Octave 3.8, jcobi/3

Percentage Accurate: 94.5% → 99.8%

Time: 17.9s

Alternatives: 20

Speedup: 1.4×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.5% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.7%

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

   1. associate-/l/ 91.9%

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

   2. associate-+l+ 91.9%

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

   3. +-commutative 91.9%

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

   4. associate-+r+ 91.9%

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

   5. associate-+l+ 91.9%

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

   6. distribute-rgt1-in 91.9%

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

   7. *-rgt-identity 91.9%

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

   8. distribute-lft-out 91.9%

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

   9. +-commutative 91.9%

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

   10. associate-*l/ 94.3%

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

   11. *-commutative 94.3%

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

   12. associate-*r/ 91.0%

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

3. Simplified 91.0%

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

   1. associate-*r/ 94.3%

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

   2. +-commutative 94.3%

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

   3. associate-+r+ 94.3%

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

   4. +-commutative 94.3%

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

   5. associate-+r+ 94.3%

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

   6. +-commutative 94.3%

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

5. Applied egg-rr 94.3%

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

   1. times-frac 99.7%

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

   2. +-commutative 99.7%

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

   3. associate-+r+ 99.7%

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

   4. +-commutative 99.7%

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

   5. associate-+r+ 99.7%

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

   6. +-commutative 99.7%

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

7. Simplified 99.7%

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

   1. associate-*r/ 99.8%

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

   2. associate-+l+ 99.8%

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

   3. +-commutative 99.8%

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

   4. associate-+l+ 99.8%

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

   5. +-commutative 99.8%

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

9. Applied egg-rr 99.8%

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

10. Final simplification 99.8%

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

Alternative 2: 93.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 3\right)\\ t_1 := \alpha + \left(2 + \beta\right)\\ \mathbf{if}\;\beta \leq 2 \cdot 10^{+150}:\\ \;\;\;\;\left(1 + \alpha\right) \cdot \frac{\frac{1 + \beta}{t_1}}{t_0 \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{2 + \left(\alpha + \beta\right)} \cdot \left(1 - \frac{1 + \alpha}{\beta}\right)}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 3.0))) (t_1 (+ alpha (+ 2.0 beta))))
   (if (<= beta 2e+150)
     (* (+ 1.0 alpha) (/ (/ (+ 1.0 beta) t_1) (* t_0 t_1)))
     (/
      (*
       (/ (+ 1.0 alpha) (+ 2.0 (+ alpha beta)))
       (- 1.0 (/ (+ 1.0 alpha) beta)))
      t_0))))

C

double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 3.0);
	double t_1 = alpha + (2.0 + beta);
	double tmp;
	if (beta <= 2e+150) {
		tmp = (1.0 + alpha) * (((1.0 + beta) / t_1) / (t_0 * t_1));
	} else {
		tmp = (((1.0 + alpha) / (2.0 + (alpha + beta))) * (1.0 - ((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) :: t_1
    real(8) :: tmp
    t_0 = alpha + (beta + 3.0d0)
    t_1 = alpha + (2.0d0 + beta)
    if (beta <= 2d+150) then
        tmp = (1.0d0 + alpha) * (((1.0d0 + beta) / t_1) / (t_0 * t_1))
    else
        tmp = (((1.0d0 + alpha) / (2.0d0 + (alpha + beta))) * (1.0d0 - ((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 + (beta + 3.0);
	double t_1 = alpha + (2.0 + beta);
	double tmp;
	if (beta <= 2e+150) {
		tmp = (1.0 + alpha) * (((1.0 + beta) / t_1) / (t_0 * t_1));
	} else {
		tmp = (((1.0 + alpha) / (2.0 + (alpha + beta))) * (1.0 - ((1.0 + alpha) / beta))) / t_0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = alpha + (beta + 3.0)
	t_1 = alpha + (2.0 + beta)
	tmp = 0
	if beta <= 2e+150:
		tmp = (1.0 + alpha) * (((1.0 + beta) / t_1) / (t_0 * t_1))
	else:
		tmp = (((1.0 + alpha) / (2.0 + (alpha + beta))) * (1.0 - ((1.0 + alpha) / beta))) / t_0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 3.0))
	t_1 = Float64(alpha + Float64(2.0 + beta))
	tmp = 0.0
	if (beta <= 2e+150)
		tmp = Float64(Float64(1.0 + alpha) * Float64(Float64(Float64(1.0 + beta) / t_1) / Float64(t_0 * t_1)));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + alpha) / Float64(2.0 + Float64(alpha + beta))) * Float64(1.0 - Float64(Float64(1.0 + alpha) / beta))) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 3.0);
	t_1 = alpha + (2.0 + beta);
	tmp = 0.0;
	if (beta <= 2e+150)
		tmp = (1.0 + alpha) * (((1.0 + beta) / t_1) / (t_0 * t_1));
	else
		tmp = (((1.0 + alpha) / (2.0 + (alpha + beta))) * (1.0 - ((1.0 + alpha) / beta))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2e+150], N[(N[(1.0 + alpha), $MachinePrecision] * N[(N[(N[(1.0 + beta), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.99999999999999996e150

   1. Initial program 97.3%

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

      1. associate-/l/ 95.6%

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

      2. associate-+l+ 95.6%

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

      3. +-commutative 95.6%

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

      4. associate-+r+ 95.6%

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

      5. associate-+l+ 95.6%

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

      6. distribute-rgt1-in 95.6%

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

      7. *-rgt-identity 95.6%

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

      8. distribute-lft-out 95.6%

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

      9. +-commutative 95.6%

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

      10. associate-*l/ 97.2%

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

      11. *-commutative 97.2%

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

      12. associate-*r/ 93.1%

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

   3. Simplified 93.1%

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

   if 1.99999999999999996e150 < beta

   1. Initial program 80.7%

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

      1. associate-/l/ 78.5%

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

      2. associate-+l+ 78.5%

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

      3. +-commutative 78.5%

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

      4. associate-+r+ 78.5%

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

      5. associate-+l+ 78.5%

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

      6. distribute-rgt1-in 78.5%

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

      7. *-rgt-identity 78.5%

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

      8. distribute-lft-out 78.5%

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

      9. +-commutative 78.5%

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

      10. associate-*l/ 83.8%

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

      11. *-commutative 83.8%

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

      12. associate-*r/ 83.9%

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

   3. Simplified 83.9%

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

      1. associate-*r/ 83.8%

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

      2. +-commutative 83.8%

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

      3. associate-+r+ 83.8%

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

      4. +-commutative 83.8%

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

      5. associate-+r+ 83.8%

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

      6. +-commutative 83.8%

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

   5. Applied egg-rr 83.8%

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

      1. times-frac 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+r+ 99.9%

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

      4. +-commutative 99.9%

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

      5. associate-+r+ 99.9%

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

      6. +-commutative 99.9%

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

   7. Simplified 99.9%

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

      1. associate-*r/ 99.9%

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

      2. associate-+l+ 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-+l+ 99.9%

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

      5. +-commutative 99.9%

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

   9. Applied egg-rr 99.9%

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

   10. Taylor expanded in beta around inf 88.0%

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

       1. mul-1-neg 88.0%

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

       2. unsub-neg 88.0%

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

   12. Simplified 88.0%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2 \cdot 10^{+150}:\\ \;\;\;\;\left(1 + \alpha\right) \cdot \frac{\frac{1 + \beta}{\alpha + \left(2 + \beta\right)}}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \left(\alpha + \left(2 + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{2 + \left(\alpha + \beta\right)} \cdot \left(1 - \frac{1 + \alpha}{\beta}\right)}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 3: 92.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 22:\\ \;\;\;\;\frac{1}{\alpha + 2} \cdot \frac{1 + \alpha}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta + \left(\alpha + 2\right)} \cdot \left(\frac{1}{\beta} - \frac{\alpha + 2}{\beta} \cdot \frac{2}{\beta}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 22.0)
   (* (/ 1.0 (+ alpha 2.0)) (/ (+ 1.0 alpha) (* (+ alpha 2.0) (+ alpha 3.0))))
   (*
    (/ (+ 1.0 alpha) (+ beta (+ alpha 2.0)))
    (- (/ 1.0 beta) (* (/ (+ alpha 2.0) beta) (/ 2.0 beta))))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 22.0) {
		tmp = (1.0 / (alpha + 2.0)) * ((1.0 + alpha) / ((alpha + 2.0) * (alpha + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / (beta + (alpha + 2.0))) * ((1.0 / beta) - (((alpha + 2.0) / 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 <= 22.0d0) then
        tmp = (1.0d0 / (alpha + 2.0d0)) * ((1.0d0 + alpha) / ((alpha + 2.0d0) * (alpha + 3.0d0)))
    else
        tmp = ((1.0d0 + alpha) / (beta + (alpha + 2.0d0))) * ((1.0d0 / beta) - (((alpha + 2.0d0) / beta) * (2.0d0 / beta)))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 22.0) {
		tmp = (1.0 / (alpha + 2.0)) * ((1.0 + alpha) / ((alpha + 2.0) * (alpha + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / (beta + (alpha + 2.0))) * ((1.0 / beta) - (((alpha + 2.0) / beta) * (2.0 / beta)));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 22.0:
		tmp = (1.0 / (alpha + 2.0)) * ((1.0 + alpha) / ((alpha + 2.0) * (alpha + 3.0)))
	else:
		tmp = ((1.0 + alpha) / (beta + (alpha + 2.0))) * ((1.0 / beta) - (((alpha + 2.0) / beta) * (2.0 / beta)))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 22.0)
		tmp = Float64(Float64(1.0 / Float64(alpha + 2.0)) * Float64(Float64(1.0 + alpha) / Float64(Float64(alpha + 2.0) * Float64(alpha + 3.0))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(beta + Float64(alpha + 2.0))) * Float64(Float64(1.0 / beta) - Float64(Float64(Float64(alpha + 2.0) / beta) * Float64(2.0 / beta))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 22.0)
		tmp = (1.0 / (alpha + 2.0)) * ((1.0 + alpha) / ((alpha + 2.0) * (alpha + 3.0)));
	else
		tmp = ((1.0 + alpha) / (beta + (alpha + 2.0))) * ((1.0 / beta) - (((alpha + 2.0) / beta) * (2.0 / beta)));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 22.0], N[(N[(1.0 / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + alpha), $MachinePrecision] / N[(N[(alpha + 2.0), $MachinePrecision] * N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / beta), $MachinePrecision] - N[(N[(N[(alpha + 2.0), $MachinePrecision] / beta), $MachinePrecision] * N[(2.0 / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 22

   1. Initial program 99.9%

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

      1. associate-/l/ 99.4%

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

      2. associate-/r* 94.7%

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

      3. associate-+l+ 94.7%

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

      4. +-commutative 94.7%

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

      5. associate-+r+ 94.7%

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

      6. associate-+l+ 94.7%

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

      7. distribute-rgt1-in 94.7%

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

      8. *-rgt-identity 94.7%

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

      9. distribute-lft-out 94.7%

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

      10. *-commutative 94.7%

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

      11. metadata-eval 94.7%

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

      12. associate-+l+ 94.7%

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

      13. +-commutative 94.7%

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

   3. Simplified 94.7%

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

   4. Taylor expanded in beta around 0 93.6%

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

      1. *-un-lft-identity 93.6%

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

      2. times-frac 98.3%

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

      3. +-commutative 98.3%

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

      4. +-commutative 98.3%

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

      5. +-commutative 98.3%

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

      6. +-commutative 98.3%

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

      7. +-commutative 98.3%

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

   6. Applied egg-rr 98.3%

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

      1. *-lft-identity 98.3%

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

      2. +-commutative 98.3%

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

      3. associate-+r+ 98.3%

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

      4. +-commutative 98.3%

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

      5. +-commutative 98.3%

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

      6. +-commutative 98.3%

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

   8. Simplified 98.3%

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

   9. Taylor expanded in beta around 0 97.1%

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

   if 22 < beta

   1. Initial program 84.0%

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

      1. associate-/l/ 80.0%

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

      2. associate-+l+ 80.0%

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

      3. +-commutative 80.0%

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

      4. associate-+r+ 80.0%

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

      5. associate-+l+ 80.0%

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

      6. distribute-rgt1-in 80.0%

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

      7. *-rgt-identity 80.0%

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

      8. distribute-lft-out 80.0%

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

      9. +-commutative 80.0%

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

      10. associate-*l/ 86.2%

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

      11. *-commutative 86.2%

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

      12. associate-*r/ 85.3%

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

   3. Simplified 85.3%

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

      1. associate-*r/ 86.2%

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

      2. +-commutative 86.2%

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

      3. associate-+r+ 86.2%

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

      4. +-commutative 86.2%

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

      5. associate-+r+ 86.2%

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

      6. +-commutative 86.2%

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

   5. Applied egg-rr 86.2%

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

      1. times-frac 99.6%

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

      2. +-commutative 99.6%

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

      3. associate-+r+ 99.6%

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

      4. +-commutative 99.6%

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

      5. associate-+r+ 99.6%

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

      6. +-commutative 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in beta around inf 79.9%

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

      1. +-commutative 79.9%

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

      2. mul-1-neg 79.9%

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

      3. unsub-neg 79.9%

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

      4. metadata-eval 79.9%

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

      5. distribute-lft-in 79.9%

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

      6. *-commutative 79.9%

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

      7. unpow2 79.9%

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

      8. times-frac 79.7%

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

   10. Simplified 79.7%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 22:\\ \;\;\;\;\frac{1}{\alpha + 2} \cdot \frac{1 + \alpha}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta + \left(\alpha + 2\right)} \cdot \left(\frac{1}{\beta} - \frac{\alpha + 2}{\beta} \cdot \frac{2}{\beta}\right)\\ \end{array} \]

Alternative 4: 92.7% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 14.0)
   (* (/ 1.0 (+ alpha 2.0)) (/ (+ 1.0 alpha) (* (+ alpha 2.0) (+ alpha 3.0))))
   (*
    (/ (- -1.0 alpha) (+ beta (+ alpha 2.0)))
    (/ (+ (/ (+ 1.0 alpha) beta) -1.0) (+ alpha (+ beta 3.0))))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 14.0) {
		tmp = (1.0 / (alpha + 2.0)) * ((1.0 + alpha) / ((alpha + 2.0) * (alpha + 3.0)));
	} else {
		tmp = ((-1.0 - alpha) / (beta + (alpha + 2.0))) * ((((1.0 + alpha) / beta) + -1.0) / (alpha + (beta + 3.0)));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 14.0d0) then
        tmp = (1.0d0 / (alpha + 2.0d0)) * ((1.0d0 + alpha) / ((alpha + 2.0d0) * (alpha + 3.0d0)))
    else
        tmp = (((-1.0d0) - alpha) / (beta + (alpha + 2.0d0))) * ((((1.0d0 + alpha) / beta) + (-1.0d0)) / (alpha + (beta + 3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 14.0) {
		tmp = (1.0 / (alpha + 2.0)) * ((1.0 + alpha) / ((alpha + 2.0) * (alpha + 3.0)));
	} else {
		tmp = ((-1.0 - alpha) / (beta + (alpha + 2.0))) * ((((1.0 + alpha) / beta) + -1.0) / (alpha + (beta + 3.0)));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 14.0:
		tmp = (1.0 / (alpha + 2.0)) * ((1.0 + alpha) / ((alpha + 2.0) * (alpha + 3.0)))
	else:
		tmp = ((-1.0 - alpha) / (beta + (alpha + 2.0))) * ((((1.0 + alpha) / beta) + -1.0) / (alpha + (beta + 3.0)))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 14.0)
		tmp = Float64(Float64(1.0 / Float64(alpha + 2.0)) * Float64(Float64(1.0 + alpha) / Float64(Float64(alpha + 2.0) * Float64(alpha + 3.0))));
	else
		tmp = Float64(Float64(Float64(-1.0 - alpha) / Float64(beta + Float64(alpha + 2.0))) * Float64(Float64(Float64(Float64(1.0 + alpha) / beta) + -1.0) / Float64(alpha + Float64(beta + 3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 14.0)
		tmp = (1.0 / (alpha + 2.0)) * ((1.0 + alpha) / ((alpha + 2.0) * (alpha + 3.0)));
	else
		tmp = ((-1.0 - alpha) / (beta + (alpha + 2.0))) * ((((1.0 + alpha) / beta) + -1.0) / (alpha + (beta + 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 14.0], N[(N[(1.0 / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + alpha), $MachinePrecision] / N[(N[(alpha + 2.0), $MachinePrecision] * N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-1.0 - alpha), $MachinePrecision] / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] + -1.0), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 14

   1. Initial program 99.9%

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

      1. associate-/l/ 99.4%

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

      2. associate-/r* 94.7%

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

      3. associate-+l+ 94.7%

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

      4. +-commutative 94.7%

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

      5. associate-+r+ 94.7%

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

      6. associate-+l+ 94.7%

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

      7. distribute-rgt1-in 94.7%

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

      8. *-rgt-identity 94.7%

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

      9. distribute-lft-out 94.7%

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

      10. *-commutative 94.7%

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

      11. metadata-eval 94.7%

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

      12. associate-+l+ 94.7%

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

      13. +-commutative 94.7%

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

   3. Simplified 94.7%

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

   4. Taylor expanded in beta around 0 93.6%

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

      1. *-un-lft-identity 93.6%

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

      2. times-frac 98.3%

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

      3. +-commutative 98.3%

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

      4. +-commutative 98.3%

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

      5. +-commutative 98.3%

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

      6. +-commutative 98.3%

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

      7. +-commutative 98.3%

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

   6. Applied egg-rr 98.3%

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

      1. *-lft-identity 98.3%

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

      2. +-commutative 98.3%

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

      3. associate-+r+ 98.3%

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

      4. +-commutative 98.3%

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

      5. +-commutative 98.3%

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

      6. +-commutative 98.3%

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

   8. Simplified 98.3%

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

   9. Taylor expanded in beta around 0 97.1%

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

   if 14 < beta

   1. Initial program 84.0%

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

      1. associate-/l/ 80.0%

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

      2. associate-+l+ 80.0%

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

      3. +-commutative 80.0%

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

      4. associate-+r+ 80.0%

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

      5. associate-+l+ 80.0%

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

      6. distribute-rgt1-in 80.0%

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

      7. *-rgt-identity 80.0%

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

      8. distribute-lft-out 80.0%

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

      9. +-commutative 80.0%

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

      10. associate-*l/ 86.2%

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

      11. *-commutative 86.2%

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

      12. associate-*r/ 85.3%

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

   3. Simplified 85.3%

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

      1. associate-*r/ 86.2%

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

      2. +-commutative 86.2%

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

      3. associate-+r+ 86.2%

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

      4. +-commutative 86.2%

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

      5. associate-+r+ 86.2%

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

      6. +-commutative 86.2%

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

   5. Applied egg-rr 86.2%

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

      1. times-frac 99.6%

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

      2. +-commutative 99.6%

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

      3. associate-+r+ 99.6%

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

      4. +-commutative 99.6%

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

      5. associate-+r+ 99.6%

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

      6. +-commutative 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in beta around inf 79.8%

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

      1. mul-1-neg 79.8%

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

      2. unsub-neg 79.8%

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

   10. Simplified 79.8%

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

4. Final simplification 90.4%

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

Alternative 5: 92.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.8:\\ \;\;\;\;\frac{1}{\alpha + 2} \cdot \frac{1 + \alpha}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{2 + \left(\alpha + \beta\right)} \cdot \left(1 - \frac{1 + \alpha}{\beta}\right)}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.8)
   (* (/ 1.0 (+ alpha 2.0)) (/ (+ 1.0 alpha) (* (+ alpha 2.0) (+ alpha 3.0))))
   (/
    (* (/ (+ 1.0 alpha) (+ 2.0 (+ alpha beta))) (- 1.0 (/ (+ 1.0 alpha) beta)))
    (+ alpha (+ beta 3.0)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.8) {
		tmp = (1.0 / (alpha + 2.0)) * ((1.0 + alpha) / ((alpha + 2.0) * (alpha + 3.0)));
	} else {
		tmp = (((1.0 + alpha) / (2.0 + (alpha + beta))) * (1.0 - ((1.0 + alpha) / beta))) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.8d0) then
        tmp = (1.0d0 / (alpha + 2.0d0)) * ((1.0d0 + alpha) / ((alpha + 2.0d0) * (alpha + 3.0d0)))
    else
        tmp = (((1.0d0 + alpha) / (2.0d0 + (alpha + beta))) * (1.0d0 - ((1.0d0 + alpha) / beta))) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.8) {
		tmp = (1.0 / (alpha + 2.0)) * ((1.0 + alpha) / ((alpha + 2.0) * (alpha + 3.0)));
	} else {
		tmp = (((1.0 + alpha) / (2.0 + (alpha + beta))) * (1.0 - ((1.0 + alpha) / beta))) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 1.8:
		tmp = (1.0 / (alpha + 2.0)) * ((1.0 + alpha) / ((alpha + 2.0) * (alpha + 3.0)))
	else:
		tmp = (((1.0 + alpha) / (2.0 + (alpha + beta))) * (1.0 - ((1.0 + alpha) / beta))) / (alpha + (beta + 3.0))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.8)
		tmp = Float64(Float64(1.0 / Float64(alpha + 2.0)) * Float64(Float64(1.0 + alpha) / Float64(Float64(alpha + 2.0) * Float64(alpha + 3.0))));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + alpha) / Float64(2.0 + Float64(alpha + beta))) * Float64(1.0 - Float64(Float64(1.0 + alpha) / beta))) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.8)
		tmp = (1.0 / (alpha + 2.0)) * ((1.0 + alpha) / ((alpha + 2.0) * (alpha + 3.0)));
	else
		tmp = (((1.0 + alpha) / (2.0 + (alpha + beta))) * (1.0 - ((1.0 + alpha) / beta))) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 1.8], N[(N[(1.0 / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + alpha), $MachinePrecision] / N[(N[(alpha + 2.0), $MachinePrecision] * N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.80000000000000004

   1. Initial program 99.9%

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

      1. associate-/l/ 99.4%

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

      2. associate-/r* 94.7%

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

      3. associate-+l+ 94.7%

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

      4. +-commutative 94.7%

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

      5. associate-+r+ 94.7%

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

      6. associate-+l+ 94.7%

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

      7. distribute-rgt1-in 94.7%

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

      8. *-rgt-identity 94.7%

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

      9. distribute-lft-out 94.7%

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

      10. *-commutative 94.7%

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

      11. metadata-eval 94.7%

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

      12. associate-+l+ 94.7%

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

      13. +-commutative 94.7%

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

   3. Simplified 94.7%

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

   4. Taylor expanded in beta around 0 93.6%

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

      1. *-un-lft-identity 93.6%

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

      2. times-frac 98.3%

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

      3. +-commutative 98.3%

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

      4. +-commutative 98.3%

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

      5. +-commutative 98.3%

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

      6. +-commutative 98.3%

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

      7. +-commutative 98.3%

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

   6. Applied egg-rr 98.3%

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

      1. *-lft-identity 98.3%

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

      2. +-commutative 98.3%

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

      3. associate-+r+ 98.3%

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

      4. +-commutative 98.3%

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

      5. +-commutative 98.3%

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

      6. +-commutative 98.3%

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

   8. Simplified 98.3%

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

   9. Taylor expanded in beta around 0 97.1%

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

   if 1.80000000000000004 < beta

   1. Initial program 84.0%

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

      1. associate-/l/ 80.0%

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

      2. associate-+l+ 80.0%

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

      3. +-commutative 80.0%

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

      4. associate-+r+ 80.0%

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

      5. associate-+l+ 80.0%

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

      6. distribute-rgt1-in 80.0%

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

      7. *-rgt-identity 80.0%

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

      8. distribute-lft-out 80.0%

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

      9. +-commutative 80.0%

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

      10. associate-*l/ 86.2%

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

      11. *-commutative 86.2%

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

      12. associate-*r/ 85.3%

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

   3. Simplified 85.3%

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

      1. associate-*r/ 86.2%

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

      2. +-commutative 86.2%

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

      3. associate-+r+ 86.2%

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

      4. +-commutative 86.2%

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

      5. associate-+r+ 86.2%

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

      6. +-commutative 86.2%

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

   5. Applied egg-rr 86.2%

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

      1. times-frac 99.6%

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

      2. +-commutative 99.6%

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

      3. associate-+r+ 99.6%

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

      4. +-commutative 99.6%

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

      5. associate-+r+ 99.6%

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

      6. +-commutative 99.6%

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

   7. Simplified 99.6%

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

      1. associate-*r/ 99.7%

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

      2. associate-+l+ 99.7%

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

      3. +-commutative 99.7%

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

      4. associate-+l+ 99.7%

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

      5. +-commutative 99.7%

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

   9. Applied egg-rr 99.7%

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

   10. Taylor expanded in beta around inf 80.0%

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

       1. mul-1-neg 80.0%

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

       2. unsub-neg 80.0%

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

   12. Simplified 80.0%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.8:\\ \;\;\;\;\frac{1}{\alpha + 2} \cdot \frac{1 + \alpha}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{2 + \left(\alpha + \beta\right)} \cdot \left(1 - \frac{1 + \alpha}{\beta}\right)}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 6: 99.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 93.7%

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

   1. associate-/l/ 91.9%

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

   2. associate-+l+ 91.9%

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

   3. +-commutative 91.9%

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

   4. associate-+r+ 91.9%

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

   5. associate-+l+ 91.9%

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

   6. distribute-rgt1-in 91.9%

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

   7. *-rgt-identity 91.9%

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

   8. distribute-lft-out 91.9%

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

   9. +-commutative 91.9%

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

   10. associate-*l/ 94.3%

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

   11. *-commutative 94.3%

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

   12. associate-*r/ 91.0%

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

3. Simplified 91.0%

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

   1. associate-*r/ 94.3%

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

   2. +-commutative 94.3%

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

   3. associate-+r+ 94.3%

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

   4. +-commutative 94.3%

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

   5. associate-+r+ 94.3%

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

   6. +-commutative 94.3%

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

5. Applied egg-rr 94.3%

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

   1. times-frac 99.7%

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

   2. +-commutative 99.7%

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

   3. associate-+r+ 99.7%

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

   4. +-commutative 99.7%

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

   5. associate-+r+ 99.7%

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

   6. +-commutative 99.7%

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

7. Simplified 99.7%

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

8. Final simplification 99.7%

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

Alternative 7: 92.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.4:\\ \;\;\;\;\frac{1}{\alpha + 2} \cdot \frac{1 + \alpha}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{2 + \left(\alpha + \beta\right)}}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.4)
   (* (/ 1.0 (+ alpha 2.0)) (/ (+ 1.0 alpha) (* (+ alpha 2.0) (+ alpha 3.0))))
   (/ (/ (+ 1.0 alpha) (+ 2.0 (+ alpha beta))) beta)))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.4) {
		tmp = (1.0 / (alpha + 2.0)) * ((1.0 + alpha) / ((alpha + 2.0) * (alpha + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / (2.0 + (alpha + 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 <= 4.4d0) then
        tmp = (1.0d0 / (alpha + 2.0d0)) * ((1.0d0 + alpha) / ((alpha + 2.0d0) * (alpha + 3.0d0)))
    else
        tmp = ((1.0d0 + alpha) / (2.0d0 + (alpha + beta))) / beta
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 4.4], N[(N[(1.0 / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + alpha), $MachinePrecision] / N[(N[(alpha + 2.0), $MachinePrecision] * N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.4000000000000004

   1. Initial program 99.9%

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

      1. associate-/l/ 99.4%

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

      2. associate-/r* 94.7%

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

      3. associate-+l+ 94.7%

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

      4. +-commutative 94.7%

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

      5. associate-+r+ 94.7%

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

      6. associate-+l+ 94.7%

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

      7. distribute-rgt1-in 94.7%

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

      8. *-rgt-identity 94.7%

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

      9. distribute-lft-out 94.7%

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

      10. *-commutative 94.7%

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

      11. metadata-eval 94.7%

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

      12. associate-+l+ 94.7%

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

      13. +-commutative 94.7%

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

   3. Simplified 94.7%

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

   4. Taylor expanded in beta around 0 93.6%

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

      1. *-un-lft-identity 93.6%

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

      2. times-frac 98.3%

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

      3. +-commutative 98.3%

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

      4. +-commutative 98.3%

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

      5. +-commutative 98.3%

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

      6. +-commutative 98.3%

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

      7. +-commutative 98.3%

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

   6. Applied egg-rr 98.3%

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

      1. *-lft-identity 98.3%

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

      2. +-commutative 98.3%

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

      3. associate-+r+ 98.3%

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

      4. +-commutative 98.3%

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

      5. +-commutative 98.3%

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

      6. +-commutative 98.3%

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

   8. Simplified 98.3%

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

   9. Taylor expanded in beta around 0 97.1%

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

   if 4.4000000000000004 < beta

   1. Initial program 84.0%

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

      1. associate-/l/ 80.0%

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

      2. associate-+l+ 80.0%

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

      3. +-commutative 80.0%

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

      4. associate-+r+ 80.0%

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

      5. associate-+l+ 80.0%

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

      6. distribute-rgt1-in 80.0%

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

      7. *-rgt-identity 80.0%

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

      8. distribute-lft-out 80.0%

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

      9. +-commutative 80.0%

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

      10. associate-*l/ 86.2%

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

      11. *-commutative 86.2%

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

      12. associate-*r/ 85.3%

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

   3. Simplified 85.3%

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

      1. associate-*r/ 86.2%

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

      2. +-commutative 86.2%

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

      3. associate-+r+ 86.2%

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

      4. +-commutative 86.2%

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

      5. associate-+r+ 86.2%

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

      6. +-commutative 86.2%

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

   5. Applied egg-rr 86.2%

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

      1. times-frac 99.6%

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

      2. +-commutative 99.6%

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

      3. associate-+r+ 99.6%

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

      4. +-commutative 99.6%

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

      5. associate-+r+ 99.6%

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

      6. +-commutative 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in beta around inf 79.7%

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

      1. un-div-inv 79.9%

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

      2. associate-+r+ 79.9%

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

   10. Applied egg-rr 79.9%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.4:\\ \;\;\;\;\frac{1}{\alpha + 2} \cdot \frac{1 + \alpha}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{2 + \left(\alpha + \beta\right)}}{\beta}\\ \end{array} \]

Alternative 8: 92.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if beta < 6

   1. Initial program 99.9%

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

      1. associate-/l/ 99.4%

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

      2. associate-/r* 94.7%

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

      3. associate-+l+ 94.7%

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

      4. +-commutative 94.7%

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

      5. associate-+r+ 94.7%

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

      6. associate-+l+ 94.7%

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

      7. distribute-rgt1-in 94.7%

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

      8. *-rgt-identity 94.7%

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

      9. distribute-lft-out 94.7%

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

      10. *-commutative 94.7%

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

      11. metadata-eval 94.7%

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

      12. associate-+l+ 94.7%

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

      13. +-commutative 94.7%

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

   3. Simplified 94.7%

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

   4. Taylor expanded in beta around 0 93.6%

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

      1. *-un-lft-identity 93.6%

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

      2. times-frac 98.3%

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

      3. +-commutative 98.3%

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

      4. +-commutative 98.3%

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

      5. +-commutative 98.3%

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

      6. +-commutative 98.3%

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

      7. +-commutative 98.3%

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

   6. Applied egg-rr 98.3%

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

      1. *-lft-identity 98.3%

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

      2. +-commutative 98.3%

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

      3. associate-+r+ 98.3%

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

      4. +-commutative 98.3%

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

      5. +-commutative 98.3%

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

      6. +-commutative 98.3%

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

   8. Simplified 98.3%

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

   9. Taylor expanded in beta around 0 97.1%

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

   if 6 < beta

   1. Initial program 84.0%

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

   2. Taylor expanded in beta around -inf 79.9%

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

4. Final simplification 90.4%

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

Alternative 9: 81.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 7.8:\\ \;\;\;\;\frac{1 + \alpha}{\left(\alpha + 2\right) \cdot \left(6 + \alpha \cdot 5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{2 + \left(\alpha + \beta\right)}}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 7.8)
   (/ (+ 1.0 alpha) (* (+ alpha 2.0) (+ 6.0 (* alpha 5.0))))
   (/ (/ (+ 1.0 alpha) (+ 2.0 (+ alpha beta))) beta)))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 7.8) {
		tmp = (1.0 + alpha) / ((alpha + 2.0) * (6.0 + (alpha * 5.0)));
	} else {
		tmp = ((1.0 + alpha) / (2.0 + (alpha + 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 <= 7.8d0) then
        tmp = (1.0d0 + alpha) / ((alpha + 2.0d0) * (6.0d0 + (alpha * 5.0d0)))
    else
        tmp = ((1.0d0 + alpha) / (2.0d0 + (alpha + beta))) / beta
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 7.8) {
		tmp = (1.0 + alpha) / ((alpha + 2.0) * (6.0 + (alpha * 5.0)));
	} else {
		tmp = ((1.0 + alpha) / (2.0 + (alpha + beta))) / beta;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 7.8:
		tmp = (1.0 + alpha) / ((alpha + 2.0) * (6.0 + (alpha * 5.0)))
	else:
		tmp = ((1.0 + alpha) / (2.0 + (alpha + beta))) / beta
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 7.8)
		tmp = Float64(Float64(1.0 + alpha) / Float64(Float64(alpha + 2.0) * Float64(6.0 + Float64(alpha * 5.0))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(2.0 + Float64(alpha + beta))) / beta);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 7.8)
		tmp = (1.0 + alpha) / ((alpha + 2.0) * (6.0 + (alpha * 5.0)));
	else
		tmp = ((1.0 + alpha) / (2.0 + (alpha + beta))) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 7.8], N[(N[(1.0 + alpha), $MachinePrecision] / N[(N[(alpha + 2.0), $MachinePrecision] * N[(6.0 + N[(alpha * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 7.79999999999999982

   1. Initial program 99.9%

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

      1. associate-/l/ 99.4%

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

      2. associate-/r* 94.7%

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

      3. associate-+l+ 94.7%

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

      4. +-commutative 94.7%

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

      5. associate-+r+ 94.7%

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

      6. associate-+l+ 94.7%

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

      7. distribute-rgt1-in 94.7%

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

      8. *-rgt-identity 94.7%

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

      9. distribute-lft-out 94.7%

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

      10. *-commutative 94.7%

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

      11. metadata-eval 94.7%

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

      12. associate-+l+ 94.7%

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

      13. +-commutative 94.7%

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

   3. Simplified 94.7%

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

   4. Taylor expanded in beta around 0 93.6%

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

   5. Taylor expanded in alpha around 0 85.5%

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

      1. *-commutative 85.5%

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

   7. Simplified 85.5%

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

   8. Taylor expanded in beta around 0 85.5%

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

   if 7.79999999999999982 < beta

   1. Initial program 84.0%

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

      1. associate-/l/ 80.0%

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

      2. associate-+l+ 80.0%

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

      3. +-commutative 80.0%

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

      4. associate-+r+ 80.0%

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

      5. associate-+l+ 80.0%

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

      6. distribute-rgt1-in 80.0%

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

      7. *-rgt-identity 80.0%

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

      8. distribute-lft-out 80.0%

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

      9. +-commutative 80.0%

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

      10. associate-*l/ 86.2%

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

      11. *-commutative 86.2%

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

      12. associate-*r/ 85.3%

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

   3. Simplified 85.3%

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

      1. associate-*r/ 86.2%

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

      2. +-commutative 86.2%

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

      3. associate-+r+ 86.2%

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

      4. +-commutative 86.2%

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

      5. associate-+r+ 86.2%

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

      6. +-commutative 86.2%

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

   5. Applied egg-rr 86.2%

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

      1. times-frac 99.6%

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

      2. +-commutative 99.6%

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

      3. associate-+r+ 99.6%

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

      4. +-commutative 99.6%

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

      5. associate-+r+ 99.6%

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

      6. +-commutative 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in beta around inf 79.7%

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

      1. un-div-inv 79.9%

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

      2. associate-+r+ 79.9%

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

   10. Applied egg-rr 79.9%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 7.8:\\ \;\;\;\;\frac{1 + \alpha}{\left(\alpha + 2\right) \cdot \left(6 + \alpha \cdot 5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{2 + \left(\alpha + \beta\right)}}{\beta}\\ \end{array} \]

Alternative 10: 71.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.8:\\ \;\;\;\;\frac{1 + \beta}{\beta + \left(\alpha + 2\right)} \cdot 0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.8)
   (* (/ (+ 1.0 beta) (+ beta (+ alpha 2.0))) 0.16666666666666666)
   (/ (/ (+ 1.0 alpha) beta) beta)))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.8) {
		tmp = ((1.0 + beta) / (beta + (alpha + 2.0))) * 0.16666666666666666;
	} else {
		tmp = ((1.0 + alpha) / 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 <= 3.8d0) then
        tmp = ((1.0d0 + beta) / (beta + (alpha + 2.0d0))) * 0.16666666666666666d0
    else
        tmp = ((1.0d0 + alpha) / beta) / beta
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.8) {
		tmp = ((1.0 + beta) / (beta + (alpha + 2.0))) * 0.16666666666666666;
	} else {
		tmp = ((1.0 + alpha) / beta) / beta;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 3.8:
		tmp = ((1.0 + beta) / (beta + (alpha + 2.0))) * 0.16666666666666666
	else:
		tmp = ((1.0 + alpha) / beta) / beta
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.8)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(beta + Float64(alpha + 2.0))) * 0.16666666666666666);
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / beta);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.8)
		tmp = ((1.0 + beta) / (beta + (alpha + 2.0))) * 0.16666666666666666;
	else
		tmp = ((1.0 + alpha) / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 3.8], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.16666666666666666), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.7999999999999998

   1. Initial program 99.9%

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

      1. associate-/l/ 99.4%

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

      2. associate-/r* 94.7%

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

      3. associate-+l+ 94.7%

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

      4. +-commutative 94.7%

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

      5. associate-+r+ 94.7%

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

      6. associate-+l+ 94.7%

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

      7. distribute-rgt1-in 94.7%

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

      8. *-rgt-identity 94.7%

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

      9. distribute-lft-out 94.7%

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

      10. *-commutative 94.7%

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

      11. metadata-eval 94.7%

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

      12. associate-+l+ 94.7%

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

      13. +-commutative 94.7%

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

   3. Simplified 94.7%

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

   4. Taylor expanded in beta around 0 93.6%

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

      1. *-un-lft-identity 93.6%

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

      2. times-frac 98.3%

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

      3. +-commutative 98.3%

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

      4. +-commutative 98.3%

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

      5. +-commutative 98.3%

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

      6. +-commutative 98.3%

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

      7. +-commutative 98.3%

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

   6. Applied egg-rr 98.3%

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

      1. *-lft-identity 98.3%

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

      2. +-commutative 98.3%

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

      3. associate-+r+ 98.3%

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

      4. +-commutative 98.3%

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

      5. +-commutative 98.3%

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

      6. +-commutative 98.3%

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

   8. Simplified 98.3%

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

   9. Taylor expanded in alpha around 0 70.7%

      \[\leadsto \frac{\beta + 1}{\left(2 + \alpha\right) + \beta} \cdot \color{blue}{0.16666666666666666} \]

   if 3.7999999999999998 < beta

   1. Initial program 84.0%

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

      1. associate-/l/ 80.0%

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

      2. associate-+l+ 80.0%

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

      3. +-commutative 80.0%

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

      4. associate-+r+ 80.0%

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

      5. associate-+l+ 80.0%

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

      6. distribute-rgt1-in 80.0%

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

      7. *-rgt-identity 80.0%

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

      8. distribute-lft-out 80.0%

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

      9. +-commutative 80.0%

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

      10. associate-*l/ 86.2%

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

      11. *-commutative 86.2%

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

      12. associate-*r/ 85.3%

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

   3. Simplified 85.3%

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

      1. associate-*r/ 86.2%

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

      2. +-commutative 86.2%

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

      3. associate-+r+ 86.2%

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

      4. +-commutative 86.2%

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

      5. associate-+r+ 86.2%

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

      6. +-commutative 86.2%

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

   5. Applied egg-rr 86.2%

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

      1. times-frac 99.6%

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

      2. +-commutative 99.6%

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

      3. associate-+r+ 99.6%

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

      4. +-commutative 99.6%

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

      5. associate-+r+ 99.6%

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

      6. +-commutative 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in beta around inf 79.7%

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

   9. Taylor expanded in beta around inf 79.5%

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

       1. un-div-inv 79.7%

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

   11. Applied egg-rr 79.7%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.8:\\ \;\;\;\;\frac{1 + \beta}{\beta + \left(\alpha + 2\right)} \cdot 0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \]

Alternative 11: 71.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\beta \leq 2.2:\\ \;\;\;\;\frac{1 + \beta}{t_0} \cdot 0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ beta (+ alpha 2.0))))
   (if (<= beta 2.2)
     (* (/ (+ 1.0 beta) t_0) 0.16666666666666666)
     (/ (/ (+ 1.0 alpha) beta) t_0))))

C

double code(double alpha, double beta) {
	double t_0 = beta + (alpha + 2.0);
	double tmp;
	if (beta <= 2.2) {
		tmp = ((1.0 + beta) / t_0) * 0.16666666666666666;
	} 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 = beta + (alpha + 2.0d0)
    if (beta <= 2.2d0) then
        tmp = ((1.0d0 + beta) / t_0) * 0.16666666666666666d0
    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 = beta + (alpha + 2.0);
	double tmp;
	if (beta <= 2.2) {
		tmp = ((1.0 + beta) / t_0) * 0.16666666666666666;
	} else {
		tmp = ((1.0 + alpha) / beta) / t_0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = beta + (alpha + 2.0)
	tmp = 0
	if beta <= 2.2:
		tmp = ((1.0 + beta) / t_0) * 0.16666666666666666
	else:
		tmp = ((1.0 + alpha) / beta) / t_0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(beta + Float64(alpha + 2.0))
	tmp = 0.0
	if (beta <= 2.2)
		tmp = Float64(Float64(Float64(1.0 + beta) / t_0) * 0.16666666666666666);
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = beta + (alpha + 2.0);
	tmp = 0.0;
	if (beta <= 2.2)
		tmp = ((1.0 + beta) / t_0) * 0.16666666666666666;
	else
		tmp = ((1.0 + alpha) / beta) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2.2], N[(N[(N[(1.0 + beta), $MachinePrecision] / t$95$0), $MachinePrecision] * 0.16666666666666666), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.2000000000000002

   1. Initial program 99.9%

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

      1. associate-/l/ 99.4%

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

      2. associate-/r* 94.7%

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

      3. associate-+l+ 94.7%

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

      4. +-commutative 94.7%

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

      5. associate-+r+ 94.7%

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

      6. associate-+l+ 94.7%

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

      7. distribute-rgt1-in 94.7%

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

      8. *-rgt-identity 94.7%

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

      9. distribute-lft-out 94.7%

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

      10. *-commutative 94.7%

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

      11. metadata-eval 94.7%

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

      12. associate-+l+ 94.7%

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

      13. +-commutative 94.7%

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

   3. Simplified 94.7%

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

   4. Taylor expanded in beta around 0 93.6%

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

      1. *-un-lft-identity 93.6%

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

      2. times-frac 98.3%

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

      3. +-commutative 98.3%

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

      4. +-commutative 98.3%

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

      5. +-commutative 98.3%

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

      6. +-commutative 98.3%

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

      7. +-commutative 98.3%

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

   6. Applied egg-rr 98.3%

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

      1. *-lft-identity 98.3%

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

      2. +-commutative 98.3%

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

      3. associate-+r+ 98.3%

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

      4. +-commutative 98.3%

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

      5. +-commutative 98.3%

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

      6. +-commutative 98.3%

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

   8. Simplified 98.3%

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

   9. Taylor expanded in alpha around 0 70.7%

      \[\leadsto \frac{\beta + 1}{\left(2 + \alpha\right) + \beta} \cdot \color{blue}{0.16666666666666666} \]

   if 2.2000000000000002 < beta

   1. Initial program 84.0%

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

      1. associate-/l/ 80.0%

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

      2. associate-+l+ 80.0%

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

      3. +-commutative 80.0%

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

      4. associate-+r+ 80.0%

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

      5. associate-+l+ 80.0%

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

      6. distribute-rgt1-in 80.0%

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

      7. *-rgt-identity 80.0%

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

      8. distribute-lft-out 80.0%

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

      9. +-commutative 80.0%

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

      10. associate-*l/ 86.2%

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

      11. *-commutative 86.2%

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

      12. associate-*r/ 85.3%

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

   3. Simplified 85.3%

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

      1. associate-*r/ 86.2%

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

      2. +-commutative 86.2%

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

      3. associate-+r+ 86.2%

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

      4. +-commutative 86.2%

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

      5. associate-+r+ 86.2%

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

      6. +-commutative 86.2%

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

   5. Applied egg-rr 86.2%

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

      1. times-frac 99.6%

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

      2. +-commutative 99.6%

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

      3. associate-+r+ 99.6%

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

      4. +-commutative 99.6%

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

      5. associate-+r+ 99.6%

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

      6. +-commutative 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in beta around inf 79.7%

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

      1. associate-+r+ 79.7%

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

      2. associate-*l/ 79.9%

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

   10. Applied egg-rr 79.9%

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

       1. associate-*r/ 79.9%

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

       2. *-rgt-identity 79.9%

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

       3. +-commutative 79.9%

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

       4. associate-+r+ 79.9%

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

       5. +-commutative 79.9%

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

       6. associate-+r+ 79.9%

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

   12. Simplified 79.9%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.2:\\ \;\;\;\;\frac{1 + \beta}{\beta + \left(\alpha + 2\right)} \cdot 0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + \left(\alpha + 2\right)}\\ \end{array} \]

Alternative 12: 71.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.05:\\ \;\;\;\;\frac{1 + \beta}{\beta + \left(\alpha + 2\right)} \cdot 0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{2 + \left(\alpha + \beta\right)}}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.05)
   (* (/ (+ 1.0 beta) (+ beta (+ alpha 2.0))) 0.16666666666666666)
   (/ (/ (+ 1.0 alpha) (+ 2.0 (+ alpha beta))) beta)))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.05) {
		tmp = ((1.0 + beta) / (beta + (alpha + 2.0))) * 0.16666666666666666;
	} else {
		tmp = ((1.0 + alpha) / (2.0 + (alpha + 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 <= 2.05d0) then
        tmp = ((1.0d0 + beta) / (beta + (alpha + 2.0d0))) * 0.16666666666666666d0
    else
        tmp = ((1.0d0 + alpha) / (2.0d0 + (alpha + beta))) / beta
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.05) {
		tmp = ((1.0 + beta) / (beta + (alpha + 2.0))) * 0.16666666666666666;
	} else {
		tmp = ((1.0 + alpha) / (2.0 + (alpha + beta))) / beta;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.05:
		tmp = ((1.0 + beta) / (beta + (alpha + 2.0))) * 0.16666666666666666
	else:
		tmp = ((1.0 + alpha) / (2.0 + (alpha + beta))) / beta
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.05)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(beta + Float64(alpha + 2.0))) * 0.16666666666666666);
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(2.0 + Float64(alpha + beta))) / beta);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.05)
		tmp = ((1.0 + beta) / (beta + (alpha + 2.0))) * 0.16666666666666666;
	else
		tmp = ((1.0 + alpha) / (2.0 + (alpha + beta))) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.05], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.16666666666666666), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.0499999999999998

   1. Initial program 99.9%

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

      1. associate-/l/ 99.4%

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

      2. associate-/r* 94.7%

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

      3. associate-+l+ 94.7%

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

      4. +-commutative 94.7%

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

      5. associate-+r+ 94.7%

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

      6. associate-+l+ 94.7%

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

      7. distribute-rgt1-in 94.7%

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

      8. *-rgt-identity 94.7%

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

      9. distribute-lft-out 94.7%

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

      10. *-commutative 94.7%

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

      11. metadata-eval 94.7%

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

      12. associate-+l+ 94.7%

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

      13. +-commutative 94.7%

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

   3. Simplified 94.7%

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

   4. Taylor expanded in beta around 0 93.6%

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

      1. *-un-lft-identity 93.6%

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

      2. times-frac 98.3%

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

      3. +-commutative 98.3%

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

      4. +-commutative 98.3%

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

      5. +-commutative 98.3%

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

      6. +-commutative 98.3%

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

      7. +-commutative 98.3%

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

   6. Applied egg-rr 98.3%

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

      1. *-lft-identity 98.3%

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

      2. +-commutative 98.3%

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

      3. associate-+r+ 98.3%

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

      4. +-commutative 98.3%

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

      5. +-commutative 98.3%

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

      6. +-commutative 98.3%

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

   8. Simplified 98.3%

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

   9. Taylor expanded in alpha around 0 70.7%

      \[\leadsto \frac{\beta + 1}{\left(2 + \alpha\right) + \beta} \cdot \color{blue}{0.16666666666666666} \]

   if 2.0499999999999998 < beta

   1. Initial program 84.0%

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

      1. associate-/l/ 80.0%

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

      2. associate-+l+ 80.0%

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

      3. +-commutative 80.0%

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

      4. associate-+r+ 80.0%

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

      5. associate-+l+ 80.0%

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

      6. distribute-rgt1-in 80.0%

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

      7. *-rgt-identity 80.0%

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

      8. distribute-lft-out 80.0%

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

      9. +-commutative 80.0%

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

      10. associate-*l/ 86.2%

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

      11. *-commutative 86.2%

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

      12. associate-*r/ 85.3%

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

   3. Simplified 85.3%

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

      1. associate-*r/ 86.2%

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

      2. +-commutative 86.2%

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

      3. associate-+r+ 86.2%

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

      4. +-commutative 86.2%

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

      5. associate-+r+ 86.2%

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

      6. +-commutative 86.2%

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

   5. Applied egg-rr 86.2%

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

      1. times-frac 99.6%

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

      2. +-commutative 99.6%

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

      3. associate-+r+ 99.6%

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

      4. +-commutative 99.6%

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

      5. associate-+r+ 99.6%

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

      6. +-commutative 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in beta around inf 79.7%

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

      1. un-div-inv 79.9%

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

      2. associate-+r+ 79.9%

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

   10. Applied egg-rr 79.9%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.05:\\ \;\;\;\;\frac{1 + \beta}{\beta + \left(\alpha + 2\right)} \cdot 0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{2 + \left(\alpha + \beta\right)}}{\beta}\\ \end{array} \]

Alternative 13: 70.4% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.8:\\ \;\;\;\;0.16666666666666666 \cdot \frac{1 + \beta}{2 + \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.8)
   (* 0.16666666666666666 (/ (+ 1.0 beta) (+ 2.0 beta)))
   (/ (/ (+ 1.0 alpha) beta) beta)))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.8) {
		tmp = 0.16666666666666666 * ((1.0 + beta) / (2.0 + beta));
	} else {
		tmp = ((1.0 + alpha) / 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 <= 2.8d0) then
        tmp = 0.16666666666666666d0 * ((1.0d0 + beta) / (2.0d0 + beta))
    else
        tmp = ((1.0d0 + alpha) / beta) / beta
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.8) {
		tmp = 0.16666666666666666 * ((1.0 + beta) / (2.0 + beta));
	} else {
		tmp = ((1.0 + alpha) / beta) / beta;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.8:
		tmp = 0.16666666666666666 * ((1.0 + beta) / (2.0 + beta))
	else:
		tmp = ((1.0 + alpha) / beta) / beta
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.8)
		tmp = Float64(0.16666666666666666 * Float64(Float64(1.0 + beta) / Float64(2.0 + beta)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / beta);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.8)
		tmp = 0.16666666666666666 * ((1.0 + beta) / (2.0 + beta));
	else
		tmp = ((1.0 + alpha) / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.8], N[(0.16666666666666666 * N[(N[(1.0 + beta), $MachinePrecision] / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.7999999999999998

   1. Initial program 99.9%

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

      1. associate-/l/ 99.4%

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

      2. associate-/r* 94.7%

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

      3. associate-+l+ 94.7%

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

      4. +-commutative 94.7%

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

      5. associate-+r+ 94.7%

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

      6. associate-+l+ 94.7%

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

      7. distribute-rgt1-in 94.7%

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

      8. *-rgt-identity 94.7%

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

      9. distribute-lft-out 94.7%

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

      10. *-commutative 94.7%

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

      11. metadata-eval 94.7%

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

      12. associate-+l+ 94.7%

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

      13. +-commutative 94.7%

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

   3. Simplified 94.7%

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

   4. Taylor expanded in beta around 0 93.6%

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

   5. Taylor expanded in alpha around 0 69.9%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot \frac{\beta + 1}{\beta + 2}} \]

   if 2.7999999999999998 < beta

   1. Initial program 84.0%

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

      1. associate-/l/ 80.0%

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

      2. associate-+l+ 80.0%

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

      3. +-commutative 80.0%

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

      4. associate-+r+ 80.0%

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

      5. associate-+l+ 80.0%

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

      6. distribute-rgt1-in 80.0%

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

      7. *-rgt-identity 80.0%

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

      8. distribute-lft-out 80.0%

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

      9. +-commutative 80.0%

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

      10. associate-*l/ 86.2%

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

      11. *-commutative 86.2%

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

      12. associate-*r/ 85.3%

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

   3. Simplified 85.3%

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

      1. associate-*r/ 86.2%

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

      2. +-commutative 86.2%

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

      3. associate-+r+ 86.2%

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

      4. +-commutative 86.2%

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

      5. associate-+r+ 86.2%

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

      6. +-commutative 86.2%

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

   5. Applied egg-rr 86.2%

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

      1. times-frac 99.6%

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

      2. +-commutative 99.6%

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

      3. associate-+r+ 99.6%

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

      4. +-commutative 99.6%

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

      5. associate-+r+ 99.6%

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

      6. +-commutative 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in beta around inf 79.7%

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

   9. Taylor expanded in beta around inf 79.5%

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

       1. un-div-inv 79.7%

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

   11. Applied egg-rr 79.7%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.8:\\ \;\;\;\;0.16666666666666666 \cdot \frac{1 + \beta}{2 + \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \]

Alternative 14: 28.6% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta} \cdot \frac{\alpha}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 1.0) (/ (/ 1.0 beta) beta) (* (/ 1.0 beta) (/ alpha beta))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if alpha < 1

   1. Initial program 99.8%

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

      1. associate-/l/ 99.8%

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

      2. associate-+l+ 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+r+ 99.8%

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

      5. associate-+l+ 99.8%

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

      6. distribute-rgt1-in 99.8%

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

      7. *-rgt-identity 99.8%

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

      8. distribute-lft-out 99.8%

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

      9. +-commutative 99.8%

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

      10. associate-*l/ 99.7%

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

      11. *-commutative 99.7%

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

      12. associate-*r/ 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in beta around inf 39.5%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 39.5%

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

   6. Simplified 39.5%

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

   7. Taylor expanded in alpha around 0 39.1%

      \[\leadsto \color{blue}{\frac{1}{{\beta}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 39.1%

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

   9. Simplified 39.1%

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

   10. Taylor expanded in beta around 0 39.1%

       \[\leadsto \color{blue}{\frac{1}{{\beta}^{2}}} \]
   11. Step-by-step derivation

       1. unpow2 39.1%

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

       2. associate-/r* 39.2%

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

   12. Simplified 39.2%

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

   if 1 < alpha

   1. Initial program 79.4%

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

      1. associate-/l/ 73.5%

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

      2. associate-+l+ 73.5%

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

      3. +-commutative 73.5%

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

      4. associate-+r+ 73.5%

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

      5. associate-+l+ 73.5%

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

      6. distribute-rgt1-in 73.5%

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

      7. *-rgt-identity 73.5%

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

      8. distribute-lft-out 73.5%

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

      9. +-commutative 73.5%

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

      10. associate-*l/ 81.6%

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

      11. *-commutative 81.6%

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

      12. associate-*r/ 70.8%

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

   3. Simplified 70.8%

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

      1. associate-*r/ 81.6%

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

      2. +-commutative 81.6%

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

      3. associate-+r+ 81.6%

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

      4. +-commutative 81.6%

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

      5. associate-+r+ 81.6%

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

      6. +-commutative 81.6%

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{1 + \beta}{2 + \left(\alpha + \beta\right)}}{\color{blue}{\left(2 + \left(\alpha + \beta\right)\right)} \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

   5. Applied egg-rr 81.6%

      \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \frac{1 + \beta}{2 + \left(\alpha + \beta\right)}}{\left(2 + \left(\alpha + \beta\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]
   6. Step-by-step derivation

      1. times-frac 99.7%

         \[\leadsto \color{blue}{\frac{\alpha + 1}{2 + \left(\alpha + \beta\right)} \cdot \frac{\frac{1 + \beta}{2 + \left(\alpha + \beta\right)}}{\alpha + \left(\beta + 3\right)}} \]

      2. +-commutative 99.7%

         \[\leadsto \frac{\color{blue}{1 + \alpha}}{2 + \left(\alpha + \beta\right)} \cdot \frac{\frac{1 + \beta}{2 + \left(\alpha + \beta\right)}}{\alpha + \left(\beta + 3\right)} \]

      3. associate-+r+ 99.7%

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\left(2 + \alpha\right) + \beta}} \cdot \frac{\frac{1 + \beta}{2 + \left(\alpha + \beta\right)}}{\alpha + \left(\beta + 3\right)} \]

      4. +-commutative 99.7%

         \[\leadsto \frac{1 + \alpha}{\left(2 + \alpha\right) + \beta} \cdot \frac{\frac{\color{blue}{\beta + 1}}{2 + \left(\alpha + \beta\right)}}{\alpha + \left(\beta + 3\right)} \]

      5. associate-+r+ 99.7%

         \[\leadsto \frac{1 + \alpha}{\left(2 + \alpha\right) + \beta} \cdot \frac{\frac{\beta + 1}{\color{blue}{\left(2 + \alpha\right) + \beta}}}{\alpha + \left(\beta + 3\right)} \]

      6. +-commutative 99.7%

         \[\leadsto \frac{1 + \alpha}{\left(2 + \alpha\right) + \beta} \cdot \frac{\frac{\beta + 1}{\left(2 + \alpha\right) + \beta}}{\color{blue}{\left(\beta + 3\right) + \alpha}} \]

   7. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(2 + \alpha\right) + \beta} \cdot \frac{\frac{\beta + 1}{\left(2 + \alpha\right) + \beta}}{\left(\beta + 3\right) + \alpha}} \]

   8. Taylor expanded in beta around inf 19.2%

      \[\leadsto \frac{1 + \alpha}{\left(2 + \alpha\right) + \beta} \cdot \color{blue}{\frac{1}{\beta}} \]

   9. Taylor expanded in beta around inf 18.8%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta}} \cdot \frac{1}{\beta} \]

   10. Taylor expanded in alpha around inf 18.7%

       \[\leadsto \color{blue}{\frac{\alpha}{\beta}} \cdot \frac{1}{\beta} \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta} \cdot \frac{\alpha}{\beta}\\ \end{array} \]

Alternative 15: 28.8% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.85 \cdot 10^{+153}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta} \cdot \frac{\alpha}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.85e+153)
   (/ (+ 1.0 alpha) (* beta beta))
   (* (/ 1.0 beta) (/ alpha beta))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.85e+153) {
		tmp = (1.0 + alpha) / (beta * beta);
	} else {
		tmp = (1.0 / beta) * (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 <= 2.85d+153) then
        tmp = (1.0d0 + alpha) / (beta * beta)
    else
        tmp = (1.0d0 / beta) * (alpha / beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.85e+153) {
		tmp = (1.0 + alpha) / (beta * beta);
	} else {
		tmp = (1.0 / beta) * (alpha / beta);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.85e+153:
		tmp = (1.0 + alpha) / (beta * beta)
	else:
		tmp = (1.0 / beta) * (alpha / beta)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.85e+153)
		tmp = Float64(Float64(1.0 + alpha) / Float64(beta * beta));
	else
		tmp = Float64(Float64(1.0 / beta) * Float64(alpha / beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.85e+153)
		tmp = (1.0 + alpha) / (beta * beta);
	else
		tmp = (1.0 / beta) * (alpha / beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.85e+153], N[(N[(1.0 + alpha), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / beta), $MachinePrecision] * N[(alpha / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.84999999999999993e153

   1. Initial program 97.3%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Step-by-step derivation

      1. associate-/l/ 95.6%

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. associate-+l+ 95.6%

         \[\leadsto \frac{\frac{\color{blue}{\left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)} + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      3. +-commutative 95.6%

         \[\leadsto \frac{\frac{\left(\alpha + \color{blue}{\left(\beta \cdot \alpha + \beta\right)}\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      4. associate-+r+ 95.6%

         \[\leadsto \frac{\frac{\color{blue}{\left(\left(\alpha + \beta \cdot \alpha\right) + \beta\right)} + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      5. associate-+l+ 95.6%

         \[\leadsto \frac{\frac{\color{blue}{\left(\alpha + \beta \cdot \alpha\right) + \left(\beta + 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      6. distribute-rgt1-in 95.6%

         \[\leadsto \frac{\frac{\color{blue}{\left(\beta + 1\right) \cdot \alpha} + \left(\beta + 1\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      7. *-rgt-identity 95.6%

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \alpha + \color{blue}{\left(\beta + 1\right) \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      8. distribute-lft-out 95.6%

         \[\leadsto \frac{\frac{\color{blue}{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      9. +-commutative 95.6%

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \color{blue}{\left(1 + \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      10. associate-*l/ 97.2%

          \[\leadsto \frac{\color{blue}{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      11. *-commutative 97.2%

          \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right) \cdot \frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      12. associate-*r/ 93.1%

          \[\leadsto \color{blue}{\left(1 + \alpha\right) \cdot \frac{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   3. Simplified 93.1%

      \[\leadsto \color{blue}{\left(\alpha + 1\right) \cdot \frac{\frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]

   4. Taylor expanded in beta around inf 17.9%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 17.9%

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

   6. Simplified 17.9%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]

   if 2.84999999999999993e153 < beta

   1. Initial program 80.7%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Step-by-step derivation

      1. associate-/l/ 78.5%

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. associate-+l+ 78.5%

         \[\leadsto \frac{\frac{\color{blue}{\left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)} + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      3. +-commutative 78.5%

         \[\leadsto \frac{\frac{\left(\alpha + \color{blue}{\left(\beta \cdot \alpha + \beta\right)}\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      4. associate-+r+ 78.5%

         \[\leadsto \frac{\frac{\color{blue}{\left(\left(\alpha + \beta \cdot \alpha\right) + \beta\right)} + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      5. associate-+l+ 78.5%

         \[\leadsto \frac{\frac{\color{blue}{\left(\alpha + \beta \cdot \alpha\right) + \left(\beta + 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      6. distribute-rgt1-in 78.5%

         \[\leadsto \frac{\frac{\color{blue}{\left(\beta + 1\right) \cdot \alpha} + \left(\beta + 1\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      7. *-rgt-identity 78.5%

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \alpha + \color{blue}{\left(\beta + 1\right) \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      8. distribute-lft-out 78.5%

         \[\leadsto \frac{\frac{\color{blue}{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      9. +-commutative 78.5%

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \color{blue}{\left(1 + \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      10. associate-*l/ 83.8%

          \[\leadsto \frac{\color{blue}{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      11. *-commutative 83.8%

          \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right) \cdot \frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      12. associate-*r/ 83.9%

          \[\leadsto \color{blue}{\left(1 + \alpha\right) \cdot \frac{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   3. Simplified 83.9%

      \[\leadsto \color{blue}{\left(\alpha + 1\right) \cdot \frac{\frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ 83.8%

         \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]

      2. +-commutative 83.8%

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\color{blue}{1 + \beta}}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

      3. associate-+r+ 83.8%

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{1 + \beta}{\color{blue}{\left(\alpha + \beta\right) + 2}}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

      4. +-commutative 83.8%

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{1 + \beta}{\color{blue}{2 + \left(\alpha + \beta\right)}}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

      5. associate-+r+ 83.8%

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{1 + \beta}{2 + \left(\alpha + \beta\right)}}{\color{blue}{\left(\left(\alpha + \beta\right) + 2\right)} \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

      6. +-commutative 83.8%

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{1 + \beta}{2 + \left(\alpha + \beta\right)}}{\color{blue}{\left(2 + \left(\alpha + \beta\right)\right)} \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

   5. Applied egg-rr 83.8%

      \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \frac{1 + \beta}{2 + \left(\alpha + \beta\right)}}{\left(2 + \left(\alpha + \beta\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]
   6. Step-by-step derivation

      1. times-frac 99.9%

         \[\leadsto \color{blue}{\frac{\alpha + 1}{2 + \left(\alpha + \beta\right)} \cdot \frac{\frac{1 + \beta}{2 + \left(\alpha + \beta\right)}}{\alpha + \left(\beta + 3\right)}} \]

      2. +-commutative 99.9%

         \[\leadsto \frac{\color{blue}{1 + \alpha}}{2 + \left(\alpha + \beta\right)} \cdot \frac{\frac{1 + \beta}{2 + \left(\alpha + \beta\right)}}{\alpha + \left(\beta + 3\right)} \]

      3. associate-+r+ 99.9%

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\left(2 + \alpha\right) + \beta}} \cdot \frac{\frac{1 + \beta}{2 + \left(\alpha + \beta\right)}}{\alpha + \left(\beta + 3\right)} \]

      4. +-commutative 99.9%

         \[\leadsto \frac{1 + \alpha}{\left(2 + \alpha\right) + \beta} \cdot \frac{\frac{\color{blue}{\beta + 1}}{2 + \left(\alpha + \beta\right)}}{\alpha + \left(\beta + 3\right)} \]

      5. associate-+r+ 99.9%

         \[\leadsto \frac{1 + \alpha}{\left(2 + \alpha\right) + \beta} \cdot \frac{\frac{\beta + 1}{\color{blue}{\left(2 + \alpha\right) + \beta}}}{\alpha + \left(\beta + 3\right)} \]

      6. +-commutative 99.9%

         \[\leadsto \frac{1 + \alpha}{\left(2 + \alpha\right) + \beta} \cdot \frac{\frac{\beta + 1}{\left(2 + \alpha\right) + \beta}}{\color{blue}{\left(\beta + 3\right) + \alpha}} \]

   7. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(2 + \alpha\right) + \beta} \cdot \frac{\frac{\beta + 1}{\left(2 + \alpha\right) + \beta}}{\left(\beta + 3\right) + \alpha}} \]

   8. Taylor expanded in beta around inf 88.4%

      \[\leadsto \frac{1 + \alpha}{\left(2 + \alpha\right) + \beta} \cdot \color{blue}{\frac{1}{\beta}} \]

   9. Taylor expanded in beta around inf 88.3%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta}} \cdot \frac{1}{\beta} \]

   10. Taylor expanded in alpha around inf 88.1%

       \[\leadsto \color{blue}{\frac{\alpha}{\beta}} \cdot \frac{1}{\beta} \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.85 \cdot 10^{+153}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta} \cdot \frac{\alpha}{\beta}\\ \end{array} \]

Alternative 16: 27.4% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 1.0) (/ 1.0 (* beta beta)) (/ alpha (* beta beta))))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1.0) {
		tmp = 1.0 / (beta * beta);
	} else {
		tmp = alpha / (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 (alpha <= 1.0d0) then
        tmp = 1.0d0 / (beta * beta)
    else
        tmp = alpha / (beta * beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1.0) {
		tmp = 1.0 / (beta * beta);
	} else {
		tmp = alpha / (beta * beta);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 1.0:
		tmp = 1.0 / (beta * beta)
	else:
		tmp = alpha / (beta * beta)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 1.0)
		tmp = Float64(1.0 / Float64(beta * beta));
	else
		tmp = Float64(alpha / Float64(beta * beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 1.0)
		tmp = 1.0 / (beta * beta);
	else
		tmp = alpha / (beta * beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 1.0], N[(1.0 / N[(beta * beta), $MachinePrecision]), $MachinePrecision], N[(alpha / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1

   1. Initial program 99.8%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Step-by-step derivation

      1. associate-/l/ 99.8%

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. associate-+l+ 99.8%

         \[\leadsto \frac{\frac{\color{blue}{\left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)} + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      3. +-commutative 99.8%

         \[\leadsto \frac{\frac{\left(\alpha + \color{blue}{\left(\beta \cdot \alpha + \beta\right)}\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      4. associate-+r+ 99.8%

         \[\leadsto \frac{\frac{\color{blue}{\left(\left(\alpha + \beta \cdot \alpha\right) + \beta\right)} + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      5. associate-+l+ 99.8%

         \[\leadsto \frac{\frac{\color{blue}{\left(\alpha + \beta \cdot \alpha\right) + \left(\beta + 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      6. distribute-rgt1-in 99.8%

         \[\leadsto \frac{\frac{\color{blue}{\left(\beta + 1\right) \cdot \alpha} + \left(\beta + 1\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      7. *-rgt-identity 99.8%

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \alpha + \color{blue}{\left(\beta + 1\right) \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      8. distribute-lft-out 99.8%

         \[\leadsto \frac{\frac{\color{blue}{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      9. +-commutative 99.8%

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \color{blue}{\left(1 + \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      10. associate-*l/ 99.7%

          \[\leadsto \frac{\color{blue}{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      11. *-commutative 99.7%

          \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right) \cdot \frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      12. associate-*r/ 99.7%

          \[\leadsto \color{blue}{\left(1 + \alpha\right) \cdot \frac{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\left(\alpha + 1\right) \cdot \frac{\frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]

   4. Taylor expanded in beta around inf 39.5%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 39.5%

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

   6. Simplified 39.5%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]

   7. Taylor expanded in alpha around 0 39.1%

      \[\leadsto \color{blue}{\frac{1}{{\beta}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 39.1%

         \[\leadsto \frac{1}{\color{blue}{\beta \cdot \beta}} \]

   9. Simplified 39.1%

      \[\leadsto \color{blue}{\frac{1}{\beta \cdot \beta}} \]

   if 1 < alpha

   1. Initial program 79.4%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Step-by-step derivation

      1. associate-/l/ 73.5%

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. associate-+l+ 73.5%

         \[\leadsto \frac{\frac{\color{blue}{\left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)} + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      3. +-commutative 73.5%

         \[\leadsto \frac{\frac{\left(\alpha + \color{blue}{\left(\beta \cdot \alpha + \beta\right)}\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      4. associate-+r+ 73.5%

         \[\leadsto \frac{\frac{\color{blue}{\left(\left(\alpha + \beta \cdot \alpha\right) + \beta\right)} + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      5. associate-+l+ 73.5%

         \[\leadsto \frac{\frac{\color{blue}{\left(\alpha + \beta \cdot \alpha\right) + \left(\beta + 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      6. distribute-rgt1-in 73.5%

         \[\leadsto \frac{\frac{\color{blue}{\left(\beta + 1\right) \cdot \alpha} + \left(\beta + 1\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      7. *-rgt-identity 73.5%

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \alpha + \color{blue}{\left(\beta + 1\right) \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      8. distribute-lft-out 73.5%

         \[\leadsto \frac{\frac{\color{blue}{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      9. +-commutative 73.5%

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \color{blue}{\left(1 + \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      10. associate-*l/ 81.6%

          \[\leadsto \frac{\color{blue}{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      11. *-commutative 81.6%

          \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right) \cdot \frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      12. associate-*r/ 70.8%

          \[\leadsto \color{blue}{\left(1 + \alpha\right) \cdot \frac{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   3. Simplified 70.8%

      \[\leadsto \color{blue}{\left(\alpha + 1\right) \cdot \frac{\frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]

   4. Taylor expanded in beta around inf 15.7%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 15.7%

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

   6. Simplified 15.7%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]

   7. Taylor expanded in alpha around inf 15.6%

      \[\leadsto \color{blue}{\frac{\alpha}{{\beta}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 15.6%

         \[\leadsto \frac{\alpha}{\color{blue}{\beta \cdot \beta}} \]

   9. Simplified 15.6%

      \[\leadsto \color{blue}{\frac{\alpha}{\beta \cdot \beta}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 32.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha}{\beta \cdot \beta}\\ \end{array} \]

Alternative 17: 27.6% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.45:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 1.45) (/ (/ 1.0 beta) beta) (/ alpha (* beta beta))))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1.45) {
		tmp = (1.0 / beta) / beta;
	} else {
		tmp = alpha / (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 (alpha <= 1.45d0) then
        tmp = (1.0d0 / beta) / beta
    else
        tmp = alpha / (beta * beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1.45) {
		tmp = (1.0 / beta) / beta;
	} else {
		tmp = alpha / (beta * beta);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 1.45:
		tmp = (1.0 / beta) / beta
	else:
		tmp = alpha / (beta * beta)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 1.45)
		tmp = Float64(Float64(1.0 / beta) / beta);
	else
		tmp = Float64(alpha / Float64(beta * beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 1.45)
		tmp = (1.0 / beta) / beta;
	else
		tmp = alpha / (beta * beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 1.45], N[(N[(1.0 / beta), $MachinePrecision] / beta), $MachinePrecision], N[(alpha / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.44999999999999996

   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/ 99.8%

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. associate-+l+ 99.8%

         \[\leadsto \frac{\frac{\color{blue}{\left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)} + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      3. +-commutative 99.8%

         \[\leadsto \frac{\frac{\left(\alpha + \color{blue}{\left(\beta \cdot \alpha + \beta\right)}\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      4. associate-+r+ 99.8%

         \[\leadsto \frac{\frac{\color{blue}{\left(\left(\alpha + \beta \cdot \alpha\right) + \beta\right)} + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      5. associate-+l+ 99.8%

         \[\leadsto \frac{\frac{\color{blue}{\left(\alpha + \beta \cdot \alpha\right) + \left(\beta + 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      6. distribute-rgt1-in 99.8%

         \[\leadsto \frac{\frac{\color{blue}{\left(\beta + 1\right) \cdot \alpha} + \left(\beta + 1\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      7. *-rgt-identity 99.8%

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \alpha + \color{blue}{\left(\beta + 1\right) \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      8. distribute-lft-out 99.8%

         \[\leadsto \frac{\frac{\color{blue}{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      9. +-commutative 99.8%

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \color{blue}{\left(1 + \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      10. associate-*l/ 99.7%

          \[\leadsto \frac{\color{blue}{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      11. *-commutative 99.7%

          \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right) \cdot \frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      12. associate-*r/ 99.7%

          \[\leadsto \color{blue}{\left(1 + \alpha\right) \cdot \frac{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\left(\alpha + 1\right) \cdot \frac{\frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]

   4. Taylor expanded in beta around inf 39.5%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 39.5%

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

   6. Simplified 39.5%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]

   7. Taylor expanded in alpha around 0 39.1%

      \[\leadsto \color{blue}{\frac{1}{{\beta}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 39.1%

         \[\leadsto \frac{1}{\color{blue}{\beta \cdot \beta}} \]

   9. Simplified 39.1%

      \[\leadsto \color{blue}{\frac{1}{\beta \cdot \beta}} \]

   10. Taylor expanded in beta around 0 39.1%

       \[\leadsto \color{blue}{\frac{1}{{\beta}^{2}}} \]
   11. Step-by-step derivation

       1. unpow2 39.1%

          \[\leadsto \frac{1}{\color{blue}{\beta \cdot \beta}} \]

       2. associate-/r* 39.2%

          \[\leadsto \color{blue}{\frac{\frac{1}{\beta}}{\beta}} \]

   12. Simplified 39.2%

       \[\leadsto \color{blue}{\frac{\frac{1}{\beta}}{\beta}} \]

   if 1.44999999999999996 < alpha

   1. Initial program 79.4%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Step-by-step derivation

      1. associate-/l/ 73.5%

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. associate-+l+ 73.5%

         \[\leadsto \frac{\frac{\color{blue}{\left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)} + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      3. +-commutative 73.5%

         \[\leadsto \frac{\frac{\left(\alpha + \color{blue}{\left(\beta \cdot \alpha + \beta\right)}\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      4. associate-+r+ 73.5%

         \[\leadsto \frac{\frac{\color{blue}{\left(\left(\alpha + \beta \cdot \alpha\right) + \beta\right)} + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      5. associate-+l+ 73.5%

         \[\leadsto \frac{\frac{\color{blue}{\left(\alpha + \beta \cdot \alpha\right) + \left(\beta + 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      6. distribute-rgt1-in 73.5%

         \[\leadsto \frac{\frac{\color{blue}{\left(\beta + 1\right) \cdot \alpha} + \left(\beta + 1\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      7. *-rgt-identity 73.5%

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \alpha + \color{blue}{\left(\beta + 1\right) \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      8. distribute-lft-out 73.5%

         \[\leadsto \frac{\frac{\color{blue}{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      9. +-commutative 73.5%

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \color{blue}{\left(1 + \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      10. associate-*l/ 81.6%

          \[\leadsto \frac{\color{blue}{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      11. *-commutative 81.6%

          \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right) \cdot \frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      12. associate-*r/ 70.8%

          \[\leadsto \color{blue}{\left(1 + \alpha\right) \cdot \frac{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   3. Simplified 70.8%

      \[\leadsto \color{blue}{\left(\alpha + 1\right) \cdot \frac{\frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]

   4. Taylor expanded in beta around inf 15.7%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 15.7%

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

   6. Simplified 15.7%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]

   7. Taylor expanded in alpha around inf 15.6%

      \[\leadsto \color{blue}{\frac{\alpha}{{\beta}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 15.6%

         \[\leadsto \frac{\alpha}{\color{blue}{\beta \cdot \beta}} \]

   9. Simplified 15.6%

      \[\leadsto \color{blue}{\frac{\alpha}{\beta \cdot \beta}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 32.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.45:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha}{\beta \cdot \beta}\\ \end{array} \]

Alternative 18: 29.0% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1 + \alpha}{\beta}}{\beta} \end{array} \]

FPCore

(FPCore (alpha beta) :precision binary64 (/ (/ (+ 1.0 alpha) beta) beta))

C

double code(double alpha, double beta) {
	return ((1.0 + alpha) / beta) / beta;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = ((1.0d0 + alpha) / beta) / beta
end function

Java

public static double code(double alpha, double beta) {
	return ((1.0 + alpha) / beta) / beta;
}

Python

def code(alpha, beta):
	return ((1.0 + alpha) / beta) / beta

Julia

function code(alpha, beta)
	return Float64(Float64(Float64(1.0 + alpha) / beta) / beta)
end

MATLAB

function tmp = code(alpha, beta)
	tmp = ((1.0 + alpha) / beta) / beta;
end

Wolfram

code[alpha_, beta_] := N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]

Derivation

1. Initial program 93.7%

   \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
2. Step-by-step derivation

   1. associate-/l/ 91.9%

      \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   2. associate-+l+ 91.9%

      \[\leadsto \frac{\frac{\color{blue}{\left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)} + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   3. +-commutative 91.9%

      \[\leadsto \frac{\frac{\left(\alpha + \color{blue}{\left(\beta \cdot \alpha + \beta\right)}\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   4. associate-+r+ 91.9%

      \[\leadsto \frac{\frac{\color{blue}{\left(\left(\alpha + \beta \cdot \alpha\right) + \beta\right)} + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   5. associate-+l+ 91.9%

      \[\leadsto \frac{\frac{\color{blue}{\left(\alpha + \beta \cdot \alpha\right) + \left(\beta + 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   6. distribute-rgt1-in 91.9%

      \[\leadsto \frac{\frac{\color{blue}{\left(\beta + 1\right) \cdot \alpha} + \left(\beta + 1\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   7. *-rgt-identity 91.9%

      \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \alpha + \color{blue}{\left(\beta + 1\right) \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   8. distribute-lft-out 91.9%

      \[\leadsto \frac{\frac{\color{blue}{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   9. +-commutative 91.9%

      \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \color{blue}{\left(1 + \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   10. associate-*l/ 94.3%

       \[\leadsto \frac{\color{blue}{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   11. *-commutative 94.3%

       \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right) \cdot \frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   12. associate-*r/ 91.0%

       \[\leadsto \color{blue}{\left(1 + \alpha\right) \cdot \frac{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

3. Simplified 91.0%

   \[\leadsto \color{blue}{\left(\alpha + 1\right) \cdot \frac{\frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]
4. Step-by-step derivation

   1. associate-*r/ 94.3%

      \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]

   2. +-commutative 94.3%

      \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\color{blue}{1 + \beta}}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

   3. associate-+r+ 94.3%

      \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{1 + \beta}{\color{blue}{\left(\alpha + \beta\right) + 2}}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

   4. +-commutative 94.3%

      \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{1 + \beta}{\color{blue}{2 + \left(\alpha + \beta\right)}}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

   5. associate-+r+ 94.3%

      \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{1 + \beta}{2 + \left(\alpha + \beta\right)}}{\color{blue}{\left(\left(\alpha + \beta\right) + 2\right)} \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

   6. +-commutative 94.3%

      \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{1 + \beta}{2 + \left(\alpha + \beta\right)}}{\color{blue}{\left(2 + \left(\alpha + \beta\right)\right)} \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

5. Applied egg-rr 94.3%

   \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \frac{1 + \beta}{2 + \left(\alpha + \beta\right)}}{\left(2 + \left(\alpha + \beta\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]
6. Step-by-step derivation

   1. times-frac 99.7%

      \[\leadsto \color{blue}{\frac{\alpha + 1}{2 + \left(\alpha + \beta\right)} \cdot \frac{\frac{1 + \beta}{2 + \left(\alpha + \beta\right)}}{\alpha + \left(\beta + 3\right)}} \]

   2. +-commutative 99.7%

      \[\leadsto \frac{\color{blue}{1 + \alpha}}{2 + \left(\alpha + \beta\right)} \cdot \frac{\frac{1 + \beta}{2 + \left(\alpha + \beta\right)}}{\alpha + \left(\beta + 3\right)} \]

   3. associate-+r+ 99.7%

      \[\leadsto \frac{1 + \alpha}{\color{blue}{\left(2 + \alpha\right) + \beta}} \cdot \frac{\frac{1 + \beta}{2 + \left(\alpha + \beta\right)}}{\alpha + \left(\beta + 3\right)} \]

   4. +-commutative 99.7%

      \[\leadsto \frac{1 + \alpha}{\left(2 + \alpha\right) + \beta} \cdot \frac{\frac{\color{blue}{\beta + 1}}{2 + \left(\alpha + \beta\right)}}{\alpha + \left(\beta + 3\right)} \]

   5. associate-+r+ 99.7%

      \[\leadsto \frac{1 + \alpha}{\left(2 + \alpha\right) + \beta} \cdot \frac{\frac{\beta + 1}{\color{blue}{\left(2 + \alpha\right) + \beta}}}{\alpha + \left(\beta + 3\right)} \]

   6. +-commutative 99.7%

      \[\leadsto \frac{1 + \alpha}{\left(2 + \alpha\right) + \beta} \cdot \frac{\frac{\beta + 1}{\left(2 + \alpha\right) + \beta}}{\color{blue}{\left(\beta + 3\right) + \alpha}} \]

7. Simplified 99.7%

   \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(2 + \alpha\right) + \beta} \cdot \frac{\frac{\beta + 1}{\left(2 + \alpha\right) + \beta}}{\left(\beta + 3\right) + \alpha}} \]

8. Taylor expanded in beta around inf 32.8%

   \[\leadsto \frac{1 + \alpha}{\left(2 + \alpha\right) + \beta} \cdot \color{blue}{\frac{1}{\beta}} \]

9. Taylor expanded in beta around inf 33.2%

   \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta}} \cdot \frac{1}{\beta} \]
10. Step-by-step derivation

    1. un-div-inv 33.3%

       \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\beta}}{\beta}} \]

11. Applied egg-rr 33.3%

    \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\beta}}{\beta}} \]

12. Final simplification 33.3%

    \[\leadsto \frac{\frac{1 + \alpha}{\beta}}{\beta} \]

Alternative 19: 26.4% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\beta \cdot \beta} \end{array} \]

FPCore

(FPCore (alpha beta) :precision binary64 (/ 1.0 (* beta beta)))

C

double code(double alpha, double beta) {
	return 1.0 / (beta * beta);
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = 1.0d0 / (beta * beta)
end function

Java

public static double code(double alpha, double beta) {
	return 1.0 / (beta * beta);
}

Python

def code(alpha, beta):
	return 1.0 / (beta * beta)

Julia

function code(alpha, beta)
	return Float64(1.0 / Float64(beta * beta))
end

MATLAB

function tmp = code(alpha, beta)
	tmp = 1.0 / (beta * beta);
end

Wolfram

code[alpha_, beta_] := N[(1.0 / N[(beta * beta), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.7%

   \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
2. Step-by-step derivation

   1. associate-/l/ 91.9%

      \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   2. associate-+l+ 91.9%

      \[\leadsto \frac{\frac{\color{blue}{\left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)} + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   3. +-commutative 91.9%

      \[\leadsto \frac{\frac{\left(\alpha + \color{blue}{\left(\beta \cdot \alpha + \beta\right)}\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   4. associate-+r+ 91.9%

      \[\leadsto \frac{\frac{\color{blue}{\left(\left(\alpha + \beta \cdot \alpha\right) + \beta\right)} + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   5. associate-+l+ 91.9%

      \[\leadsto \frac{\frac{\color{blue}{\left(\alpha + \beta \cdot \alpha\right) + \left(\beta + 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   6. distribute-rgt1-in 91.9%

      \[\leadsto \frac{\frac{\color{blue}{\left(\beta + 1\right) \cdot \alpha} + \left(\beta + 1\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   7. *-rgt-identity 91.9%

      \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \alpha + \color{blue}{\left(\beta + 1\right) \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   8. distribute-lft-out 91.9%

      \[\leadsto \frac{\frac{\color{blue}{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   9. +-commutative 91.9%

      \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \color{blue}{\left(1 + \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   10. associate-*l/ 94.3%

       \[\leadsto \frac{\color{blue}{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   11. *-commutative 94.3%

       \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right) \cdot \frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   12. associate-*r/ 91.0%

       \[\leadsto \color{blue}{\left(1 + \alpha\right) \cdot \frac{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

3. Simplified 91.0%

   \[\leadsto \color{blue}{\left(\alpha + 1\right) \cdot \frac{\frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]

4. Taylor expanded in beta around inf 32.3%

   \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
5. Step-by-step derivation

   1. unpow2 32.3%

      \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

6. Simplified 32.3%

   \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]

7. Taylor expanded in alpha around 0 30.9%

   \[\leadsto \color{blue}{\frac{1}{{\beta}^{2}}} \]
8. Step-by-step derivation

   1. unpow2 30.9%

      \[\leadsto \frac{1}{\color{blue}{\beta \cdot \beta}} \]

9. Simplified 30.9%

   \[\leadsto \color{blue}{\frac{1}{\beta \cdot \beta}} \]

10. Final simplification 30.9%

    \[\leadsto \frac{1}{\beta \cdot \beta} \]

Alternative 20: 4.2% accurate, 11.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\beta} \end{array} \]

FPCore

(FPCore (alpha beta) :precision binary64 (/ 1.0 beta))

C

double code(double alpha, double beta) {
	return 1.0 / beta;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = 1.0d0 / beta
end function

Java

public static double code(double alpha, double beta) {
	return 1.0 / beta;
}

Python

def code(alpha, beta):
	return 1.0 / beta

Julia

function code(alpha, beta)
	return Float64(1.0 / beta)
end

MATLAB

function tmp = code(alpha, beta)
	tmp = 1.0 / beta;
end

Wolfram

code[alpha_, beta_] := N[(1.0 / beta), $MachinePrecision]

Derivation

1. Initial program 93.7%

   \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
2. Step-by-step derivation

   1. associate-/l/ 91.9%

      \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   2. associate-+l+ 91.9%

      \[\leadsto \frac{\frac{\color{blue}{\left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)} + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   3. +-commutative 91.9%

      \[\leadsto \frac{\frac{\left(\alpha + \color{blue}{\left(\beta \cdot \alpha + \beta\right)}\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   4. associate-+r+ 91.9%

      \[\leadsto \frac{\frac{\color{blue}{\left(\left(\alpha + \beta \cdot \alpha\right) + \beta\right)} + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   5. associate-+l+ 91.9%

      \[\leadsto \frac{\frac{\color{blue}{\left(\alpha + \beta \cdot \alpha\right) + \left(\beta + 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   6. distribute-rgt1-in 91.9%

      \[\leadsto \frac{\frac{\color{blue}{\left(\beta + 1\right) \cdot \alpha} + \left(\beta + 1\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   7. *-rgt-identity 91.9%

      \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \alpha + \color{blue}{\left(\beta + 1\right) \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   8. distribute-lft-out 91.9%

      \[\leadsto \frac{\frac{\color{blue}{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   9. +-commutative 91.9%

      \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \color{blue}{\left(1 + \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   10. associate-*l/ 94.3%

       \[\leadsto \frac{\color{blue}{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   11. *-commutative 94.3%

       \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right) \cdot \frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   12. associate-*r/ 91.0%

       \[\leadsto \color{blue}{\left(1 + \alpha\right) \cdot \frac{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

3. Simplified 91.0%

   \[\leadsto \color{blue}{\left(\alpha + 1\right) \cdot \frac{\frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]
4. Step-by-step derivation

   1. associate-*r/ 94.3%

      \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]

   2. +-commutative 94.3%

      \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\color{blue}{1 + \beta}}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

   3. associate-+r+ 94.3%

      \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{1 + \beta}{\color{blue}{\left(\alpha + \beta\right) + 2}}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

   4. +-commutative 94.3%

      \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{1 + \beta}{\color{blue}{2 + \left(\alpha + \beta\right)}}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

   5. associate-+r+ 94.3%

      \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{1 + \beta}{2 + \left(\alpha + \beta\right)}}{\color{blue}{\left(\left(\alpha + \beta\right) + 2\right)} \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

   6. +-commutative 94.3%

      \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{1 + \beta}{2 + \left(\alpha + \beta\right)}}{\color{blue}{\left(2 + \left(\alpha + \beta\right)\right)} \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

5. Applied egg-rr 94.3%

   \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \frac{1 + \beta}{2 + \left(\alpha + \beta\right)}}{\left(2 + \left(\alpha + \beta\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]
6. Step-by-step derivation

   1. times-frac 99.7%

      \[\leadsto \color{blue}{\frac{\alpha + 1}{2 + \left(\alpha + \beta\right)} \cdot \frac{\frac{1 + \beta}{2 + \left(\alpha + \beta\right)}}{\alpha + \left(\beta + 3\right)}} \]

   2. +-commutative 99.7%

      \[\leadsto \frac{\color{blue}{1 + \alpha}}{2 + \left(\alpha + \beta\right)} \cdot \frac{\frac{1 + \beta}{2 + \left(\alpha + \beta\right)}}{\alpha + \left(\beta + 3\right)} \]

   3. associate-+r+ 99.7%

      \[\leadsto \frac{1 + \alpha}{\color{blue}{\left(2 + \alpha\right) + \beta}} \cdot \frac{\frac{1 + \beta}{2 + \left(\alpha + \beta\right)}}{\alpha + \left(\beta + 3\right)} \]

   4. +-commutative 99.7%

      \[\leadsto \frac{1 + \alpha}{\left(2 + \alpha\right) + \beta} \cdot \frac{\frac{\color{blue}{\beta + 1}}{2 + \left(\alpha + \beta\right)}}{\alpha + \left(\beta + 3\right)} \]

   5. associate-+r+ 99.7%

      \[\leadsto \frac{1 + \alpha}{\left(2 + \alpha\right) + \beta} \cdot \frac{\frac{\beta + 1}{\color{blue}{\left(2 + \alpha\right) + \beta}}}{\alpha + \left(\beta + 3\right)} \]

   6. +-commutative 99.7%

      \[\leadsto \frac{1 + \alpha}{\left(2 + \alpha\right) + \beta} \cdot \frac{\frac{\beta + 1}{\left(2 + \alpha\right) + \beta}}{\color{blue}{\left(\beta + 3\right) + \alpha}} \]

7. Simplified 99.7%

   \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(2 + \alpha\right) + \beta} \cdot \frac{\frac{\beta + 1}{\left(2 + \alpha\right) + \beta}}{\left(\beta + 3\right) + \alpha}} \]

8. Taylor expanded in beta around inf 32.8%

   \[\leadsto \frac{1 + \alpha}{\left(2 + \alpha\right) + \beta} \cdot \color{blue}{\frac{1}{\beta}} \]

9. Taylor expanded in alpha around inf 4.3%

   \[\leadsto \color{blue}{\frac{1}{\beta}} \]

10. Final simplification 4.3%

    \[\leadsto \frac{1}{\beta} \]

Reproduce

herbie shell --seed 2023229 
(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)))