = (1.0 + beta) / ((2.0 + beta) * ((2.0 + beta) * (beta + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / beta) * (1.0 / (alpha + (beta + 3.0)));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 3d+15) then
        tmp = (1.0d0 + beta) / ((2.0d0 + beta) * ((2.0d0 + beta) * (beta + 3.0d0)))
    else
        tmp = ((1.0d0 + alpha) / beta) * (1.0d0 / (alpha + (beta + 3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3e+15) {
		tmp = (1.0 + beta) / ((2.0 + beta) * ((2.0 + beta) * (beta + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / beta) * (1.0 / (alpha + (beta + 3.0)));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 3e+15:
		tmp = (1.0 + beta) / ((2.0 + beta) * ((2.0 + beta) * (beta + 3.0)))
	else:
		tmp = ((1.0 + alpha) / beta) * (1.0 / (alpha + (beta + 3.0)))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3e+15)
		tmp = Float64(Float64(1.0 + beta) / Float64(Float64(2.0 + beta) * Float64(Float64(2.0 + beta) * Float64(beta + 3.0))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) * Float64(1.0 / Float64(alpha + Float64(beta + 3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3e+15)
		tmp = (1.0 + beta) / ((2.0 + beta) * ((2.0 + beta) * (beta + 3.0)));
	else
		tmp = ((1.0 + alpha) / beta) * (1.0 / (alpha + (beta + 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 3e+15], N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(2.0 + beta), $MachinePrecision] * N[(N[(2.0 + beta), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] * N[(1.0 / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3e15

   1. Initial program 99.9%

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

   3. Simplified 96.2%

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

      1. times-frac 99.6%

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

      2. +-commutative 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in alpha around 0 64.3%

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

      1. +-commutative 64.3%

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

      2. +-commutative 64.3%

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

   9. Simplified 64.3%

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

   10. Taylor expanded in alpha around 0 64.3%

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

       1. frac-times 64.3%

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

       2. *-un-lft-identity 64.3%

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

       3. +-commutative 64.3%

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

   12. Applied egg-rr 64.3%

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

   if 3e15 < beta

   1. Initial program 79.8%

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

   3. Taylor expanded in beta around inf 82.9%

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

      1. div-inv 82.8%

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

      2. metadata-eval 82.8%

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

      3. associate-+l+ 82.8%

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

      4. metadata-eval 82.8%

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

      5. associate-+r+ 82.8%

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

   5. Applied egg-rr 82.8%

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

4. Final simplification 70.7%

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

Alternative 10: 71.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 9.5 \cdot 10^{+14}:\\ \;\;\;\;\frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(\left(2 + \beta\right) \cdot \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 9.5e+14)
   (/ (+ 1.0 beta) (* (+ 2.0 beta) (* (+ 2.0 beta) (+ beta 3.0))))
   (/ (/ (+ 1.0 alpha) beta) (+ 1.0 (+ 2.0 (+ alpha beta))))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 9.5e+14) {
		tmp = (1.0 + beta) / ((2.0 + beta) * ((2.0 + beta) * (beta + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / beta) / (1.0 + (2.0 + (alpha + beta)));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 9.5d+14) then
        tmp = (1.0d0 + beta) / ((2.0d0 + beta) * ((2.0d0 + beta) * (beta + 3.0d0)))
    else
        tmp = ((1.0d0 + alpha) / beta) / (1.0d0 + (2.0d0 + (alpha + beta)))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 9.5e+14) {
		tmp = (1.0 + beta) / ((2.0 + beta) * ((2.0 + beta) * (beta + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / beta) / (1.0 + (2.0 + (alpha + beta)));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 9.5e+14:
		tmp = (1.0 + beta) / ((2.0 + beta) * ((2.0 + beta) * (beta + 3.0)))
	else:
		tmp = ((1.0 + alpha) / beta) / (1.0 + (2.0 + (alpha + beta)))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 9.5e+14)
		tmp = Float64(Float64(1.0 + beta) / Float64(Float64(2.0 + beta) * Float64(Float64(2.0 + beta) * Float64(beta + 3.0))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(1.0 + Float64(2.0 + Float64(alpha + beta))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 9.5e+14)
		tmp = (1.0 + beta) / ((2.0 + beta) * ((2.0 + beta) * (beta + 3.0)));
	else
		tmp = ((1.0 + alpha) / beta) / (1.0 + (2.0 + (alpha + beta)));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 9.5e+14], N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(2.0 + beta), $MachinePrecision] * N[(N[(2.0 + beta), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(1.0 + N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 9.5e14

   1. Initial program 99.9%

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

   3. Simplified 96.2%

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

      1. times-frac 99.6%

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

      2. +-commutative 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in alpha around 0 64.3%

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

      1. +-commutative 64.3%

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

      2. +-commutative 64.3%

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

   9. Simplified 64.3%

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

   10. Taylor expanded in alpha around 0 64.3%

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

       1. frac-times 64.3%

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

       2. *-un-lft-identity 64.3%

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

       3. +-commutative 64.3%

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

   12. Applied egg-rr 64.3%

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

   if 9.5e14 < beta

   1. Initial program 79.8%

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

   3. Taylor expanded in beta around inf 82.9%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 9.5 \cdot 10^{+14}:\\ \;\;\;\;\frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(\left(2 + \beta\right) \cdot \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 69.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.52:\\ \;\;\;\;0.08333333333333333 + \beta \cdot \left(\beta \cdot -0.011574074074074073 - 0.027777777777777776\right)\\ \mathbf{elif}\;\beta \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.52)
   (+
    0.08333333333333333
    (* beta (- (* beta -0.011574074074074073) 0.027777777777777776)))
   (if (<= beta 1.4e+154)
     (/ 1.0 (* beta (+ beta 3.0)))
     (/ (/ alpha beta) (+ alpha (+ beta 3.0))))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.52) {
		tmp = 0.08333333333333333 + (beta * ((beta * -0.011574074074074073) - 0.027777777777777776));
	} else if (beta <= 1.4e+154) {
		tmp = 1.0 / (beta * (beta + 3.0));
	} else {
		tmp = (alpha / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.52d0) then
        tmp = 0.08333333333333333d0 + (beta * ((beta * (-0.011574074074074073d0)) - 0.027777777777777776d0))
    else if (beta <= 1.4d+154) then
        tmp = 1.0d0 / (beta * (beta + 3.0d0))
    else
        tmp = (alpha / beta) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.52) {
		tmp = 0.08333333333333333 + (beta * ((beta * -0.011574074074074073) - 0.027777777777777776));
	} else if (beta <= 1.4e+154) {
		tmp = 1.0 / (beta * (beta + 3.0));
	} else {
		tmp = (alpha / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 1.52:
		tmp = 0.08333333333333333 + (beta * ((beta * -0.011574074074074073) - 0.027777777777777776))
	elif beta <= 1.4e+154:
		tmp = 1.0 / (beta * (beta + 3.0))
	else:
		tmp = (alpha / beta) / (alpha + (beta + 3.0))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.52)
		tmp = Float64(0.08333333333333333 + Float64(beta * Float64(Float64(beta * -0.011574074074074073) - 0.027777777777777776)));
	elseif (beta <= 1.4e+154)
		tmp = Float64(1.0 / Float64(beta * Float64(beta + 3.0)));
	else
		tmp = Float64(Float64(alpha / beta) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.52)
		tmp = 0.08333333333333333 + (beta * ((beta * -0.011574074074074073) - 0.027777777777777776));
	elseif (beta <= 1.4e+154)
		tmp = 1.0 / (beta * (beta + 3.0));
	else
		tmp = (alpha / beta) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 1.52], N[(0.08333333333333333 + N[(beta * N[(N[(beta * -0.011574074074074073), $MachinePrecision] - 0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 1.4e+154], N[(1.0 / N[(beta * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if beta < 1.52

   1. Initial program 99.9%

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

   3. Simplified 96.5%

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

      1. times-frac 99.9%

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

      2. +-commutative 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in alpha around 0 65.0%

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

      1. +-commutative 65.0%

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

      2. +-commutative 65.0%

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

   9. Simplified 65.0%

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

   10. Taylor expanded in alpha around 0 65.1%

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

   11. Taylor expanded in beta around 0 64.6%

       \[\leadsto \color{blue}{0.08333333333333333 + \beta \cdot \left(-0.011574074074074073 \cdot \beta - 0.027777777777777776\right)} \]

   if 1.52 < beta < 1.4e154

   1. Initial program 92.9%

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

   3. Taylor expanded in beta around inf 78.6%

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

   4. Taylor expanded in alpha around 0 66.6%

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

   if 1.4e154 < beta

   1. Initial program 67.9%

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

   3. Taylor expanded in beta around inf 83.6%

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

   4. Taylor expanded in alpha around inf 83.6%

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

      1. *-un-lft-identity 83.6%

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

      2. metadata-eval 83.6%

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

      3. associate-+l+ 83.6%

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

      4. metadata-eval 83.6%

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

      5. associate-+l+ 83.6%

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

   6. Applied egg-rr 83.6%

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

      1. *-lft-identity 83.6%

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

      2. +-commutative 83.6%

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

      3. +-commutative 83.6%

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

      4. +-commutative 83.6%

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

   8. Simplified 83.6%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.52:\\ \;\;\;\;0.08333333333333333 + \beta \cdot \left(\beta \cdot -0.011574074074074073 - 0.027777777777777776\right)\\ \mathbf{elif}\;\beta \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 70.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.2:\\ \;\;\;\;\frac{0.25}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \mathbf{elif}\;\beta \leq 6.2 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.2)
   (/ 0.25 (+ 1.0 (+ 2.0 (+ alpha beta))))
   (if (<= beta 6.2e+154)
     (/ 1.0 (* beta (+ beta 3.0)))
     (/ (/ alpha beta) (+ alpha (+ beta 3.0))))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.2) {
		tmp = 0.25 / (1.0 + (2.0 + (alpha + beta)));
	} else if (beta <= 6.2e+154) {
		tmp = 1.0 / (beta * (beta + 3.0));
	} else {
		tmp = (alpha / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 4.2d0) then
        tmp = 0.25d0 / (1.0d0 + (2.0d0 + (alpha + beta)))
    else if (beta <= 6.2d+154) then
        tmp = 1.0d0 / (beta * (beta + 3.0d0))
    else
        tmp = (alpha / beta) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.2) {
		tmp = 0.25 / (1.0 + (2.0 + (alpha + beta)));
	} else if (beta <= 6.2e+154) {
		tmp = 1.0 / (beta * (beta + 3.0));
	} else {
		tmp = (alpha / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 4.2:
		tmp = 0.25 / (1.0 + (2.0 + (alpha + beta)))
	elif beta <= 6.2e+154:
		tmp = 1.0 / (beta * (beta + 3.0))
	else:
		tmp = (alpha / beta) / (alpha + (beta + 3.0))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.2)
		tmp = Float64(0.25 / Float64(1.0 + Float64(2.0 + Float64(alpha + beta))));
	elseif (beta <= 6.2e+154)
		tmp = Float64(1.0 / Float64(beta * Float64(beta + 3.0)));
	else
		tmp = Float64(Float64(alpha / beta) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4.2)
		tmp = 0.25 / (1.0 + (2.0 + (alpha + beta)));
	elseif (beta <= 6.2e+154)
		tmp = 1.0 / (beta * (beta + 3.0));
	else
		tmp = (alpha / beta) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 4.2], N[(0.25 / N[(1.0 + N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 6.2e+154], N[(1.0 / N[(beta * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if beta < 4.20000000000000018

   1. Initial program 99.9%

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

   3. Taylor expanded in beta around 0 98.6%

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

   4. Taylor expanded in alpha around 0 65.8%

      \[\leadsto \frac{\color{blue}{0.25}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   if 4.20000000000000018 < beta < 6.2000000000000003e154

   1. Initial program 92.9%

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

   3. Taylor expanded in beta around inf 78.6%

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

   4. Taylor expanded in alpha around 0 66.6%

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

   if 6.2000000000000003e154 < beta

   1. Initial program 67.9%

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

   3. Taylor expanded in beta around inf 83.6%

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

   4. Taylor expanded in alpha around inf 83.6%

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

      1. *-un-lft-identity 83.6%

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

      2. metadata-eval 83.6%

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

      3. associate-+l+ 83.6%

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

      4. metadata-eval 83.6%

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

      5. associate-+l+ 83.6%

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

   6. Applied egg-rr 83.6%

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

      1. *-lft-identity 83.6%

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

      2. +-commutative 83.6%

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

      3. +-commutative 83.6%

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

      4. +-commutative 83.6%

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

   8. Simplified 83.6%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.2:\\ \;\;\;\;\frac{0.25}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \mathbf{elif}\;\beta \leq 6.2 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 71.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4:\\ \;\;\;\;\frac{0.25}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta} \cdot \frac{1}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.0)
   (/ 0.25 (+ 1.0 (+ 2.0 (+ alpha beta))))
   (* (/ (+ 1.0 alpha) beta) (/ 1.0 (+ alpha (+ beta 3.0))))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.0) {
		tmp = 0.25 / (1.0 + (2.0 + (alpha + beta)));
	} else {
		tmp = ((1.0 + alpha) / beta) * (1.0 / (alpha + (beta + 3.0)));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 4.0d0) then
        tmp = 0.25d0 / (1.0d0 + (2.0d0 + (alpha + beta)))
    else
        tmp = ((1.0d0 + alpha) / beta) * (1.0d0 / (alpha + (beta + 3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.0) {
		tmp = 0.25 / (1.0 + (2.0 + (alpha + beta)));
	} else {
		tmp = ((1.0 + alpha) / beta) * (1.0 / (alpha + (beta + 3.0)));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 4.0:
		tmp = 0.25 / (1.0 + (2.0 + (alpha + beta)))
	else:
		tmp = ((1.0 + alpha) / beta) * (1.0 / (alpha + (beta + 3.0)))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.0)
		tmp = Float64(0.25 / Float64(1.0 + Float64(2.0 + Float64(alpha + beta))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) * Float64(1.0 / Float64(alpha + Float64(beta + 3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4.0)
		tmp = 0.25 / (1.0 + (2.0 + (alpha + beta)));
	else
		tmp = ((1.0 + alpha) / beta) * (1.0 / (alpha + (beta + 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 4.0], N[(0.25 / N[(1.0 + N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] * N[(1.0 / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4

   1. Initial program 99.9%

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

   3. Taylor expanded in beta around 0 98.6%

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

   4. Taylor expanded in alpha around 0 65.8%

      \[\leadsto \frac{\color{blue}{0.25}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   if 4 < beta

   1. Initial program 80.3%

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

   3. Taylor expanded in beta around inf 81.1%

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

      1. div-inv 81.0%

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

      2. metadata-eval 81.0%

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

      3. associate-+l+ 81.0%

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

      4. metadata-eval 81.0%

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

      5. associate-+r+ 81.0%

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

   5. Applied egg-rr 81.0%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4:\\ \;\;\;\;\frac{0.25}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta} \cdot \frac{1}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 68.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.52:\\ \;\;\;\;0.08333333333333333 + \beta \cdot \left(\beta \cdot -0.011574074074074073 - 0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.52)
   (+
    0.08333333333333333
    (* beta (- (* beta -0.011574074074074073) 0.027777777777777776)))
   (/ 1.0 (* beta (+ beta 3.0)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.52) {
		tmp = 0.08333333333333333 + (beta * ((beta * -0.011574074074074073) - 0.027777777777777776));
	} else {
		tmp = 1.0 / (beta * (beta + 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.52d0) then
        tmp = 0.08333333333333333d0 + (beta * ((beta * (-0.011574074074074073d0)) - 0.027777777777777776d0))
    else
        tmp = 1.0d0 / (beta * (beta + 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.52) {
		tmp = 0.08333333333333333 + (beta * ((beta * -0.011574074074074073) - 0.027777777777777776));
	} else {
		tmp = 1.0 / (beta * (beta + 3.0));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 1.52:
		tmp = 0.08333333333333333 + (beta * ((beta * -0.011574074074074073) - 0.027777777777777776))
	else:
		tmp = 1.0 / (beta * (beta + 3.0))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.52)
		tmp = Float64(0.08333333333333333 + Float64(beta * Float64(Float64(beta * -0.011574074074074073) - 0.027777777777777776)));
	else
		tmp = Float64(1.0 / Float64(beta * Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.52)
		tmp = 0.08333333333333333 + (beta * ((beta * -0.011574074074074073) - 0.027777777777777776));
	else
		tmp = 1.0 / (beta * (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 1.52], N[(0.08333333333333333 + N[(beta * N[(N[(beta * -0.011574074074074073), $MachinePrecision] - 0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(beta * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.52

   1. Initial program 99.9%

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

   3. Simplified 96.5%

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

      1. times-frac 99.9%

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

      2. +-commutative 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in alpha around 0 65.0%

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

      1. +-commutative 65.0%

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

      2. +-commutative 65.0%

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

   9. Simplified 65.0%

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

   10. Taylor expanded in alpha around 0 65.1%

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

   11. Taylor expanded in beta around 0 64.6%

       \[\leadsto \color{blue}{0.08333333333333333 + \beta \cdot \left(-0.011574074074074073 \cdot \beta - 0.027777777777777776\right)} \]

   if 1.52 < beta

   1. Initial program 80.3%

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

   3. Taylor expanded in beta around inf 81.1%

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

   4. Taylor expanded in alpha around 0 75.8%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.52:\\ \;\;\;\;0.08333333333333333 + \beta \cdot \left(\beta \cdot -0.011574074074074073 - 0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 69.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.4:\\ \;\;\;\;\frac{0.25}{\alpha + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.4) (/ 0.25 (+ alpha 3.0)) (/ 1.0 (* beta (+ beta 3.0)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.4) {
		tmp = 0.25 / (alpha + 3.0);
	} else {
		tmp = 1.0 / (beta * (beta + 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.4d0) then
        tmp = 0.25d0 / (alpha + 3.0d0)
    else
        tmp = 1.0d0 / (beta * (beta + 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.4) {
		tmp = 0.25 / (alpha + 3.0);
	} else {
		tmp = 1.0 / (beta * (beta + 3.0));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.4:
		tmp = 0.25 / (alpha + 3.0)
	else:
		tmp = 1.0 / (beta * (beta + 3.0))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.4)
		tmp = Float64(0.25 / Float64(alpha + 3.0));
	else
		tmp = Float64(1.0 / Float64(beta * Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.4)
		tmp = 0.25 / (alpha + 3.0);
	else
		tmp = 1.0 / (beta * (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.4], N[(0.25 / N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(beta * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.39999999999999991

   1. Initial program 99.9%

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

   3. Taylor expanded in beta around 0 98.6%

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

   4. Taylor expanded in alpha around 0 65.8%

      \[\leadsto \frac{\color{blue}{0.25}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   5. Taylor expanded in beta around 0 65.2%

      \[\leadsto \color{blue}{\frac{0.25}{3 + \alpha}} \]

   if 2.39999999999999991 < beta

   1. Initial program 80.3%

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

   3. Taylor expanded in beta around inf 81.1%

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

   4. Taylor expanded in alpha around 0 75.8%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.4:\\ \;\;\;\;\frac{0.25}{\alpha + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 46.7% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.65:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.65)
   (+ 0.08333333333333333 (* beta -0.027777777777777776))
   (/ 0.25 beta)))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.65) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = 0.25 / 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 <= 2.65d0) then
        tmp = 0.08333333333333333d0 + (beta * (-0.027777777777777776d0))
    else
        tmp = 0.25d0 / beta
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.65) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = 0.25 / beta;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.65:
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776)
	else:
		tmp = 0.25 / beta
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.65)
		tmp = Float64(0.08333333333333333 + Float64(beta * -0.027777777777777776));
	else
		tmp = Float64(0.25 / beta);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.65)
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	else
		tmp = 0.25 / beta;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.65], N[(0.08333333333333333 + N[(beta * -0.027777777777777776), $MachinePrecision]), $MachinePrecision], N[(0.25 / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.64999999999999991

   1. Initial program 99.9%

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

   3. Taylor expanded in beta around 0 98.6%

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

   4. Taylor expanded in alpha around 0 64.1%

      \[\leadsto \color{blue}{\frac{0.25}{3 + \beta}} \]

   5. Taylor expanded in beta around 0 64.1%

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

      1. *-commutative 64.1%

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

   7. Simplified 64.1%

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

   if 2.64999999999999991 < beta

   1. Initial program 80.3%

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

   3. Taylor expanded in beta around 0 17.0%

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

   4. Taylor expanded in alpha around 0 6.8%

      \[\leadsto \color{blue}{\frac{0.25}{3 + \beta}} \]

   5. Taylor expanded in beta around inf 6.8%

      \[\leadsto \color{blue}{\frac{0.25}{\beta}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.65:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 47.4% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4:\\ \;\;\;\;\frac{0.25}{\alpha + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.0) (/ 0.25 (+ alpha 3.0)) (/ 0.25 beta)))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.0) {
		tmp = 0.25 / (alpha + 3.0);
	} else {
		tmp = 0.25 / 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 = 0.25d0 / (alpha + 3.0d0)
    else
        tmp = 0.25d0 / beta
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.0) {
		tmp = 0.25 / (alpha + 3.0);
	} else {
		tmp = 0.25 / beta;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 4.0:
		tmp = 0.25 / (alpha + 3.0)
	else:
		tmp = 0.25 / beta
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.0)
		tmp = Float64(0.25 / Float64(alpha + 3.0));
	else
		tmp = Float64(0.25 / beta);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4.0)
		tmp = 0.25 / (alpha + 3.0);
	else
		tmp = 0.25 / beta;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 4.0], N[(0.25 / N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision], N[(0.25 / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4

   1. Initial program 99.9%

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

   3. Taylor expanded in beta around 0 98.6%

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

   4. Taylor expanded in alpha around 0 65.8%

      \[\leadsto \frac{\color{blue}{0.25}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   5. Taylor expanded in beta around 0 65.2%

      \[\leadsto \color{blue}{\frac{0.25}{3 + \alpha}} \]

   if 4 < beta

   1. Initial program 80.3%

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

   3. Taylor expanded in beta around 0 17.0%

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

   4. Taylor expanded in alpha around 0 6.8%

      \[\leadsto \color{blue}{\frac{0.25}{3 + \beta}} \]

   5. Taylor expanded in beta around inf 6.8%

      \[\leadsto \color{blue}{\frac{0.25}{\beta}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4:\\ \;\;\;\;\frac{0.25}{\alpha + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 46.3% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.0) 0.08333333333333333 (/ 0.25 beta)))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.0) {
		tmp = 0.08333333333333333;
	} else {
		tmp = 0.25 / 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 <= 3.0d0) then
        tmp = 0.08333333333333333d0
    else
        tmp = 0.25d0 / beta
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.0) {
		tmp = 0.08333333333333333;
	} else {
		tmp = 0.25 / beta;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 3.0:
		tmp = 0.08333333333333333
	else:
		tmp = 0.25 / beta
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.0)
		tmp = 0.08333333333333333;
	else
		tmp = Float64(0.25 / beta);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.0)
		tmp = 0.08333333333333333;
	else
		tmp = 0.25 / beta;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 3.0], 0.08333333333333333, N[(0.25 / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3

   1. Initial program 99.9%

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

   3. Taylor expanded in beta around 0 98.6%

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

   4. Taylor expanded in alpha around 0 64.1%

      \[\leadsto \color{blue}{\frac{0.25}{3 + \beta}} \]

   5. Taylor expanded in beta around 0 63.5%

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

   if 3 < beta

   1. Initial program 80.3%

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

   3. Taylor expanded in beta around 0 17.0%

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

   4. Taylor expanded in alpha around 0 6.8%

      \[\leadsto \color{blue}{\frac{0.25}{3 + \beta}} \]

   5. Taylor expanded in beta around inf 6.8%

      \[\leadsto \color{blue}{\frac{0.25}{\beta}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 46.7% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.25}{\beta + 3} \end{array} \]

FPCore

(FPCore (alpha beta) :precision binary64 (/ 0.25 (+ beta 3.0)))

C

double code(double alpha, double beta) {
	return 0.25 / (beta + 3.0);
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = 0.25d0 / (beta + 3.0d0)
end function

Java

public static double code(double alpha, double beta) {
	return 0.25 / (beta + 3.0);
}

Python

def code(alpha, beta):
	return 0.25 / (beta + 3.0)

Julia

function code(alpha, beta)
	return Float64(0.25 / Float64(beta + 3.0))
end

MATLAB

function tmp = code(alpha, beta)
	tmp = 0.25 / (beta + 3.0);
end

Wolfram

code[alpha_, beta_] := N[(0.25 / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.9%

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

3. Taylor expanded in beta around 0 69.6%

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

4. Taylor expanded in alpha around 0 43.7%

   \[\leadsto \color{blue}{\frac{0.25}{3 + \beta}} \]

5. Final simplification 43.7%

   \[\leadsto \frac{0.25}{\beta + 3} \]
6. Add Preprocessing

Alternative 20: 45.4% 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 92.9%

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

3. Taylor expanded in beta around 0 69.6%

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

4. Taylor expanded in alpha around 0 43.7%

   \[\leadsto \color{blue}{\frac{0.25}{3 + \beta}} \]

5. Taylor expanded in beta around 0 42.3%

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

6. Final simplification 42.3%

   \[\leadsto 0.08333333333333333 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024131 
(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)))