Octave 3.8, jcobi/3

Percentage Accurate: 94.4% → 99.8%

Time: 19.8s

Alternatives: 19

Speedup: 1.4×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.4× speedup

Math

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

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(alpha + Float64(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[(alpha + N[(2.0 + beta), $MachinePrecision]), $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 95.2%

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

3. Simplified 96.7%

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

   1. associate-*r/ 96.8%

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

   2. +-commutative 96.8%

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

   3. associate-+r+ 96.8%

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

   4. +-commutative 96.8%

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

   5. associate-+r+ 96.8%

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

   6. associate-+r+ 96.8%

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

   7. +-commutative 96.8%

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

   8. associate-+r+ 96.8%

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

6. Applied egg-rr 96.8%

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

   1. times-frac 99.8%

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

   2. +-commutative 99.8%

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

   3. +-commutative 99.8%

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

   4. +-commutative 99.8%

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

   5. +-commutative 99.8%

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

   6. +-commutative 99.8%

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

   7. +-commutative 99.8%

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

8. Simplified 99.8%

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

   1. associate-*l/ 99.8%

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

   2. associate-+l+ 99.8%

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

   3. +-commutative 99.8%

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

   4. associate-+r+ 99.8%

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

   5. +-commutative 99.8%

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

   6. +-commutative 99.8%

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

   7. associate-+r+ 99.8%

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

   8. associate-+l+ 99.8%

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

   9. +-commutative 99.8%

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

   10. associate-+r+ 99.8%

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

   11. +-commutative 99.8%

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

10. Applied egg-rr 99.8%

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

11. Final simplification 99.8%

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

Alternative 2: 73.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \alpha + \left(2 + \beta\right)\\ \mathbf{if}\;\beta \leq 122000000:\\ \;\;\;\;\frac{\frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(2 + \beta\right)}}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{t\_0} \cdot \left(1 - \frac{\alpha + 2}{\beta}\right)}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ 2.0 beta))))
   (if (<= beta 122000000.0)
     (/
      (/ (+ 1.0 beta) (* (+ 2.0 beta) (+ 2.0 beta)))
      (+ 1.0 (+ 2.0 (+ alpha beta))))
     (/ (* (/ (+ 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 <= 122000000.0) {
		tmp = ((1.0 + beta) / ((2.0 + beta) * (2.0 + beta))) / (1.0 + (2.0 + (alpha + beta)));
	} 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 <= 122000000.0d0) then
        tmp = ((1.0d0 + beta) / ((2.0d0 + beta) * (2.0d0 + beta))) / (1.0d0 + (2.0d0 + (alpha + beta)))
    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 <= 122000000.0) {
		tmp = ((1.0 + beta) / ((2.0 + beta) * (2.0 + beta))) / (1.0 + (2.0 + (alpha + beta)));
	} 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 <= 122000000.0:
		tmp = ((1.0 + beta) / ((2.0 + beta) * (2.0 + beta))) / (1.0 + (2.0 + (alpha + beta)))
	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(alpha + Float64(2.0 + beta))
	tmp = 0.0
	if (beta <= 122000000.0)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(Float64(2.0 + beta) * Float64(2.0 + beta))) / Float64(1.0 + Float64(2.0 + Float64(alpha + beta))));
	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 <= 122000000.0)
		tmp = ((1.0 + beta) / ((2.0 + beta) * (2.0 + beta))) / (1.0 + (2.0 + (alpha + beta)));
	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[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 122000000.0], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(2.0 + beta), $MachinePrecision] * N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 + N[(alpha + beta), $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 < 1.22e8

   1. Initial program 99.8%

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

   3. Taylor expanded in alpha around 0 68.2%

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

      1. unpow2 68.2%

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

   5. Applied egg-rr 68.2%

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

   if 1.22e8 < beta

   1. Initial program 86.0%

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

   3. Simplified 91.0%

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

      1. associate-*r/ 91.1%

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

      2. +-commutative 91.1%

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

      3. associate-+r+ 91.1%

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

      4. +-commutative 91.1%

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

      5. associate-+r+ 91.1%

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

      6. associate-+r+ 91.1%

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

      7. +-commutative 91.1%

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

      8. associate-+r+ 91.1%

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

   6. Applied egg-rr 91.1%

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

      1. times-frac 99.8%

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

      2. +-commutative 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

   8. Simplified 99.8%

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

      1. associate-*l/ 99.8%

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

      2. associate-+l+ 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+r+ 99.8%

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

      5. +-commutative 99.8%

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

      6. +-commutative 99.8%

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

      7. associate-+r+ 99.8%

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

      8. associate-+l+ 99.8%

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

      9. +-commutative 99.8%

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

      10. associate-+r+ 99.8%

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

      11. +-commutative 99.8%

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

   10. Applied egg-rr 99.8%

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

   11. Taylor expanded in beta around inf 83.1%

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

       1. mul-1-neg 83.1%

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

       2. unsub-neg 83.1%

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

   13. Simplified 83.1%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 122000000:\\ \;\;\;\;\frac{\frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(2 + \beta\right)}}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \left(1 - \frac{\alpha + 2}{\beta}\right)}{\alpha + \left(2 + \beta\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \alpha + \left(2 + \beta\right)\\ \mathbf{if}\;\beta \leq 820000000:\\ \;\;\;\;\frac{\frac{1 + \left(\alpha + \beta\right)}{t\_0}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{t\_0} \cdot \left(1 - \frac{\alpha + 2}{\beta}\right)}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ 2.0 beta))))
   (if (<= beta 820000000.0)
     (/ (/ (+ 1.0 (+ alpha 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 <= 820000000.0) {
		tmp = ((1.0 + (alpha + 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 <= 820000000.0d0) then
        tmp = ((1.0d0 + (alpha + 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 <= 820000000.0) {
		tmp = ((1.0 + (alpha + 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 <= 820000000.0:
		tmp = ((1.0 + (alpha + 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(alpha + Float64(2.0 + beta))
	tmp = 0.0
	if (beta <= 820000000.0)
		tmp = Float64(Float64(Float64(1.0 + Float64(alpha + beta)) / t_0) / Float64(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 <= 820000000.0)
		tmp = ((1.0 + (alpha + 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[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 820000000.0], N[(N[(N[(1.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(N[(N[(alpha + beta), $MachinePrecision] + 3.0), $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 < 8.2e8

   1. Initial program 99.8%

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

      1. associate-/l/ 99.6%

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

      2. +-commutative 99.6%

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

      3. associate-+l+ 99.6%

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

      4. *-commutative 99.6%

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

      5. metadata-eval 99.6%

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

      6. associate-+l+ 99.6%

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

      7. metadata-eval 99.6%

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

      8. +-commutative 99.6%

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

      9. metadata-eval 99.6%

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

      10. metadata-eval 99.6%

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

      11. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in alpha around 0 99.2%

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

   if 8.2e8 < beta

   1. Initial program 86.0%

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

   3. Simplified 91.0%

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

      1. associate-*r/ 91.1%

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

      2. +-commutative 91.1%

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

      3. associate-+r+ 91.1%

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

      4. +-commutative 91.1%

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

      5. associate-+r+ 91.1%

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

      6. associate-+r+ 91.1%

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

      7. +-commutative 91.1%

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

      8. associate-+r+ 91.1%

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

   6. Applied egg-rr 91.1%

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

      1. times-frac 99.8%

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

      2. +-commutative 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

   8. Simplified 99.8%

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

      1. associate-*l/ 99.8%

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

      2. associate-+l+ 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+r+ 99.8%

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

      5. +-commutative 99.8%

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

      6. +-commutative 99.8%

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

      7. associate-+r+ 99.8%

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

      8. associate-+l+ 99.8%

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

      9. +-commutative 99.8%

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

      10. associate-+r+ 99.8%

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

      11. +-commutative 99.8%

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

   10. Applied egg-rr 99.8%

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

   11. Taylor expanded in beta around inf 83.1%

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

       1. mul-1-neg 83.1%

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

       2. unsub-neg 83.1%

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

   13. Simplified 83.1%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 820000000:\\ \;\;\;\;\frac{\frac{1 + \left(\alpha + \beta\right)}{\alpha + \left(2 + \beta\right)}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\alpha + \left(2 + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \left(1 - \frac{\alpha + 2}{\beta}\right)}{\alpha + \left(2 + \beta\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.2%

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

3. Simplified 96.7%

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

   1. associate-*r/ 96.8%

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

   2. +-commutative 96.8%

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

   3. associate-+r+ 96.8%

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

   4. +-commutative 96.8%

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

   5. associate-+r+ 96.8%

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

   6. associate-+r+ 96.8%

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

   7. +-commutative 96.8%

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

   8. associate-+r+ 96.8%

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

6. Applied egg-rr 96.8%

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

   1. times-frac 99.8%

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

   2. +-commutative 99.8%

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

   3. +-commutative 99.8%

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

   4. +-commutative 99.8%

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

   5. +-commutative 99.8%

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

   6. +-commutative 99.8%

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

   7. +-commutative 99.8%

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

8. Simplified 99.8%

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

9. Final simplification 99.8%

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

Alternative 5: 89.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \alpha + \left(2 + \beta\right)\\ \mathbf{if}\;\beta \leq 2.2:\\ \;\;\;\;\frac{1 + \alpha}{t\_0 \cdot \left(\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{t\_0} \cdot \frac{1}{4 + \left(\beta + \alpha \cdot 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ 2.0 beta))))
   (if (<= beta 2.2)
     (/ (+ 1.0 alpha) (* t_0 (* (+ alpha 2.0) (+ alpha 3.0))))
     (* (/ (+ 1.0 alpha) t_0) (/ 1.0 (+ 4.0 (+ beta (* alpha 2.0))))))))

C

double code(double alpha, double beta) {
	double t_0 = alpha + (2.0 + beta);
	double tmp;
	if (beta <= 2.2) {
		tmp = (1.0 + alpha) / (t_0 * ((alpha + 2.0) * (alpha + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / t_0) * (1.0 / (4.0 + (beta + (alpha * 2.0))));
	}
	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 <= 2.2d0) then
        tmp = (1.0d0 + alpha) / (t_0 * ((alpha + 2.0d0) * (alpha + 3.0d0)))
    else
        tmp = ((1.0d0 + alpha) / t_0) * (1.0d0 / (4.0d0 + (beta + (alpha * 2.0d0))))
    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 <= 2.2) {
		tmp = (1.0 + alpha) / (t_0 * ((alpha + 2.0) * (alpha + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / t_0) * (1.0 / (4.0 + (beta + (alpha * 2.0))));
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = alpha + (2.0 + beta)
	tmp = 0
	if beta <= 2.2:
		tmp = (1.0 + alpha) / (t_0 * ((alpha + 2.0) * (alpha + 3.0)))
	else:
		tmp = ((1.0 + alpha) / t_0) * (1.0 / (4.0 + (beta + (alpha * 2.0))))
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(alpha + Float64(2.0 + beta))
	tmp = 0.0
	if (beta <= 2.2)
		tmp = Float64(Float64(1.0 + alpha) / Float64(t_0 * Float64(Float64(alpha + 2.0) * Float64(alpha + 3.0))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / t_0) * Float64(1.0 / Float64(4.0 + Float64(beta + Float64(alpha * 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = alpha + (2.0 + beta);
	tmp = 0.0;
	if (beta <= 2.2)
		tmp = (1.0 + alpha) / (t_0 * ((alpha + 2.0) * (alpha + 3.0)));
	else
		tmp = ((1.0 + alpha) / t_0) * (1.0 / (4.0 + (beta + (alpha * 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2.2], N[(N[(1.0 + alpha), $MachinePrecision] / N[(t$95$0 * N[(N[(alpha + 2.0), $MachinePrecision] * N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(1.0 / N[(4.0 + N[(beta + N[(alpha * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.2000000000000002

   1. Initial program 99.8%

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

   3. Simplified 96.3%

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

   5. Taylor expanded in beta around 0 93.5%

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

   6. Taylor expanded in beta around 0 93.7%

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

      1. +-commutative 93.7%

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

   8. Simplified 93.7%

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

   if 2.2000000000000002 < beta

   1. Initial program 86.2%

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

   3. Simplified 91.1%

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

      1. clear-num 91.1%

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

      2. inv-pow 91.1%

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

      3. associate-+r+ 91.1%

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

      4. +-commutative 91.1%

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

      5. associate-+r+ 91.1%

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

   6. Applied egg-rr 91.1%

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

      1. unpow-1 91.1%

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

      2. associate-/l* 99.4%

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

      3. +-commutative 99.4%

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

      4. +-commutative 99.4%

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

      5. +-commutative 99.4%

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

      6. +-commutative 99.4%

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

   8. Simplified 99.4%

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

   9. Taylor expanded in beta around inf 83.8%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.2:\\ \;\;\;\;\frac{1 + \alpha}{\left(\alpha + \left(2 + \beta\right)\right) \cdot \left(\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1}{4 + \left(\beta + \alpha \cdot 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 85000000:\\ \;\;\;\;\frac{\frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(2 + \beta\right)}}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1}{4 + \left(\beta + \alpha \cdot 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 85000000.0)
   (/
    (/ (+ 1.0 beta) (* (+ 2.0 beta) (+ 2.0 beta)))
    (+ 1.0 (+ 2.0 (+ alpha beta))))
   (*
    (/ (+ 1.0 alpha) (+ alpha (+ 2.0 beta)))
    (/ 1.0 (+ 4.0 (+ beta (* alpha 2.0)))))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 85000000.0) {
		tmp = ((1.0 + beta) / ((2.0 + beta) * (2.0 + beta))) / (1.0 + (2.0 + (alpha + beta)));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) * (1.0 / (4.0 + (beta + (alpha * 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 85000000.0d0) then
        tmp = ((1.0d0 + beta) / ((2.0d0 + beta) * (2.0d0 + beta))) / (1.0d0 + (2.0d0 + (alpha + beta)))
    else
        tmp = ((1.0d0 + alpha) / (alpha + (2.0d0 + beta))) * (1.0d0 / (4.0d0 + (beta + (alpha * 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 85000000.0) {
		tmp = ((1.0 + beta) / ((2.0 + beta) * (2.0 + beta))) / (1.0 + (2.0 + (alpha + beta)));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) * (1.0 / (4.0 + (beta + (alpha * 2.0))));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 85000000.0:
		tmp = ((1.0 + beta) / ((2.0 + beta) * (2.0 + beta))) / (1.0 + (2.0 + (alpha + beta)))
	else:
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) * (1.0 / (4.0 + (beta + (alpha * 2.0))))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 85000000.0)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(Float64(2.0 + beta) * Float64(2.0 + beta))) / Float64(1.0 + Float64(2.0 + Float64(alpha + beta))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(2.0 + beta))) * Float64(1.0 / Float64(4.0 + Float64(beta + Float64(alpha * 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 85000000.0)
		tmp = ((1.0 + beta) / ((2.0 + beta) * (2.0 + beta))) / (1.0 + (2.0 + (alpha + beta)));
	else
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) * (1.0 / (4.0 + (beta + (alpha * 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 85000000.0], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(2.0 + beta), $MachinePrecision] * N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(4.0 + N[(beta + N[(alpha * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 8.5e7

   1. Initial program 99.8%

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

   3. Taylor expanded in alpha around 0 68.0%

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

      1. unpow2 68.0%

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

   5. Applied egg-rr 68.0%

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

   if 8.5e7 < beta

   1. Initial program 86.2%

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

   3. Simplified 91.1%

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

      1. clear-num 91.1%

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

      2. inv-pow 91.1%

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

      3. associate-+r+ 91.1%

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

      4. +-commutative 91.1%

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

      5. associate-+r+ 91.1%

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

   6. Applied egg-rr 91.1%

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

      1. unpow-1 91.1%

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

      2. associate-/l* 99.4%

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

      3. +-commutative 99.4%

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

      4. +-commutative 99.4%

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

      5. +-commutative 99.4%

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

      6. +-commutative 99.4%

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

   8. Simplified 99.4%

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

   9. Taylor expanded in beta around inf 83.8%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 85000000:\\ \;\;\;\;\frac{\frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(2 + \beta\right)}}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1}{4 + \left(\beta + \alpha \cdot 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 32:\\ \;\;\;\;\frac{1 + \alpha}{\left(\alpha + \left(2 + \beta\right)\right) \cdot \left(\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 32.0)
   (/ (+ 1.0 alpha) (* (+ alpha (+ 2.0 beta)) (* (+ alpha 2.0) (+ alpha 3.0))))
   (/ (/ (- alpha -1.0) beta) (+ 1.0 (+ 2.0 (+ alpha beta))))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 32.0) {
		tmp = (1.0 + alpha) / ((alpha + (2.0 + beta)) * ((alpha + 2.0) * (alpha + 3.0)));
	} else {
		tmp = ((alpha - -1.0) / 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 <= 32.0d0) then
        tmp = (1.0d0 + alpha) / ((alpha + (2.0d0 + beta)) * ((alpha + 2.0d0) * (alpha + 3.0d0)))
    else
        tmp = ((alpha - (-1.0d0)) / 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 <= 32.0) {
		tmp = (1.0 + alpha) / ((alpha + (2.0 + beta)) * ((alpha + 2.0) * (alpha + 3.0)));
	} else {
		tmp = ((alpha - -1.0) / beta) / (1.0 + (2.0 + (alpha + beta)));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 32.0:
		tmp = (1.0 + alpha) / ((alpha + (2.0 + beta)) * ((alpha + 2.0) * (alpha + 3.0)))
	else:
		tmp = ((alpha - -1.0) / beta) / (1.0 + (2.0 + (alpha + beta)))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 32.0)
		tmp = Float64(Float64(1.0 + alpha) / Float64(Float64(alpha + Float64(2.0 + beta)) * Float64(Float64(alpha + 2.0) * Float64(alpha + 3.0))));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / 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 <= 32.0)
		tmp = (1.0 + alpha) / ((alpha + (2.0 + beta)) * ((alpha + 2.0) * (alpha + 3.0)));
	else
		tmp = ((alpha - -1.0) / beta) / (1.0 + (2.0 + (alpha + beta)));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 32.0], N[(N[(1.0 + alpha), $MachinePrecision] / N[(N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] * N[(N[(alpha + 2.0), $MachinePrecision] * N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $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 < 32

   1. Initial program 99.8%

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

   3. Simplified 96.3%

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

   5. Taylor expanded in beta around 0 93.5%

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

   6. Taylor expanded in beta around 0 93.7%

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

      1. +-commutative 93.7%

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

   8. Simplified 93.7%

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

   if 32 < beta

   1. Initial program 86.2%

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

   3. Taylor expanded in beta around -inf 82.6%

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

      1. associate-*r/ 82.6%

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

      2. mul-1-neg 82.6%

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

      3. sub-neg 82.6%

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

      4. mul-1-neg 82.6%

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

      5. distribute-neg-in 82.6%

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

      6. +-commutative 82.6%

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

      7. mul-1-neg 82.6%

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

      8. distribute-lft-in 82.6%

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

      9. metadata-eval 82.6%

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

      10. mul-1-neg 82.6%

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

      11. unsub-neg 82.6%

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

   5. Simplified 82.6%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 32:\\ \;\;\;\;\frac{1 + \alpha}{\left(\alpha + \left(2 + \beta\right)\right) \cdot \left(\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \alpha + \left(2 + \beta\right)\\ \mathbf{if}\;\beta \leq 1.42 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(\beta + 3\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ 2.0 beta))))
   (if (<= beta 1.42e+16)
     (/ (/ (+ 1.0 beta) (* (+ 2.0 beta) (+ beta 3.0))) t_0)
     (/ (/ (+ 1.0 alpha) beta) t_0))))

C

double code(double alpha, double beta) {
	double t_0 = alpha + (2.0 + beta);
	double tmp;
	if (beta <= 1.42e+16) {
		tmp = ((1.0 + beta) / ((2.0 + beta) * (beta + 3.0))) / t_0;
	} else {
		tmp = ((1.0 + alpha) / 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 <= 1.42d+16) then
        tmp = ((1.0d0 + beta) / ((2.0d0 + beta) * (beta + 3.0d0))) / t_0
    else
        tmp = ((1.0d0 + alpha) / 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 <= 1.42e+16) {
		tmp = ((1.0 + beta) / ((2.0 + beta) * (beta + 3.0))) / t_0;
	} else {
		tmp = ((1.0 + alpha) / beta) / t_0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = alpha + (2.0 + beta)
	tmp = 0
	if beta <= 1.42e+16:
		tmp = ((1.0 + beta) / ((2.0 + beta) * (beta + 3.0))) / t_0
	else:
		tmp = ((1.0 + alpha) / beta) / t_0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(alpha + Float64(2.0 + beta))
	tmp = 0.0
	if (beta <= 1.42e+16)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(Float64(2.0 + beta) * Float64(beta + 3.0))) / t_0);
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / 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 <= 1.42e+16)
		tmp = ((1.0 + beta) / ((2.0 + beta) * (beta + 3.0))) / t_0;
	else
		tmp = ((1.0 + alpha) / beta) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1.42e+16], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(2.0 + beta), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.42e16

   1. Initial program 99.8%

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

   3. Simplified 99.6%

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

      1. associate-*r/ 99.6%

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

      2. +-commutative 99.6%

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

      3. associate-+r+ 99.6%

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

      4. +-commutative 99.6%

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

      5. associate-+r+ 99.6%

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

      6. associate-+r+ 99.6%

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

      7. +-commutative 99.6%

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

      8. associate-+r+ 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. times-frac 99.8%

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

      2. +-commutative 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

   8. Simplified 99.8%

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

      1. associate-*l/ 99.9%

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

      2. associate-+l+ 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-+r+ 99.9%

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

      5. +-commutative 99.9%

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

      6. +-commutative 99.9%

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

      7. associate-+r+ 99.9%

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

      8. associate-+l+ 99.9%

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

      9. +-commutative 99.9%

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

      10. associate-+r+ 99.9%

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

      11. +-commutative 99.9%

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

   10. Applied egg-rr 99.9%

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

   11. Taylor expanded in alpha around 0 67.6%

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

       1. +-commutative 67.6%

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

       2. +-commutative 67.6%

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

   13. Simplified 67.6%

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

   if 1.42e16 < beta

   1. Initial program 85.6%

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

   3. Simplified 90.7%

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

   5. Taylor expanded in beta around inf 83.7%

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

      1. associate-*l/ 83.8%

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

      2. +-commutative 83.8%

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

      3. div-inv 83.8%

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

      4. +-commutative 83.8%

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

   7. Applied egg-rr 83.8%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.42 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(\beta + 3\right)}}{\alpha + \left(2 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(2 + \beta\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 89.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 30:\\ \;\;\;\;\frac{1 + \alpha}{\left(\alpha + 2\right) \cdot \left(\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(2 + \beta\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 30.0)
   (/ (+ 1.0 alpha) (* (+ alpha 2.0) (* (+ alpha 2.0) (+ alpha 3.0))))
   (/ (/ (+ 1.0 alpha) beta) (+ alpha (+ 2.0 beta)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 30.0) {
		tmp = (1.0 + alpha) / ((alpha + 2.0) * ((alpha + 2.0) * (alpha + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (2.0 + beta));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 30.0d0) then
        tmp = (1.0d0 + alpha) / ((alpha + 2.0d0) * ((alpha + 2.0d0) * (alpha + 3.0d0)))
    else
        tmp = ((1.0d0 + alpha) / beta) / (alpha + (2.0d0 + beta))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 30.0) {
		tmp = (1.0 + alpha) / ((alpha + 2.0) * ((alpha + 2.0) * (alpha + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (2.0 + beta));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 30.0:
		tmp = (1.0 + alpha) / ((alpha + 2.0) * ((alpha + 2.0) * (alpha + 3.0)))
	else:
		tmp = ((1.0 + alpha) / beta) / (alpha + (2.0 + beta))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 30.0)
		tmp = Float64(Float64(1.0 + alpha) / Float64(Float64(alpha + 2.0) * Float64(Float64(alpha + 2.0) * Float64(alpha + 3.0))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(alpha + Float64(2.0 + beta)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 30.0)
		tmp = (1.0 + alpha) / ((alpha + 2.0) * ((alpha + 2.0) * (alpha + 3.0)));
	else
		tmp = ((1.0 + alpha) / beta) / (alpha + (2.0 + beta));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 30.0], N[(N[(1.0 + alpha), $MachinePrecision] / N[(N[(alpha + 2.0), $MachinePrecision] * N[(N[(alpha + 2.0), $MachinePrecision] * N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 30

   1. Initial program 99.8%

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

   3. Simplified 96.3%

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

   5. Taylor expanded in beta around 0 93.5%

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

   6. Taylor expanded in beta around 0 93.7%

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

      1. +-commutative 93.7%

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

   8. Simplified 93.7%

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

   9. Taylor expanded in beta around 0 93.6%

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

   if 30 < beta

   1. Initial program 86.2%

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

   3. Simplified 91.1%

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

   5. Taylor expanded in beta around inf 82.5%

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

      1. associate-*l/ 82.5%

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

      2. +-commutative 82.5%

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

      3. div-inv 82.6%

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

      4. +-commutative 82.6%

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

   7. Applied egg-rr 82.6%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 30:\\ \;\;\;\;\frac{1 + \alpha}{\left(\alpha + 2\right) \cdot \left(\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(2 + \beta\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 89.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.7:\\ \;\;\;\;\frac{1 + \alpha}{\left(\alpha + 2\right) \cdot \left(\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.7)
   (/ (+ 1.0 alpha) (* (+ alpha 2.0) (* (+ alpha 2.0) (+ alpha 3.0))))
   (/ (/ (- alpha -1.0) beta) (+ 1.0 (+ 2.0 (+ alpha beta))))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.7) {
		tmp = (1.0 + alpha) / ((alpha + 2.0) * ((alpha + 2.0) * (alpha + 3.0)));
	} else {
		tmp = ((alpha - -1.0) / beta) / (1.0 + (2.0 + (alpha + beta)));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.7d0) then
        tmp = (1.0d0 + alpha) / ((alpha + 2.0d0) * ((alpha + 2.0d0) * (alpha + 3.0d0)))
    else
        tmp = ((alpha - (-1.0d0)) / beta) / (1.0d0 + (2.0d0 + (alpha + beta)))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.7) {
		tmp = (1.0 + alpha) / ((alpha + 2.0) * ((alpha + 2.0) * (alpha + 3.0)));
	} else {
		tmp = ((alpha - -1.0) / beta) / (1.0 + (2.0 + (alpha + beta)));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.7:
		tmp = (1.0 + alpha) / ((alpha + 2.0) * ((alpha + 2.0) * (alpha + 3.0)))
	else:
		tmp = ((alpha - -1.0) / beta) / (1.0 + (2.0 + (alpha + beta)))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.7)
		tmp = Float64(Float64(1.0 + alpha) / Float64(Float64(alpha + 2.0) * Float64(Float64(alpha + 2.0) * Float64(alpha + 3.0))));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / beta) / Float64(1.0 + Float64(2.0 + Float64(alpha + beta))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.7)
		tmp = (1.0 + alpha) / ((alpha + 2.0) * ((alpha + 2.0) * (alpha + 3.0)));
	else
		tmp = ((alpha - -1.0) / beta) / (1.0 + (2.0 + (alpha + beta)));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.7], N[(N[(1.0 + alpha), $MachinePrecision] / N[(N[(alpha + 2.0), $MachinePrecision] * N[(N[(alpha + 2.0), $MachinePrecision] * N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(1.0 + N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.7000000000000002

   1. Initial program 99.8%

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

   3. Simplified 96.3%

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

   5. Taylor expanded in beta around 0 93.5%

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

   6. Taylor expanded in beta around 0 93.7%

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

      1. +-commutative 93.7%

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

   8. Simplified 93.7%

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

   9. Taylor expanded in beta around 0 93.6%

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

   if 2.7000000000000002 < beta

   1. Initial program 86.2%

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

   3. Taylor expanded in beta around -inf 82.6%

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

      1. associate-*r/ 82.6%

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

      2. mul-1-neg 82.6%

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

      3. sub-neg 82.6%

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

      4. mul-1-neg 82.6%

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

      5. distribute-neg-in 82.6%

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

      6. +-commutative 82.6%

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

      7. mul-1-neg 82.6%

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

      8. distribute-lft-in 82.6%

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

      9. metadata-eval 82.6%

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

      10. mul-1-neg 82.6%

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

      11. unsub-neg 82.6%

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

   5. Simplified 82.6%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.7:\\ \;\;\;\;\frac{1 + \alpha}{\left(\alpha + 2\right) \cdot \left(\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 72.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.2:\\ \;\;\;\;\frac{0.25}{\alpha + \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(2 + \beta\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5.2)
   (/ 0.25 (+ alpha (+ beta 3.0)))
   (/ (/ (+ 1.0 alpha) beta) (+ alpha (+ 2.0 beta)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.2) {
		tmp = 0.25 / (alpha + (beta + 3.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (2.0 + beta));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 5.2d0) then
        tmp = 0.25d0 / (alpha + (beta + 3.0d0))
    else
        tmp = ((1.0d0 + alpha) / beta) / (alpha + (2.0d0 + beta))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.2) {
		tmp = 0.25 / (alpha + (beta + 3.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (2.0 + beta));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 5.2:
		tmp = 0.25 / (alpha + (beta + 3.0))
	else:
		tmp = ((1.0 + alpha) / beta) / (alpha + (2.0 + beta))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 5.2)
		tmp = Float64(0.25 / Float64(alpha + Float64(beta + 3.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(alpha + Float64(2.0 + beta)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 5.2)
		tmp = 0.25 / (alpha + (beta + 3.0));
	else
		tmp = ((1.0 + alpha) / beta) / (alpha + (2.0 + beta));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 5.2], N[(0.25 / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.20000000000000018

   1. Initial program 99.8%

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

   3. Taylor expanded in alpha around 0 68.0%

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

   4. Taylor expanded in beta around 0 66.8%

      \[\leadsto \frac{\color{blue}{0.25}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   5. Step-by-step derivation

      1. expm1-log1p-u 66.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.25}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\right)\right)} \]

      2. expm1-udef 81.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{0.25}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\right)} - 1} \]

      3. metadata-eval 81.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{0.25}{\left(\left(\alpha + \beta\right) + \color{blue}{2}\right) + 1}\right)} - 1 \]

      4. associate-+l+ 81.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{0.25}{\color{blue}{\left(\alpha + \beta\right) + \left(2 + 1\right)}}\right)} - 1 \]

      5. metadata-eval 81.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{0.25}{\left(\alpha + \beta\right) + \color{blue}{3}}\right)} - 1 \]

      6. associate-+r+ 81.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{0.25}{\color{blue}{\alpha + \left(\beta + 3\right)}}\right)} - 1 \]

   6. Applied egg-rr 81.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{0.25}{\alpha + \left(\beta + 3\right)}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 66.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.25}{\alpha + \left(\beta + 3\right)}\right)\right)} \]

      2. expm1-log1p 66.8%

         \[\leadsto \color{blue}{\frac{0.25}{\alpha + \left(\beta + 3\right)}} \]

      3. +-commutative 66.8%

         \[\leadsto \frac{0.25}{\alpha + \color{blue}{\left(3 + \beta\right)}} \]

      4. +-commutative 66.8%

         \[\leadsto \frac{0.25}{\color{blue}{\left(3 + \beta\right) + \alpha}} \]

      5. +-commutative 66.8%

         \[\leadsto \frac{0.25}{\color{blue}{\left(\beta + 3\right)} + \alpha} \]

   8. Simplified 66.8%

      \[\leadsto \color{blue}{\frac{0.25}{\left(\beta + 3\right) + \alpha}} \]

   if 5.20000000000000018 < beta

   1. Initial program 86.2%

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

   3. Simplified 91.1%

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

   5. Taylor expanded in beta around inf 82.5%

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

      1. associate-*l/ 82.5%

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

      2. +-commutative 82.5%

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

      3. div-inv 82.6%

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

      4. +-commutative 82.6%

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

   7. Applied egg-rr 82.6%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.2:\\ \;\;\;\;\frac{0.25}{\alpha + \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(2 + \beta\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 72.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.4:\\ \;\;\;\;\frac{0.25}{\alpha + \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta} \cdot \frac{1}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.4)
   (/ 0.25 (+ alpha (+ beta 3.0)))
   (* (/ (+ 1.0 alpha) beta) (/ 1.0 beta))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.4) {
		tmp = 0.25 / (alpha + (beta + 3.0));
	} else {
		tmp = ((1.0 + alpha) / beta) * (1.0 / beta);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 6.4d0) then
        tmp = 0.25d0 / (alpha + (beta + 3.0d0))
    else
        tmp = ((1.0d0 + alpha) / beta) * (1.0d0 / beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.4) {
		tmp = 0.25 / (alpha + (beta + 3.0));
	} else {
		tmp = ((1.0 + alpha) / beta) * (1.0 / beta);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 6.4:
		tmp = 0.25 / (alpha + (beta + 3.0))
	else:
		tmp = ((1.0 + alpha) / beta) * (1.0 / beta)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.4)
		tmp = Float64(0.25 / Float64(alpha + Float64(beta + 3.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) * Float64(1.0 / beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 6.4)
		tmp = 0.25 / (alpha + (beta + 3.0));
	else
		tmp = ((1.0 + alpha) / beta) * (1.0 / beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 6.4], N[(0.25 / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] * N[(1.0 / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.4000000000000004

   1. Initial program 99.8%

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

   3. Taylor expanded in alpha around 0 68.0%

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

   4. Taylor expanded in beta around 0 66.8%

      \[\leadsto \frac{\color{blue}{0.25}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   5. Step-by-step derivation

      1. expm1-log1p-u 66.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.25}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\right)\right)} \]

      2. expm1-udef 81.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{0.25}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\right)} - 1} \]

      3. metadata-eval 81.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{0.25}{\left(\left(\alpha + \beta\right) + \color{blue}{2}\right) + 1}\right)} - 1 \]

      4. associate-+l+ 81.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{0.25}{\color{blue}{\left(\alpha + \beta\right) + \left(2 + 1\right)}}\right)} - 1 \]

      5. metadata-eval 81.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{0.25}{\left(\alpha + \beta\right) + \color{blue}{3}}\right)} - 1 \]

      6. associate-+r+ 81.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{0.25}{\color{blue}{\alpha + \left(\beta + 3\right)}}\right)} - 1 \]

   6. Applied egg-rr 81.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{0.25}{\alpha + \left(\beta + 3\right)}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 66.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.25}{\alpha + \left(\beta + 3\right)}\right)\right)} \]

      2. expm1-log1p 66.8%

         \[\leadsto \color{blue}{\frac{0.25}{\alpha + \left(\beta + 3\right)}} \]

      3. +-commutative 66.8%

         \[\leadsto \frac{0.25}{\alpha + \color{blue}{\left(3 + \beta\right)}} \]

      4. +-commutative 66.8%

         \[\leadsto \frac{0.25}{\color{blue}{\left(3 + \beta\right) + \alpha}} \]

      5. +-commutative 66.8%

         \[\leadsto \frac{0.25}{\color{blue}{\left(\beta + 3\right)} + \alpha} \]

   8. Simplified 66.8%

      \[\leadsto \color{blue}{\frac{0.25}{\left(\beta + 3\right) + \alpha}} \]

   if 6.4000000000000004 < beta

   1. Initial program 86.2%

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

   3. Simplified 91.1%

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

   5. Taylor expanded in beta around inf 82.5%

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

   6. Taylor expanded in beta around inf 82.3%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.4:\\ \;\;\;\;\frac{0.25}{\alpha + \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta} \cdot \frac{1}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 70.7% accurate, 2.9× speedup

Math

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

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5.0)
   (/ 0.25 (+ alpha (+ beta 3.0)))
   (/ 1.0 (* beta (+ 2.0 beta)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.0) {
		tmp = 0.25 / (alpha + (beta + 3.0));
	} else {
		tmp = 1.0 / (beta * (2.0 + beta));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 5.0d0) then
        tmp = 0.25d0 / (alpha + (beta + 3.0d0))
    else
        tmp = 1.0d0 / (beta * (2.0d0 + beta))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.0) {
		tmp = 0.25 / (alpha + (beta + 3.0));
	} else {
		tmp = 1.0 / (beta * (2.0 + beta));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 5.0:
		tmp = 0.25 / (alpha + (beta + 3.0))
	else:
		tmp = 1.0 / (beta * (2.0 + beta))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 5.0)
		tmp = Float64(0.25 / Float64(alpha + Float64(beta + 3.0)));
	else
		tmp = Float64(1.0 / Float64(beta * Float64(2.0 + beta)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 5.0)
		tmp = 0.25 / (alpha + (beta + 3.0));
	else
		tmp = 1.0 / (beta * (2.0 + beta));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 5.0], N[(0.25 / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(beta * N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5

   1. Initial program 99.8%

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

   3. Taylor expanded in alpha around 0 68.0%

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

   4. Taylor expanded in beta around 0 66.8%

      \[\leadsto \frac{\color{blue}{0.25}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   5. Step-by-step derivation

      1. expm1-log1p-u 66.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.25}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\right)\right)} \]

      2. expm1-udef 81.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{0.25}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\right)} - 1} \]

      3. metadata-eval 81.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{0.25}{\left(\left(\alpha + \beta\right) + \color{blue}{2}\right) + 1}\right)} - 1 \]

      4. associate-+l+ 81.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{0.25}{\color{blue}{\left(\alpha + \beta\right) + \left(2 + 1\right)}}\right)} - 1 \]

      5. metadata-eval 81.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{0.25}{\left(\alpha + \beta\right) + \color{blue}{3}}\right)} - 1 \]

      6. associate-+r+ 81.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{0.25}{\color{blue}{\alpha + \left(\beta + 3\right)}}\right)} - 1 \]

   6. Applied egg-rr 81.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{0.25}{\alpha + \left(\beta + 3\right)}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 66.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.25}{\alpha + \left(\beta + 3\right)}\right)\right)} \]

      2. expm1-log1p 66.8%

         \[\leadsto \color{blue}{\frac{0.25}{\alpha + \left(\beta + 3\right)}} \]

      3. +-commutative 66.8%

         \[\leadsto \frac{0.25}{\alpha + \color{blue}{\left(3 + \beta\right)}} \]

      4. +-commutative 66.8%

         \[\leadsto \frac{0.25}{\color{blue}{\left(3 + \beta\right) + \alpha}} \]

      5. +-commutative 66.8%

         \[\leadsto \frac{0.25}{\color{blue}{\left(\beta + 3\right)} + \alpha} \]

   8. Simplified 66.8%

      \[\leadsto \color{blue}{\frac{0.25}{\left(\beta + 3\right) + \alpha}} \]

   if 5 < beta

   1. Initial program 86.2%

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

   3. Simplified 91.1%

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

   5. Taylor expanded in beta around inf 82.5%

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

   6. Taylor expanded in alpha around 0 73.3%

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

4. Final simplification 69.0%

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

Alternative 14: 70.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.8:\\ \;\;\;\;\frac{0.25}{\alpha + \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2 + \beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.8)
   (/ 0.25 (+ alpha (+ beta 3.0)))
   (/ (/ 1.0 (+ 2.0 beta)) beta)))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.8) {
		tmp = 0.25 / (alpha + (beta + 3.0));
	} else {
		tmp = (1.0 / (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 <= 4.8d0) then
        tmp = 0.25d0 / (alpha + (beta + 3.0d0))
    else
        tmp = (1.0d0 / (2.0d0 + beta)) / beta
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.8) {
		tmp = 0.25 / (alpha + (beta + 3.0));
	} else {
		tmp = (1.0 / (2.0 + beta)) / beta;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 4.8:
		tmp = 0.25 / (alpha + (beta + 3.0))
	else:
		tmp = (1.0 / (2.0 + beta)) / beta
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.8)
		tmp = Float64(0.25 / Float64(alpha + Float64(beta + 3.0)));
	else
		tmp = Float64(Float64(1.0 / Float64(2.0 + beta)) / beta);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4.8)
		tmp = 0.25 / (alpha + (beta + 3.0));
	else
		tmp = (1.0 / (2.0 + beta)) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 4.8], N[(0.25 / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.79999999999999982

   1. Initial program 99.8%

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

   3. Taylor expanded in alpha around 0 68.0%

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

   4. Taylor expanded in beta around 0 66.8%

      \[\leadsto \frac{\color{blue}{0.25}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   5. Step-by-step derivation

      1. expm1-log1p-u 66.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.25}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\right)\right)} \]

      2. expm1-udef 81.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{0.25}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\right)} - 1} \]

      3. metadata-eval 81.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{0.25}{\left(\left(\alpha + \beta\right) + \color{blue}{2}\right) + 1}\right)} - 1 \]

      4. associate-+l+ 81.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{0.25}{\color{blue}{\left(\alpha + \beta\right) + \left(2 + 1\right)}}\right)} - 1 \]

      5. metadata-eval 81.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{0.25}{\left(\alpha + \beta\right) + \color{blue}{3}}\right)} - 1 \]

      6. associate-+r+ 81.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{0.25}{\color{blue}{\alpha + \left(\beta + 3\right)}}\right)} - 1 \]

   6. Applied egg-rr 81.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{0.25}{\alpha + \left(\beta + 3\right)}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 66.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.25}{\alpha + \left(\beta + 3\right)}\right)\right)} \]

      2. expm1-log1p 66.8%

         \[\leadsto \color{blue}{\frac{0.25}{\alpha + \left(\beta + 3\right)}} \]

      3. +-commutative 66.8%

         \[\leadsto \frac{0.25}{\alpha + \color{blue}{\left(3 + \beta\right)}} \]

      4. +-commutative 66.8%

         \[\leadsto \frac{0.25}{\color{blue}{\left(3 + \beta\right) + \alpha}} \]

      5. +-commutative 66.8%

         \[\leadsto \frac{0.25}{\color{blue}{\left(\beta + 3\right)} + \alpha} \]

   8. Simplified 66.8%

      \[\leadsto \color{blue}{\frac{0.25}{\left(\beta + 3\right) + \alpha}} \]

   if 4.79999999999999982 < beta

   1. Initial program 86.2%

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

   3. Simplified 91.1%

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

   5. Taylor expanded in beta around inf 82.5%

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

   6. Taylor expanded in alpha around 0 74.7%

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

      1. un-div-inv 74.7%

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

      2. +-commutative 74.7%

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

   8. Applied egg-rr 74.7%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.8:\\ \;\;\;\;\frac{0.25}{\alpha + \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2 + \beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 47.9% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.25}{\alpha + \left(\beta + 3\right)} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.2%

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

3. Taylor expanded in alpha around 0 71.1%

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

4. Taylor expanded in beta around 0 46.7%

   \[\leadsto \frac{\color{blue}{0.25}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
5. Step-by-step derivation

   1. expm1-log1p-u 46.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.25}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\right)\right)} \]

   2. expm1-udef 69.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{0.25}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\right)} - 1} \]

   3. metadata-eval 69.4%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{0.25}{\left(\left(\alpha + \beta\right) + \color{blue}{2}\right) + 1}\right)} - 1 \]

   4. associate-+l+ 69.4%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{0.25}{\color{blue}{\left(\alpha + \beta\right) + \left(2 + 1\right)}}\right)} - 1 \]

   5. metadata-eval 69.4%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{0.25}{\left(\alpha + \beta\right) + \color{blue}{3}}\right)} - 1 \]

   6. associate-+r+ 69.4%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{0.25}{\color{blue}{\alpha + \left(\beta + 3\right)}}\right)} - 1 \]

6. Applied egg-rr 69.4%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{0.25}{\alpha + \left(\beta + 3\right)}\right)} - 1} \]
7. Step-by-step derivation

   1. expm1-def 46.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.25}{\alpha + \left(\beta + 3\right)}\right)\right)} \]

   2. expm1-log1p 46.7%

      \[\leadsto \color{blue}{\frac{0.25}{\alpha + \left(\beta + 3\right)}} \]

   3. +-commutative 46.7%

      \[\leadsto \frac{0.25}{\alpha + \color{blue}{\left(3 + \beta\right)}} \]

   4. +-commutative 46.7%

      \[\leadsto \frac{0.25}{\color{blue}{\left(3 + \beta\right) + \alpha}} \]

   5. +-commutative 46.7%

      \[\leadsto \frac{0.25}{\color{blue}{\left(\beta + 3\right)} + \alpha} \]

8. Simplified 46.7%

   \[\leadsto \color{blue}{\frac{0.25}{\left(\beta + 3\right) + \alpha}} \]

9. Final simplification 46.7%

   \[\leadsto \frac{0.25}{\alpha + \left(\beta + 3\right)} \]
10. Add Preprocessing

Alternative 16: 46.3% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.16666666666666666}{2 + \beta} \end{array} \]

FPCore

(FPCore (alpha beta) :precision binary64 (/ 0.16666666666666666 (+ 2.0 beta)))

C

double code(double alpha, double beta) {
	return 0.16666666666666666 / (2.0 + beta);
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = 0.16666666666666666d0 / (2.0d0 + beta)
end function

Java

public static double code(double alpha, double beta) {
	return 0.16666666666666666 / (2.0 + beta);
}

Python

def code(alpha, beta):
	return 0.16666666666666666 / (2.0 + beta)

Julia

function code(alpha, beta)
	return Float64(0.16666666666666666 / Float64(2.0 + beta))
end

MATLAB

function tmp = code(alpha, beta)
	tmp = 0.16666666666666666 / (2.0 + beta);
end

Wolfram

code[alpha_, beta_] := N[(0.16666666666666666 / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.2%

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

3. Simplified 87.0%

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

5. Taylor expanded in beta around 0 77.5%

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

6. Taylor expanded in beta around 0 68.9%

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

   1. +-commutative 68.9%

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

8. Simplified 68.9%

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

9. Taylor expanded in alpha around 0 44.8%

   \[\leadsto \color{blue}{\frac{0.16666666666666666}{2 + \beta}} \]
10. Step-by-step derivation

    1. +-commutative 44.8%

       \[\leadsto \frac{0.16666666666666666}{\color{blue}{\beta + 2}} \]

11. Simplified 44.8%

    \[\leadsto \color{blue}{\frac{0.16666666666666666}{\beta + 2}} \]

12. Final simplification 44.8%

    \[\leadsto \frac{0.16666666666666666}{2 + \beta} \]
13. Add Preprocessing

Alternative 17: 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 95.2%

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

3. Taylor expanded in alpha around 0 71.1%

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

4. Taylor expanded in beta around 0 46.7%

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

5. Taylor expanded in alpha around 0 45.4%

   \[\leadsto \color{blue}{\frac{0.25}{3 + \beta}} \]
6. Step-by-step derivation

   1. +-commutative 45.4%

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

7. Simplified 45.4%

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

8. Final simplification 45.4%

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

Alternative 18: 4.2% accurate, 11.7× speedup

Math

\[\begin{array}{l} \\ \frac{0.25}{\alpha} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[alpha_, beta_] := N[(0.25 / alpha), $MachinePrecision]

Derivation

1. Initial program 95.2%

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

3. Taylor expanded in alpha around 0 71.1%

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

4. Taylor expanded in beta around 0 46.7%

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

5. Taylor expanded in alpha around inf 4.2%

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

6. Final simplification 4.2%

   \[\leadsto \frac{0.25}{\alpha} \]
7. Add Preprocessing

Alternative 19: 4.3% accurate, 11.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.2%

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

3. Taylor expanded in alpha around 0 71.1%

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

4. Taylor expanded in beta around 0 46.7%

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

5. Taylor expanded in beta around inf 4.4%

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

6. Final simplification 4.4%

   \[\leadsto \frac{0.25}{\beta} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024030 
(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)))