Octave 3.8, jcobi/3

Percentage Accurate: 94.7% → 99.8%

Time: 23.5s

Alternatives: 14

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: 94.7% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.8%

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

   1. associate-/l/ 92.7%

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

   2. associate-/r* 83.0%

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

   3. +-commutative 83.0%

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

   4. associate-+r+ 83.0%

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

   5. +-commutative 83.0%

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

   6. associate-+r+ 83.0%

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

   7. associate-+r+ 83.0%

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

   8. distribute-rgt1-in 83.0%

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

   9. +-commutative 83.0%

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

   10. *-commutative 83.0%

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

   11. distribute-rgt1-in 83.0%

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

   12. +-commutative 83.0%

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

   13. times-frac 97.4%

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

3. Simplified 97.4%

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

   1. distribute-lft-in 97.4%

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

5. Applied egg-rr 97.4%

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

   1. div-inv 97.4%

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

   2. distribute-lft-in 97.4%

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

   3. expm1-log1p-u 97.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \left(\left(\beta + 1\right) \cdot \frac{1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}\right)\right)\right)} \]

   4. expm1-udef 71.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \left(\left(\beta + 1\right) \cdot \frac{1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}\right)\right)} - 1} \]

7. Applied egg-rr 71.5%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2} \cdot \frac{\beta + 1}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\left(\beta + \alpha\right) + 3\right)}\right)} - 1} \]
8. Step-by-step derivation

   1. expm1-def 97.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2} \cdot \frac{\beta + 1}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\left(\beta + \alpha\right) + 3\right)}\right)\right)} \]

   2. expm1-log1p 97.4%

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

   3. *-commutative 97.4%

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

   4. associate-/r* 99.8%

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

   5. +-commutative 99.8%

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

   6. +-commutative 99.8%

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

   7. +-commutative 99.8%

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

   8. +-commutative 99.8%

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

   9. +-commutative 99.8%

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

9. Simplified 99.8%

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

10. Final simplification 99.8%

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

Alternative 2: 73.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.6 \cdot 10^{+15}:\\ \;\;\;\;\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(3 \cdot \left(\beta + 2\right) + \beta \cdot \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.6e+15)
   (/
    (+ 1.0 beta)
    (* (+ beta 2.0) (+ (* 3.0 (+ beta 2.0)) (* beta (+ beta 2.0)))))
   (/ (/ (- alpha -1.0) beta) (+ alpha (+ beta 3.0)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.6e+15) {
		tmp = (1.0 + beta) / ((beta + 2.0) * ((3.0 * (beta + 2.0)) + (beta * (beta + 2.0))));
	} else {
		tmp = ((alpha - -1.0) / 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 <= 3.6d+15) then
        tmp = (1.0d0 + beta) / ((beta + 2.0d0) * ((3.0d0 * (beta + 2.0d0)) + (beta * (beta + 2.0d0))))
    else
        tmp = ((alpha - (-1.0d0)) / beta) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.6e+15) {
		tmp = (1.0 + beta) / ((beta + 2.0) * ((3.0 * (beta + 2.0)) + (beta * (beta + 2.0))));
	} else {
		tmp = ((alpha - -1.0) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 3.6e+15:
		tmp = (1.0 + beta) / ((beta + 2.0) * ((3.0 * (beta + 2.0)) + (beta * (beta + 2.0))))
	else:
		tmp = ((alpha - -1.0) / beta) / (alpha + (beta + 3.0))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.6e+15)
		tmp = Float64(Float64(1.0 + beta) / Float64(Float64(beta + 2.0) * Float64(Float64(3.0 * Float64(beta + 2.0)) + Float64(beta * Float64(beta + 2.0)))));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / beta) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.6e+15)
		tmp = (1.0 + beta) / ((beta + 2.0) * ((3.0 * (beta + 2.0)) + (beta * (beta + 2.0))));
	else
		tmp = ((alpha - -1.0) / beta) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 3.6e+15], N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(N[(3.0 * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] + N[(beta * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.6e15

   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/ 99.7%

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

      2. associate-/r* 93.7%

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

      3. +-commutative 93.7%

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

      4. associate-+r+ 93.7%

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

      5. +-commutative 93.7%

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

      6. associate-+r+ 93.7%

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

      7. associate-+r+ 93.7%

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

      8. distribute-rgt1-in 93.7%

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

      9. +-commutative 93.7%

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

      10. *-commutative 93.7%

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

      11. distribute-rgt1-in 93.7%

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

      12. +-commutative 93.7%

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

      13. times-frac 99.7%

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

   3. Simplified 99.7%

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

      1. distribute-lft-in 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in alpha around 0 67.6%

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

   if 3.6e15 < beta

   1. Initial program 85.3%

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

   2. Taylor expanded in beta around -inf 83.7%

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

      1. associate-*r/ 83.7%

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

      2. mul-1-neg 83.7%

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

      3. sub-neg 83.7%

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

      4. mul-1-neg 83.7%

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

      5. distribute-neg-in 83.7%

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

      6. +-commutative 83.7%

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

      7. mul-1-neg 83.7%

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

      8. distribute-lft-in 83.7%

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

      9. metadata-eval 83.7%

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

      10. mul-1-neg 83.7%

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

      11. unsub-neg 83.7%

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

   4. Simplified 83.7%

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

   5. Taylor expanded in alpha around 0 83.7%

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

      1. +-commutative 83.7%

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

      2. associate-+r+ 83.7%

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

   7. Simplified 83.7%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.6 \cdot 10^{+15}:\\ \;\;\;\;\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(3 \cdot \left(\beta + 2\right) + \beta \cdot \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 3: 73.0% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.8%

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

   1. associate-/l/ 92.7%

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

   2. associate-/r* 83.0%

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

   3. +-commutative 83.0%

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

   4. associate-+r+ 83.0%

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

   5. +-commutative 83.0%

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

   6. associate-+r+ 83.0%

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

   7. associate-+r+ 83.0%

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

   8. distribute-rgt1-in 83.0%

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

   9. +-commutative 83.0%

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

   10. *-commutative 83.0%

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

   11. distribute-rgt1-in 83.0%

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

   12. +-commutative 83.0%

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

   13. times-frac 97.4%

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

3. Simplified 97.4%

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

   1. distribute-lft-in 97.4%

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

5. Applied egg-rr 97.4%

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

   1. div-inv 97.4%

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

   2. distribute-lft-in 97.4%

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

   3. expm1-log1p-u 97.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \left(\left(\beta + 1\right) \cdot \frac{1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}\right)\right)\right)} \]

   4. expm1-udef 71.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \left(\left(\beta + 1\right) \cdot \frac{1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}\right)\right)} - 1} \]

7. Applied egg-rr 71.5%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2} \cdot \frac{\beta + 1}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\left(\beta + \alpha\right) + 3\right)}\right)} - 1} \]
8. Step-by-step derivation

   1. expm1-def 97.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2} \cdot \frac{\beta + 1}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\left(\beta + \alpha\right) + 3\right)}\right)\right)} \]

   2. expm1-log1p 97.4%

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

   3. *-commutative 97.4%

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

   4. associate-/r* 99.8%

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

   5. +-commutative 99.8%

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

   6. +-commutative 99.8%

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

   7. +-commutative 99.8%

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

   8. +-commutative 99.8%

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

   9. +-commutative 99.8%

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

9. Simplified 99.8%

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

    1. associate-*r/ 99.8%

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

    2. associate-/l/ 97.4%

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

    3. +-commutative 97.4%

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

    4. +-commutative 97.4%

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

    5. +-commutative 97.4%

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

    6. +-commutative 97.4%

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

11. Applied egg-rr 97.4%

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

    1. associate-/l* 97.4%

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

    2. *-commutative 97.4%

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

    3. +-commutative 97.4%

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

    4. +-commutative 97.4%

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

    5. associate-+r+ 97.4%

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

    6. +-commutative 97.4%

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

    7. +-commutative 97.4%

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

    8. associate-+r+ 97.4%

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

    9. +-commutative 97.4%

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

    10. +-commutative 97.4%

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

    11. associate-+r+ 97.4%

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

    12. +-commutative 97.4%

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

13. Simplified 97.4%

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

14. Taylor expanded in alpha around 0 72.6%

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

    1. associate-/r* 73.2%

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

16. Simplified 73.2%

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

17. Final simplification 73.2%

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

Alternative 4: 73.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 9.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{1}{\beta + 2} \cdot \frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 9.5e+15)
   (* (/ 1.0 (+ beta 2.0)) (/ (+ 1.0 beta) (* (+ beta 2.0) (+ beta 3.0))))
   (/ (/ (- alpha -1.0) beta) (+ alpha (+ beta 3.0)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 9.5e+15) {
		tmp = (1.0 / (beta + 2.0)) * ((1.0 + beta) / ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = ((alpha - -1.0) / 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 <= 9.5d+15) then
        tmp = (1.0d0 / (beta + 2.0d0)) * ((1.0d0 + beta) / ((beta + 2.0d0) * (beta + 3.0d0)))
    else
        tmp = ((alpha - (-1.0d0)) / beta) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 9.5e+15) {
		tmp = (1.0 / (beta + 2.0)) * ((1.0 + beta) / ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = ((alpha - -1.0) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 9.5e+15:
		tmp = (1.0 / (beta + 2.0)) * ((1.0 + beta) / ((beta + 2.0) * (beta + 3.0)))
	else:
		tmp = ((alpha - -1.0) / beta) / (alpha + (beta + 3.0))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 9.5e+15)
		tmp = Float64(Float64(1.0 / Float64(beta + 2.0)) * Float64(Float64(1.0 + beta) / Float64(Float64(beta + 2.0) * Float64(beta + 3.0))));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / beta) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 9.5e+15)
		tmp = (1.0 / (beta + 2.0)) * ((1.0 + beta) / ((beta + 2.0) * (beta + 3.0)));
	else
		tmp = ((alpha - -1.0) / beta) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 9.5e+15], N[(N[(1.0 / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 9.5e15

   1. Initial program 99.8%

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

      1. associate-/l/ 99.7%

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

      2. associate-/r* 93.7%

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

      3. +-commutative 93.7%

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

      4. associate-+r+ 93.7%

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

      5. +-commutative 93.7%

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

      6. associate-+r+ 93.7%

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

      7. associate-+r+ 93.7%

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

      8. distribute-rgt1-in 93.7%

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

      9. +-commutative 93.7%

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

      10. *-commutative 93.7%

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

      11. distribute-rgt1-in 93.7%

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

      12. +-commutative 93.7%

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

      13. times-frac 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in alpha around 0 67.5%

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

   5. Taylor expanded in alpha around 0 67.6%

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

   if 9.5e15 < beta

   1. Initial program 85.3%

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

   2. Taylor expanded in beta around -inf 83.7%

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

      1. associate-*r/ 83.7%

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

      2. mul-1-neg 83.7%

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

      3. sub-neg 83.7%

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

      4. mul-1-neg 83.7%

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

      5. distribute-neg-in 83.7%

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

      6. +-commutative 83.7%

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

      7. mul-1-neg 83.7%

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

      8. distribute-lft-in 83.7%

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

      9. metadata-eval 83.7%

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

      10. mul-1-neg 83.7%

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

      11. unsub-neg 83.7%

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

   4. Simplified 83.7%

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

   5. Taylor expanded in alpha around 0 83.7%

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

      1. +-commutative 83.7%

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

      2. associate-+r+ 83.7%

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

   7. Simplified 83.7%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 9.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{1}{\beta + 2} \cdot \frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 5: 73.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.3 \cdot 10^{+16}:\\ \;\;\;\;\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.3e+16)
   (/ (+ 1.0 beta) (* (+ beta 2.0) (* (+ beta 2.0) (+ beta 3.0))))
   (/ (/ (- alpha -1.0) beta) (+ alpha (+ beta 3.0)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.3e+16) {
		tmp = (1.0 + beta) / ((beta + 2.0) * ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = ((alpha - -1.0) / 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 <= 1.3d+16) then
        tmp = (1.0d0 + beta) / ((beta + 2.0d0) * ((beta + 2.0d0) * (beta + 3.0d0)))
    else
        tmp = ((alpha - (-1.0d0)) / beta) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.3e+16) {
		tmp = (1.0 + beta) / ((beta + 2.0) * ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = ((alpha - -1.0) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 1.3e+16:
		tmp = (1.0 + beta) / ((beta + 2.0) * ((beta + 2.0) * (beta + 3.0)))
	else:
		tmp = ((alpha - -1.0) / beta) / (alpha + (beta + 3.0))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.3e+16)
		tmp = Float64(Float64(1.0 + beta) / Float64(Float64(beta + 2.0) * Float64(Float64(beta + 2.0) * Float64(beta + 3.0))));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / beta) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.3e+16)
		tmp = (1.0 + beta) / ((beta + 2.0) * ((beta + 2.0) * (beta + 3.0)));
	else
		tmp = ((alpha - -1.0) / beta) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 1.3e+16], N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.3e16

   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/ 99.7%

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

      2. associate-/r* 93.7%

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

      3. +-commutative 93.7%

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

      4. associate-+r+ 93.7%

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

      5. +-commutative 93.7%

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

      6. associate-+r+ 93.7%

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

      7. associate-+r+ 93.7%

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

      8. distribute-rgt1-in 93.7%

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

      9. +-commutative 93.7%

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

      10. *-commutative 93.7%

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

      11. distribute-rgt1-in 93.7%

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

      12. +-commutative 93.7%

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

      13. times-frac 99.7%

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

   3. Simplified 99.7%

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

      1. distribute-lft-in 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in alpha around 0 67.6%

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

      1. distribute-rgt-in 67.6%

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

   8. Simplified 67.6%

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

   if 1.3e16 < beta

   1. Initial program 85.3%

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

   2. Taylor expanded in beta around -inf 83.7%

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

      1. associate-*r/ 83.7%

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

      2. mul-1-neg 83.7%

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

      3. sub-neg 83.7%

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

      4. mul-1-neg 83.7%

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

      5. distribute-neg-in 83.7%

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

      6. +-commutative 83.7%

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

      7. mul-1-neg 83.7%

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

      8. distribute-lft-in 83.7%

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

      9. metadata-eval 83.7%

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

      10. mul-1-neg 83.7%

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

      11. unsub-neg 83.7%

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

   4. Simplified 83.7%

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

   5. Taylor expanded in alpha around 0 83.7%

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

      1. +-commutative 83.7%

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

      2. associate-+r+ 83.7%

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

   7. Simplified 83.7%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.3 \cdot 10^{+16}:\\ \;\;\;\;\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 6: 72.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.5:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 2}}{6 + \beta \cdot 5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.5)
   (/ (/ (+ 1.0 beta) (+ beta 2.0)) (+ 6.0 (* beta 5.0)))
   (/ (/ (- alpha -1.0) beta) (+ alpha (+ beta 3.0)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.5) {
		tmp = ((1.0 + beta) / (beta + 2.0)) / (6.0 + (beta * 5.0));
	} else {
		tmp = ((alpha - -1.0) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 4.5d0) then
        tmp = ((1.0d0 + beta) / (beta + 2.0d0)) / (6.0d0 + (beta * 5.0d0))
    else
        tmp = ((alpha - (-1.0d0)) / beta) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.5) {
		tmp = ((1.0 + beta) / (beta + 2.0)) / (6.0 + (beta * 5.0));
	} else {
		tmp = ((alpha - -1.0) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 4.5:
		tmp = ((1.0 + beta) / (beta + 2.0)) / (6.0 + (beta * 5.0))
	else:
		tmp = ((alpha - -1.0) / beta) / (alpha + (beta + 3.0))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.5)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(beta + 2.0)) / Float64(6.0 + Float64(beta * 5.0)));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / beta) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4.5)
		tmp = ((1.0 + beta) / (beta + 2.0)) / (6.0 + (beta * 5.0));
	else
		tmp = ((alpha - -1.0) / beta) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 4.5], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(6.0 + N[(beta * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.5

   1. Initial program 99.8%

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

      1. associate-/l/ 99.7%

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

      2. associate-/r* 93.6%

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

      3. +-commutative 93.6%

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

      4. associate-+r+ 93.6%

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

      5. +-commutative 93.6%

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

      6. associate-+r+ 93.6%

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

      7. associate-+r+ 93.6%

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

      8. distribute-rgt1-in 93.6%

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

      9. +-commutative 93.6%

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

      10. *-commutative 93.6%

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

      11. distribute-rgt1-in 93.6%

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

      12. +-commutative 93.6%

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

      13. times-frac 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in alpha around 0 67.3%

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

      1. associate-*r/ 67.3%

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

      2. +-commutative 67.3%

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

      3. associate-+r+ 67.3%

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

      4. +-commutative 67.3%

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

      5. +-commutative 67.3%

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

   6. Applied egg-rr 67.3%

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

   7. Taylor expanded in beta around 0 66.9%

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

      1. *-commutative 66.9%

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

   9. Simplified 66.9%

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

   10. Taylor expanded in alpha around 0 67.0%

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

       1. associate-/r* 67.0%

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

       2. *-commutative 67.0%

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

   12. Simplified 67.0%

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

   if 4.5 < beta

   1. Initial program 85.5%

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

   2. Taylor expanded in beta around -inf 83.2%

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

      1. associate-*r/ 83.2%

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

      2. mul-1-neg 83.2%

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

      3. sub-neg 83.2%

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

      4. mul-1-neg 83.2%

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

      5. distribute-neg-in 83.2%

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

      6. +-commutative 83.2%

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

      7. mul-1-neg 83.2%

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

      8. distribute-lft-in 83.2%

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

      9. metadata-eval 83.2%

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

      10. mul-1-neg 83.2%

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

      11. unsub-neg 83.2%

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

   4. Simplified 83.2%

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

   5. Taylor expanded in alpha around 0 83.2%

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

      1. +-commutative 83.2%

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

      2. associate-+r+ 83.2%

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

   7. Simplified 83.2%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.5:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 2}}{6 + \beta \cdot 5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 7: 72.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.3:\\ \;\;\;\;\frac{0.16666666666666666 + \beta \cdot 0.027777777777777776}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.3)
   (/ (+ 0.16666666666666666 (* beta 0.027777777777777776)) (+ beta 2.0))
   (/ (/ (- alpha -1.0) beta) (+ alpha (+ beta 3.0)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.3) {
		tmp = (0.16666666666666666 + (beta * 0.027777777777777776)) / (beta + 2.0);
	} else {
		tmp = ((alpha - -1.0) / 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 <= 3.3d0) then
        tmp = (0.16666666666666666d0 + (beta * 0.027777777777777776d0)) / (beta + 2.0d0)
    else
        tmp = ((alpha - (-1.0d0)) / beta) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.3) {
		tmp = (0.16666666666666666 + (beta * 0.027777777777777776)) / (beta + 2.0);
	} else {
		tmp = ((alpha - -1.0) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 3.3:
		tmp = (0.16666666666666666 + (beta * 0.027777777777777776)) / (beta + 2.0)
	else:
		tmp = ((alpha - -1.0) / beta) / (alpha + (beta + 3.0))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.3)
		tmp = Float64(Float64(0.16666666666666666 + Float64(beta * 0.027777777777777776)) / Float64(beta + 2.0));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / beta) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.3)
		tmp = (0.16666666666666666 + (beta * 0.027777777777777776)) / (beta + 2.0);
	else
		tmp = ((alpha - -1.0) / beta) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 3.3], N[(N[(0.16666666666666666 + N[(beta * 0.027777777777777776), $MachinePrecision]), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.2999999999999998

   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/ 99.7%

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

      2. associate-/r* 93.6%

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

      3. +-commutative 93.6%

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

      4. associate-+r+ 93.6%

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

      5. +-commutative 93.6%

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

      6. associate-+r+ 93.6%

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

      7. associate-+r+ 93.6%

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

      8. distribute-rgt1-in 93.6%

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

      9. +-commutative 93.6%

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

      10. *-commutative 93.6%

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

      11. distribute-rgt1-in 93.6%

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

      12. +-commutative 93.6%

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

      13. times-frac 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in alpha around 0 67.3%

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

   5. Taylor expanded in beta around 0 66.9%

      \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \color{blue}{\left(0.16666666666666666 + 0.027777777777777776 \cdot \beta\right)} \]
   6. Step-by-step derivation

      1. *-commutative 66.9%

         \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \left(0.16666666666666666 + \color{blue}{\beta \cdot 0.027777777777777776}\right) \]

   7. Simplified 66.9%

      \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \color{blue}{\left(0.16666666666666666 + \beta \cdot 0.027777777777777776\right)} \]

   8. Taylor expanded in alpha around 0 67.0%

      \[\leadsto \color{blue}{\frac{0.16666666666666666 + 0.027777777777777776 \cdot \beta}{2 + \beta}} \]
   9. Step-by-step derivation

      1. *-commutative 67.0%

         \[\leadsto \frac{0.16666666666666666 + \color{blue}{\beta \cdot 0.027777777777777776}}{2 + \beta} \]

   10. Simplified 67.0%

       \[\leadsto \color{blue}{\frac{0.16666666666666666 + \beta \cdot 0.027777777777777776}{2 + \beta}} \]

   if 3.2999999999999998 < beta

   1. Initial program 85.5%

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

   2. Taylor expanded in beta around -inf 83.2%

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

      1. associate-*r/ 83.2%

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

      2. mul-1-neg 83.2%

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

      3. sub-neg 83.2%

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

      4. mul-1-neg 83.2%

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

      5. distribute-neg-in 83.2%

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

      6. +-commutative 83.2%

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

      7. mul-1-neg 83.2%

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

      8. distribute-lft-in 83.2%

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

      9. metadata-eval 83.2%

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

      10. mul-1-neg 83.2%

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

      11. unsub-neg 83.2%

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

   4. Simplified 83.2%

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

   5. Taylor expanded in alpha around 0 83.2%

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

      1. +-commutative 83.2%

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

      2. associate-+r+ 83.2%

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

   7. Simplified 83.2%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.3:\\ \;\;\;\;\frac{0.16666666666666666 + \beta \cdot 0.027777777777777776}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 8: 70.3% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.3:\\ \;\;\;\;0.08333333333333333\\ \mathbf{elif}\;\beta \leq 1.6 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.3)
   0.08333333333333333
   (if (<= beta 1.6e+154)
     (/ 1.0 (* beta (+ beta 3.0)))
     (/ (/ alpha beta) beta))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.3) {
		tmp = 0.08333333333333333;
	} else if (beta <= 1.6e+154) {
		tmp = 1.0 / (beta * (beta + 3.0));
	} 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 (beta <= 2.3d0) then
        tmp = 0.08333333333333333d0
    else if (beta <= 1.6d+154) then
        tmp = 1.0d0 / (beta * (beta + 3.0d0))
    else
        tmp = (alpha / beta) / beta
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.3) {
		tmp = 0.08333333333333333;
	} else if (beta <= 1.6e+154) {
		tmp = 1.0 / (beta * (beta + 3.0));
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.3:
		tmp = 0.08333333333333333
	elif beta <= 1.6e+154:
		tmp = 1.0 / (beta * (beta + 3.0))
	else:
		tmp = (alpha / beta) / beta
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.3)
		tmp = 0.08333333333333333;
	elseif (beta <= 1.6e+154)
		tmp = Float64(1.0 / Float64(beta * Float64(beta + 3.0)));
	else
		tmp = Float64(Float64(alpha / beta) / beta);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.3)
		tmp = 0.08333333333333333;
	elseif (beta <= 1.6e+154)
		tmp = 1.0 / (beta * (beta + 3.0));
	else
		tmp = (alpha / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.3], 0.08333333333333333, If[LessEqual[beta, 1.6e+154], N[(1.0 / N[(beta * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / beta), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if beta < 2.2999999999999998

   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/ 99.7%

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

      2. associate-/r* 93.6%

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

      3. +-commutative 93.6%

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

      4. associate-+r+ 93.6%

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

      5. +-commutative 93.6%

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

      6. associate-+r+ 93.6%

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

      7. associate-+r+ 93.6%

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

      8. distribute-rgt1-in 93.6%

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

      9. +-commutative 93.6%

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

      10. *-commutative 93.6%

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

      11. distribute-rgt1-in 93.6%

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

      12. +-commutative 93.6%

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

      13. times-frac 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in alpha around 0 67.3%

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

      1. associate-*r/ 67.3%

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

      2. +-commutative 67.3%

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

      3. associate-+r+ 67.3%

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

      4. +-commutative 67.3%

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

      5. +-commutative 67.3%

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

   6. Applied egg-rr 67.3%

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

   7. Taylor expanded in beta around 0 66.3%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot \frac{1 + \alpha}{2 + \alpha}} \]
   8. Step-by-step derivation

      1. *-commutative 66.3%

         \[\leadsto \color{blue}{\frac{1 + \alpha}{2 + \alpha} \cdot 0.16666666666666666} \]

      2. +-commutative 66.3%

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\alpha + 2}} \cdot 0.16666666666666666 \]

   9. Simplified 66.3%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\alpha + 2} \cdot 0.16666666666666666} \]

   10. Taylor expanded in alpha around 0 66.4%

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

   if 2.2999999999999998 < beta < 1.6e154

   1. Initial program 88.5%

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

   2. Taylor expanded in beta around -inf 70.6%

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

      1. associate-*r/ 70.6%

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

      2. mul-1-neg 70.6%

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

      3. sub-neg 70.6%

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

      4. mul-1-neg 70.6%

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

      5. distribute-neg-in 70.6%

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

      6. +-commutative 70.6%

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

      7. mul-1-neg 70.6%

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

      8. distribute-lft-in 70.6%

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

      9. metadata-eval 70.6%

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

      10. mul-1-neg 70.6%

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

      11. unsub-neg 70.6%

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

   4. Simplified 70.6%

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

   5. Taylor expanded in alpha around 0 59.4%

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

   if 1.6e154 < beta

   1. Initial program 82.5%

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

      1. associate-/l/ 77.0%

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

      2. associate-/r* 77.0%

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

      3. +-commutative 77.0%

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

      4. associate-+r+ 77.0%

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

      5. +-commutative 77.0%

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

      6. associate-+r+ 77.0%

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

      7. associate-+r+ 77.0%

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

      8. distribute-rgt1-in 77.0%

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

      9. +-commutative 77.0%

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

      10. *-commutative 77.0%

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

      11. distribute-rgt1-in 77.0%

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

      12. +-commutative 77.0%

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

      13. times-frac 92.7%

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

   3. Simplified 92.7%

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

   4. Taylor expanded in beta around inf 92.7%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 92.7%

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

   6. Simplified 92.7%

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

   7. Taylor expanded in alpha around inf 92.7%

      \[\leadsto \color{blue}{\frac{\alpha}{{\beta}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 92.7%

         \[\leadsto \frac{\alpha}{\color{blue}{\beta \cdot \beta}} \]

   9. Simplified 92.7%

      \[\leadsto \color{blue}{\frac{\alpha}{\beta \cdot \beta}} \]

   10. Taylor expanded in alpha around 0 92.7%

       \[\leadsto \color{blue}{\frac{\alpha}{{\beta}^{2}}} \]
   11. Step-by-step derivation

       1. unpow2 92.7%

          \[\leadsto \frac{\alpha}{\color{blue}{\beta \cdot \beta}} \]

       2. associate-/r* 92.7%

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

   12. Simplified 92.7%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.3:\\ \;\;\;\;0.08333333333333333\\ \mathbf{elif}\;\beta \leq 1.6 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]

Alternative 9: 71.5% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.5:\\ \;\;\;\;0.08333333333333333\\ \mathbf{elif}\;\beta \leq 1.65 \cdot 10^{+154}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.5)
   0.08333333333333333
   (if (<= beta 1.65e+154)
     (/ (+ 1.0 alpha) (* beta beta))
     (/ (/ alpha beta) beta))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.5) {
		tmp = 0.08333333333333333;
	} else if (beta <= 1.65e+154) {
		tmp = (1.0 + alpha) / (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 (beta <= 3.5d0) then
        tmp = 0.08333333333333333d0
    else if (beta <= 1.65d+154) then
        tmp = (1.0d0 + alpha) / (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 (beta <= 3.5) {
		tmp = 0.08333333333333333;
	} else if (beta <= 1.65e+154) {
		tmp = (1.0 + alpha) / (beta * beta);
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 3.5:
		tmp = 0.08333333333333333
	elif beta <= 1.65e+154:
		tmp = (1.0 + alpha) / (beta * beta)
	else:
		tmp = (alpha / beta) / beta
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.5)
		tmp = 0.08333333333333333;
	elseif (beta <= 1.65e+154)
		tmp = Float64(Float64(1.0 + alpha) / Float64(beta * beta));
	else
		tmp = Float64(Float64(alpha / beta) / beta);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.5)
		tmp = 0.08333333333333333;
	elseif (beta <= 1.65e+154)
		tmp = (1.0 + alpha) / (beta * beta);
	else
		tmp = (alpha / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 3.5], 0.08333333333333333, If[LessEqual[beta, 1.65e+154], N[(N[(1.0 + alpha), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / beta), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if beta < 3.5

   1. Initial program 99.8%

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

      1. associate-/l/ 99.7%

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

      2. associate-/r* 93.6%

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

      3. +-commutative 93.6%

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

      4. associate-+r+ 93.6%

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

      5. +-commutative 93.6%

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

      6. associate-+r+ 93.6%

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

      7. associate-+r+ 93.6%

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

      8. distribute-rgt1-in 93.6%

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

      9. +-commutative 93.6%

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

      10. *-commutative 93.6%

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

      11. distribute-rgt1-in 93.6%

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

      12. +-commutative 93.6%

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

      13. times-frac 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in alpha around 0 67.3%

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

      1. associate-*r/ 67.3%

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

      2. +-commutative 67.3%

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

      3. associate-+r+ 67.3%

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

      4. +-commutative 67.3%

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

      5. +-commutative 67.3%

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

   6. Applied egg-rr 67.3%

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

   7. Taylor expanded in beta around 0 66.3%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot \frac{1 + \alpha}{2 + \alpha}} \]
   8. Step-by-step derivation

      1. *-commutative 66.3%

         \[\leadsto \color{blue}{\frac{1 + \alpha}{2 + \alpha} \cdot 0.16666666666666666} \]

      2. +-commutative 66.3%

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\alpha + 2}} \cdot 0.16666666666666666 \]

   9. Simplified 66.3%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\alpha + 2} \cdot 0.16666666666666666} \]

   10. Taylor expanded in alpha around 0 66.4%

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

   if 3.5 < beta < 1.65e154

   1. Initial program 88.5%

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

      1. associate-/l/ 82.5%

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

      2. associate-/r* 49.6%

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

      3. +-commutative 49.6%

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

      4. associate-+r+ 49.6%

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

      5. +-commutative 49.6%

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

      6. associate-+r+ 49.6%

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

      7. associate-+r+ 49.6%

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

      8. distribute-rgt1-in 49.6%

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

      9. +-commutative 49.6%

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

      10. *-commutative 49.6%

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

      11. distribute-rgt1-in 49.6%

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

      12. +-commutative 49.6%

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

      13. times-frac 93.5%

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

   3. Simplified 93.5%

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

   4. Taylor expanded in beta around inf 70.4%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 70.4%

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

   6. Simplified 70.4%

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

   if 1.65e154 < beta

   1. Initial program 82.5%

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

      1. associate-/l/ 77.0%

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

      2. associate-/r* 77.0%

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

      3. +-commutative 77.0%

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

      4. associate-+r+ 77.0%

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

      5. +-commutative 77.0%

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

      6. associate-+r+ 77.0%

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

      7. associate-+r+ 77.0%

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

      8. distribute-rgt1-in 77.0%

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

      9. +-commutative 77.0%

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

      10. *-commutative 77.0%

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

      11. distribute-rgt1-in 77.0%

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

      12. +-commutative 77.0%

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

      13. times-frac 92.7%

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

   3. Simplified 92.7%

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

   4. Taylor expanded in beta around inf 92.7%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 92.7%

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

   6. Simplified 92.7%

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

   7. Taylor expanded in alpha around inf 92.7%

      \[\leadsto \color{blue}{\frac{\alpha}{{\beta}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 92.7%

         \[\leadsto \frac{\alpha}{\color{blue}{\beta \cdot \beta}} \]

   9. Simplified 92.7%

      \[\leadsto \color{blue}{\frac{\alpha}{\beta \cdot \beta}} \]

   10. Taylor expanded in alpha around 0 92.7%

       \[\leadsto \color{blue}{\frac{\alpha}{{\beta}^{2}}} \]
   11. Step-by-step derivation

       1. unpow2 92.7%

          \[\leadsto \frac{\alpha}{\color{blue}{\beta \cdot \beta}} \]

       2. associate-/r* 92.7%

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

   12. Simplified 92.7%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.5:\\ \;\;\;\;0.08333333333333333\\ \mathbf{elif}\;\beta \leq 1.65 \cdot 10^{+154}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]

Alternative 10: 72.2% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5:\\ \;\;\;\;\frac{0.16666666666666666 + \beta \cdot 0.027777777777777776}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5.0)
   (/ (+ 0.16666666666666666 (* beta 0.027777777777777776)) (+ beta 2.0))
   (/ (/ (- alpha -1.0) beta) beta)))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.0) {
		tmp = (0.16666666666666666 + (beta * 0.027777777777777776)) / (beta + 2.0);
	} else {
		tmp = ((alpha - -1.0) / beta) / beta;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 5.0d0) then
        tmp = (0.16666666666666666d0 + (beta * 0.027777777777777776d0)) / (beta + 2.0d0)
    else
        tmp = ((alpha - (-1.0d0)) / beta) / beta
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.0) {
		tmp = (0.16666666666666666 + (beta * 0.027777777777777776)) / (beta + 2.0);
	} else {
		tmp = ((alpha - -1.0) / beta) / beta;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 5.0:
		tmp = (0.16666666666666666 + (beta * 0.027777777777777776)) / (beta + 2.0)
	else:
		tmp = ((alpha - -1.0) / beta) / beta
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 5.0)
		tmp = Float64(Float64(0.16666666666666666 + Float64(beta * 0.027777777777777776)) / Float64(beta + 2.0));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / beta) / beta);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 5.0)
		tmp = (0.16666666666666666 + (beta * 0.027777777777777776)) / (beta + 2.0);
	else
		tmp = ((alpha - -1.0) / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 5.0], N[(N[(0.16666666666666666 + N[(beta * 0.027777777777777776), $MachinePrecision]), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5

   1. Initial program 99.8%

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

      1. associate-/l/ 99.7%

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

      2. associate-/r* 93.6%

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

      3. +-commutative 93.6%

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

      4. associate-+r+ 93.6%

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

      5. +-commutative 93.6%

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

      6. associate-+r+ 93.6%

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

      7. associate-+r+ 93.6%

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

      8. distribute-rgt1-in 93.6%

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

      9. +-commutative 93.6%

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

      10. *-commutative 93.6%

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

      11. distribute-rgt1-in 93.6%

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

      12. +-commutative 93.6%

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

      13. times-frac 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in alpha around 0 67.3%

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

   5. Taylor expanded in beta around 0 66.9%

      \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \color{blue}{\left(0.16666666666666666 + 0.027777777777777776 \cdot \beta\right)} \]
   6. Step-by-step derivation

      1. *-commutative 66.9%

         \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \left(0.16666666666666666 + \color{blue}{\beta \cdot 0.027777777777777776}\right) \]

   7. Simplified 66.9%

      \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \color{blue}{\left(0.16666666666666666 + \beta \cdot 0.027777777777777776\right)} \]

   8. Taylor expanded in alpha around 0 67.0%

      \[\leadsto \color{blue}{\frac{0.16666666666666666 + 0.027777777777777776 \cdot \beta}{2 + \beta}} \]
   9. Step-by-step derivation

      1. *-commutative 67.0%

         \[\leadsto \frac{0.16666666666666666 + \color{blue}{\beta \cdot 0.027777777777777776}}{2 + \beta} \]

   10. Simplified 67.0%

       \[\leadsto \color{blue}{\frac{0.16666666666666666 + \beta \cdot 0.027777777777777776}{2 + \beta}} \]

   if 5 < beta

   1. Initial program 85.5%

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

   2. Taylor expanded in beta around -inf 83.2%

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

      1. associate-*r/ 83.2%

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

      2. mul-1-neg 83.2%

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

      3. sub-neg 83.2%

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

      4. mul-1-neg 83.2%

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

      5. distribute-neg-in 83.2%

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

      6. +-commutative 83.2%

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

      7. mul-1-neg 83.2%

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

      8. distribute-lft-in 83.2%

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

      9. metadata-eval 83.2%

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

      10. mul-1-neg 83.2%

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

      11. unsub-neg 83.2%

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

   4. Simplified 83.2%

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

   5. Taylor expanded in beta around inf 83.0%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5:\\ \;\;\;\;\frac{0.16666666666666666 + \beta \cdot 0.027777777777777776}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{\beta}\\ \end{array} \]

Alternative 11: 70.3% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.45:\\ \;\;\;\;0.08333333333333333\\ \mathbf{elif}\;\beta \leq 10^{+154}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.45)
   0.08333333333333333
   (if (<= beta 1e+154) (/ 1.0 (* beta beta)) (/ (/ alpha beta) beta))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.45) {
		tmp = 0.08333333333333333;
	} else if (beta <= 1e+154) {
		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 (beta <= 3.45d0) then
        tmp = 0.08333333333333333d0
    else if (beta <= 1d+154) 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 (beta <= 3.45) {
		tmp = 0.08333333333333333;
	} else if (beta <= 1e+154) {
		tmp = 1.0 / (beta * beta);
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 3.45:
		tmp = 0.08333333333333333
	elif beta <= 1e+154:
		tmp = 1.0 / (beta * beta)
	else:
		tmp = (alpha / beta) / beta
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.45)
		tmp = 0.08333333333333333;
	elseif (beta <= 1e+154)
		tmp = Float64(1.0 / Float64(beta * beta));
	else
		tmp = Float64(Float64(alpha / beta) / beta);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.45)
		tmp = 0.08333333333333333;
	elseif (beta <= 1e+154)
		tmp = 1.0 / (beta * beta);
	else
		tmp = (alpha / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 3.45], 0.08333333333333333, If[LessEqual[beta, 1e+154], N[(1.0 / N[(beta * beta), $MachinePrecision]), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / beta), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if beta < 3.4500000000000002

   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/ 99.7%

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

      2. associate-/r* 93.6%

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

      3. +-commutative 93.6%

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

      4. associate-+r+ 93.6%

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

      5. +-commutative 93.6%

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

      6. associate-+r+ 93.6%

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

      7. associate-+r+ 93.6%

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

      8. distribute-rgt1-in 93.6%

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

      9. +-commutative 93.6%

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

      10. *-commutative 93.6%

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

      11. distribute-rgt1-in 93.6%

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

      12. +-commutative 93.6%

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

      13. times-frac 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in alpha around 0 67.3%

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

      1. associate-*r/ 67.3%

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

      2. +-commutative 67.3%

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

      3. associate-+r+ 67.3%

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

      4. +-commutative 67.3%

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

      5. +-commutative 67.3%

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

   6. Applied egg-rr 67.3%

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

   7. Taylor expanded in beta around 0 66.3%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot \frac{1 + \alpha}{2 + \alpha}} \]
   8. Step-by-step derivation

      1. *-commutative 66.3%

         \[\leadsto \color{blue}{\frac{1 + \alpha}{2 + \alpha} \cdot 0.16666666666666666} \]

      2. +-commutative 66.3%

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\alpha + 2}} \cdot 0.16666666666666666 \]

   9. Simplified 66.3%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\alpha + 2} \cdot 0.16666666666666666} \]

   10. Taylor expanded in alpha around 0 66.4%

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

   if 3.4500000000000002 < beta < 1.00000000000000004e154

   1. Initial program 88.5%

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

      1. associate-/l/ 82.5%

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

      2. associate-/r* 49.6%

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

      3. +-commutative 49.6%

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

      4. associate-+r+ 49.6%

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

      5. +-commutative 49.6%

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

      6. associate-+r+ 49.6%

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

      7. associate-+r+ 49.6%

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

      8. distribute-rgt1-in 49.6%

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

      9. +-commutative 49.6%

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

      10. *-commutative 49.6%

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

      11. distribute-rgt1-in 49.6%

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

      12. +-commutative 49.6%

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

      13. times-frac 93.5%

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

   3. Simplified 93.5%

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

   4. Taylor expanded in beta around inf 70.4%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 70.4%

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

   6. Simplified 70.4%

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

   7. Taylor expanded in alpha around 0 59.4%

      \[\leadsto \color{blue}{\frac{1}{{\beta}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 59.4%

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

   9. Simplified 59.4%

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

   if 1.00000000000000004e154 < beta

   1. Initial program 82.5%

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

      1. associate-/l/ 77.0%

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

      2. associate-/r* 77.0%

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

      3. +-commutative 77.0%

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

      4. associate-+r+ 77.0%

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

      5. +-commutative 77.0%

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

      6. associate-+r+ 77.0%

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

      7. associate-+r+ 77.0%

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

      8. distribute-rgt1-in 77.0%

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

      9. +-commutative 77.0%

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

      10. *-commutative 77.0%

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

      11. distribute-rgt1-in 77.0%

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

      12. +-commutative 77.0%

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

      13. times-frac 92.7%

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

   3. Simplified 92.7%

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

   4. Taylor expanded in beta around inf 92.7%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 92.7%

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

   6. Simplified 92.7%

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

   7. Taylor expanded in alpha around inf 92.7%

      \[\leadsto \color{blue}{\frac{\alpha}{{\beta}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 92.7%

         \[\leadsto \frac{\alpha}{\color{blue}{\beta \cdot \beta}} \]

   9. Simplified 92.7%

      \[\leadsto \color{blue}{\frac{\alpha}{\beta \cdot \beta}} \]

   10. Taylor expanded in alpha around 0 92.7%

       \[\leadsto \color{blue}{\frac{\alpha}{{\beta}^{2}}} \]
   11. Step-by-step derivation

       1. unpow2 92.7%

          \[\leadsto \frac{\alpha}{\color{blue}{\beta \cdot \beta}} \]

       2. associate-/r* 92.7%

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

   12. Simplified 92.7%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.45:\\ \;\;\;\;0.08333333333333333\\ \mathbf{elif}\;\beta \leq 10^{+154}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]

Alternative 12: 71.7% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.6:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.6) 0.08333333333333333 (/ (/ (- alpha -1.0) beta) beta)))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.6) {
		tmp = 0.08333333333333333;
	} else {
		tmp = ((alpha - -1.0) / beta) / beta;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 3.6d0) then
        tmp = 0.08333333333333333d0
    else
        tmp = ((alpha - (-1.0d0)) / beta) / beta
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.6) {
		tmp = 0.08333333333333333;
	} else {
		tmp = ((alpha - -1.0) / beta) / beta;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 3.6:
		tmp = 0.08333333333333333
	else:
		tmp = ((alpha - -1.0) / beta) / beta
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.6)
		tmp = 0.08333333333333333;
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / beta) / beta);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.6)
		tmp = 0.08333333333333333;
	else
		tmp = ((alpha - -1.0) / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 3.6], 0.08333333333333333, N[(N[(N[(alpha - -1.0), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.60000000000000009

   1. Initial program 99.8%

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

      1. associate-/l/ 99.7%

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

      2. associate-/r* 93.6%

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

      3. +-commutative 93.6%

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

      4. associate-+r+ 93.6%

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

      5. +-commutative 93.6%

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

      6. associate-+r+ 93.6%

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

      7. associate-+r+ 93.6%

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

      8. distribute-rgt1-in 93.6%

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

      9. +-commutative 93.6%

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

      10. *-commutative 93.6%

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

      11. distribute-rgt1-in 93.6%

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

      12. +-commutative 93.6%

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

      13. times-frac 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in alpha around 0 67.3%

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

      1. associate-*r/ 67.3%

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

      2. +-commutative 67.3%

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

      3. associate-+r+ 67.3%

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

      4. +-commutative 67.3%

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

      5. +-commutative 67.3%

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

   6. Applied egg-rr 67.3%

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

   7. Taylor expanded in beta around 0 66.3%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot \frac{1 + \alpha}{2 + \alpha}} \]
   8. Step-by-step derivation

      1. *-commutative 66.3%

         \[\leadsto \color{blue}{\frac{1 + \alpha}{2 + \alpha} \cdot 0.16666666666666666} \]

      2. +-commutative 66.3%

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\alpha + 2}} \cdot 0.16666666666666666 \]

   9. Simplified 66.3%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\alpha + 2} \cdot 0.16666666666666666} \]

   10. Taylor expanded in alpha around 0 66.4%

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

   if 3.60000000000000009 < beta

   1. Initial program 85.5%

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

   2. Taylor expanded in beta around -inf 83.2%

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

      1. associate-*r/ 83.2%

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

      2. mul-1-neg 83.2%

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

      3. sub-neg 83.2%

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

      4. mul-1-neg 83.2%

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

      5. distribute-neg-in 83.2%

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

      6. +-commutative 83.2%

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

      7. mul-1-neg 83.2%

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

      8. distribute-lft-in 83.2%

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

      9. metadata-eval 83.2%

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

      10. mul-1-neg 83.2%

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

      11. unsub-neg 83.2%

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

   4. Simplified 83.2%

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

   5. Taylor expanded in beta around inf 83.0%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.6:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{\beta}\\ \end{array} \]

Alternative 13: 69.6% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.45:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.45) 0.08333333333333333 (/ 1.0 (* beta beta))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.45) {
		tmp = 0.08333333333333333;
	} else {
		tmp = 1.0 / (beta * beta);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 3.45d0) then
        tmp = 0.08333333333333333d0
    else
        tmp = 1.0d0 / (beta * beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.45) {
		tmp = 0.08333333333333333;
	} else {
		tmp = 1.0 / (beta * beta);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 3.45:
		tmp = 0.08333333333333333
	else:
		tmp = 1.0 / (beta * beta)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.45)
		tmp = 0.08333333333333333;
	else
		tmp = Float64(1.0 / Float64(beta * beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.45)
		tmp = 0.08333333333333333;
	else
		tmp = 1.0 / (beta * beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 3.45], 0.08333333333333333, N[(1.0 / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.4500000000000002

   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/ 99.7%

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

      2. associate-/r* 93.6%

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

      3. +-commutative 93.6%

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

      4. associate-+r+ 93.6%

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

      5. +-commutative 93.6%

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

      6. associate-+r+ 93.6%

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

      7. associate-+r+ 93.6%

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

      8. distribute-rgt1-in 93.6%

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

      9. +-commutative 93.6%

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

      10. *-commutative 93.6%

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

      11. distribute-rgt1-in 93.6%

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

      12. +-commutative 93.6%

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

      13. times-frac 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in alpha around 0 67.3%

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

      1. associate-*r/ 67.3%

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

      2. +-commutative 67.3%

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

      3. associate-+r+ 67.3%

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

      4. +-commutative 67.3%

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

      5. +-commutative 67.3%

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

   6. Applied egg-rr 67.3%

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

   7. Taylor expanded in beta around 0 66.3%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot \frac{1 + \alpha}{2 + \alpha}} \]
   8. Step-by-step derivation

      1. *-commutative 66.3%

         \[\leadsto \color{blue}{\frac{1 + \alpha}{2 + \alpha} \cdot 0.16666666666666666} \]

      2. +-commutative 66.3%

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\alpha + 2}} \cdot 0.16666666666666666 \]

   9. Simplified 66.3%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\alpha + 2} \cdot 0.16666666666666666} \]

   10. Taylor expanded in alpha around 0 66.4%

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

   if 3.4500000000000002 < beta

   1. Initial program 85.5%

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

      1. associate-/l/ 79.8%

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

      2. associate-/r* 63.3%

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

      3. +-commutative 63.3%

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

      4. associate-+r+ 63.3%

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

      5. +-commutative 63.3%

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

      6. associate-+r+ 63.3%

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

      7. associate-+r+ 63.3%

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

      8. distribute-rgt1-in 63.3%

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

      9. +-commutative 63.3%

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

      10. *-commutative 63.3%

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

      11. distribute-rgt1-in 63.3%

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

      12. +-commutative 63.3%

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

      13. times-frac 93.1%

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

   3. Simplified 93.1%

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

   4. Taylor expanded in beta around inf 81.6%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 81.6%

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

   6. Simplified 81.6%

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

   7. Taylor expanded in alpha around 0 76.1%

      \[\leadsto \color{blue}{\frac{1}{{\beta}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 76.1%

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

   9. Simplified 76.1%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.45:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \]

Alternative 14: 45.5% accurate, 35.0× speedup

Math

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

FPCore

(FPCore (alpha beta) :precision binary64 0.08333333333333333)

C

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

Fortran

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

Java

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

Python

def code(alpha, beta):
	return 0.08333333333333333

Julia

function code(alpha, beta)
	return 0.08333333333333333
end

MATLAB

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

Wolfram

code[alpha_, beta_] := 0.08333333333333333

Derivation

1. Initial program 94.8%

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

   1. associate-/l/ 92.7%

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

   2. associate-/r* 83.0%

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

   3. +-commutative 83.0%

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

   4. associate-+r+ 83.0%

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

   5. +-commutative 83.0%

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

   6. associate-+r+ 83.0%

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

   7. associate-+r+ 83.0%

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

   8. distribute-rgt1-in 83.0%

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

   9. +-commutative 83.0%

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

   10. *-commutative 83.0%

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

   11. distribute-rgt1-in 83.0%

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

   12. +-commutative 83.0%

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

   13. times-frac 97.4%

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

3. Simplified 97.4%

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

4. Taylor expanded in alpha around 0 72.6%

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

   1. associate-*r/ 72.6%

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

   2. +-commutative 72.6%

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

   3. associate-+r+ 72.6%

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

   4. +-commutative 72.6%

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

   5. +-commutative 72.6%

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

6. Applied egg-rr 72.6%

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

7. Taylor expanded in beta around 0 44.3%

   \[\leadsto \color{blue}{0.16666666666666666 \cdot \frac{1 + \alpha}{2 + \alpha}} \]
8. Step-by-step derivation

   1. *-commutative 44.3%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{2 + \alpha} \cdot 0.16666666666666666} \]

   2. +-commutative 44.3%

      \[\leadsto \frac{1 + \alpha}{\color{blue}{\alpha + 2}} \cdot 0.16666666666666666 \]

9. Simplified 44.3%

   \[\leadsto \color{blue}{\frac{1 + \alpha}{\alpha + 2} \cdot 0.16666666666666666} \]

10. Taylor expanded in alpha around 0 44.3%

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

11. Final simplification 44.3%

    \[\leadsto 0.08333333333333333 \]

Reproduce

herbie shell --seed 2023285 
(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)))