Octave 3.8, jcobi/3

Percentage Accurate: 94.2% → 99.8%

Time: 22.4s

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: 94.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.4%

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

3. Simplified 85.3%

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

   1. times-frac 97.3%

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

   2. +-commutative 97.3%

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

6. Applied egg-rr 97.3%

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

   1. add-log-exp 70.6%

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

   2. exp-prod 70.6%

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

   3. +-commutative 70.6%

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

   4. +-commutative 70.6%

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

   5. associate-+l+ 70.6%

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

   6. +-commutative 70.6%

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

   7. +-commutative 70.6%

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

   8. associate-+l+ 70.6%

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

   9. +-commutative 70.6%

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

8. Applied egg-rr 70.6%

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

   1. log-pow 84.8%

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

   2. rem-log-exp 97.3%

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

   3. associate-/r* 99.7%

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

   4. +-commutative 99.7%

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

   5. +-commutative 99.7%

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

   6. +-commutative 99.7%

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

   7. +-commutative 99.7%

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

   8. +-commutative 99.7%

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

10. Simplified 99.7%

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

    1. associate-*l/ 99.8%

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

    2. +-commutative 99.8%

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

    3. +-commutative 99.8%

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

    4. +-commutative 99.8%

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

    5. +-commutative 99.8%

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

    6. +-commutative 99.8%

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

    7. +-commutative 99.8%

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

    8. +-commutative 99.8%

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

12. Applied egg-rr 99.8%

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

13. Final simplification 99.8%

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

Alternative 2: 92.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 3\right)\\ t_1 := \beta + \left(2 + \alpha\right)\\ \mathbf{if}\;\beta \leq 2 \cdot 10^{+92}:\\ \;\;\;\;\left(1 + \alpha\right) \cdot \frac{\frac{\frac{1 + \beta}{t\_1}}{t\_0}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{t\_1} \cdot \left(1 + \frac{-1 - \alpha}{\beta}\right)}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 3.0))) (t_1 (+ beta (+ 2.0 alpha))))
   (if (<= beta 2e+92)
     (* (+ 1.0 alpha) (/ (/ (/ (+ 1.0 beta) t_1) t_0) t_1))
     (/ (* (/ (+ 1.0 alpha) t_1) (+ 1.0 (/ (- -1.0 alpha) beta))) t_0))))

C

double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 3.0);
	double t_1 = beta + (2.0 + alpha);
	double tmp;
	if (beta <= 2e+92) {
		tmp = (1.0 + alpha) * ((((1.0 + beta) / t_1) / t_0) / t_1);
	} else {
		tmp = (((1.0 + alpha) / t_1) * (1.0 + ((-1.0 - alpha) / beta))) / t_0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = alpha + (beta + 3.0d0)
    t_1 = beta + (2.0d0 + alpha)
    if (beta <= 2d+92) then
        tmp = (1.0d0 + alpha) * ((((1.0d0 + beta) / t_1) / t_0) / t_1)
    else
        tmp = (((1.0d0 + alpha) / t_1) * (1.0d0 + (((-1.0d0) - alpha) / beta))) / t_0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 3.0);
	double t_1 = beta + (2.0 + alpha);
	double tmp;
	if (beta <= 2e+92) {
		tmp = (1.0 + alpha) * ((((1.0 + beta) / t_1) / t_0) / t_1);
	} else {
		tmp = (((1.0 + alpha) / t_1) * (1.0 + ((-1.0 - alpha) / beta))) / t_0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = alpha + (beta + 3.0)
	t_1 = beta + (2.0 + alpha)
	tmp = 0
	if beta <= 2e+92:
		tmp = (1.0 + alpha) * ((((1.0 + beta) / t_1) / t_0) / t_1)
	else:
		tmp = (((1.0 + alpha) / t_1) * (1.0 + ((-1.0 - alpha) / beta))) / t_0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 3.0))
	t_1 = Float64(beta + Float64(2.0 + alpha))
	tmp = 0.0
	if (beta <= 2e+92)
		tmp = Float64(Float64(1.0 + alpha) * Float64(Float64(Float64(Float64(1.0 + beta) / t_1) / t_0) / t_1));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + alpha) / t_1) * Float64(1.0 + Float64(Float64(-1.0 - alpha) / beta))) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 3.0);
	t_1 = beta + (2.0 + alpha);
	tmp = 0.0;
	if (beta <= 2e+92)
		tmp = (1.0 + alpha) * ((((1.0 + beta) / t_1) / t_0) / t_1);
	else
		tmp = (((1.0 + alpha) / t_1) * (1.0 + ((-1.0 - alpha) / beta))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(beta + N[(2.0 + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2e+92], N[(N[(1.0 + alpha), $MachinePrecision] * N[(N[(N[(N[(1.0 + beta), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$1), $MachinePrecision] * N[(1.0 + N[(N[(-1.0 - alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.0000000000000001e92

   1. Initial program 99.8%

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

   3. Simplified 94.8%

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

      1. times-frac 99.0%

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

      2. +-commutative 99.0%

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

   6. Applied egg-rr 99.0%

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

      1. associate-*l/ 99.1%

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

      2. +-commutative 99.1%

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

      3. +-commutative 99.1%

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

      4. associate-+l+ 99.1%

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

      5. +-commutative 99.1%

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

      6. +-commutative 99.1%

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

      7. associate-+l+ 99.1%

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

      8. +-commutative 99.1%

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

   8. Applied egg-rr 99.1%

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

      1. associate-/l* 95.3%

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

      2. associate-/r* 95.3%

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

      3. +-commutative 95.3%

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

      4. +-commutative 95.3%

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

      5. +-commutative 95.3%

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

      6. +-commutative 95.3%

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

      7. +-commutative 95.3%

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

   10. Simplified 95.3%

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

   if 2.0000000000000001e92 < beta

   1. Initial program 86.0%

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

   3. Simplified 56.8%

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

      1. times-frac 91.9%

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

      2. +-commutative 91.9%

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

   6. Applied egg-rr 91.9%

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

      1. add-log-exp 65.0%

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

      2. exp-prod 65.0%

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

      3. +-commutative 65.0%

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

      4. +-commutative 65.0%

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

      5. associate-+l+ 65.0%

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

      6. +-commutative 65.0%

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

      7. +-commutative 65.0%

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

      8. associate-+l+ 65.0%

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

      9. +-commutative 65.0%

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

   8. Applied egg-rr 65.0%

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

      1. log-pow 65.0%

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

      2. rem-log-exp 91.9%

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

      3. associate-/r* 99.7%

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

      4. +-commutative 99.7%

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

      5. +-commutative 99.7%

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

      6. +-commutative 99.7%

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

      7. +-commutative 99.7%

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

      8. +-commutative 99.7%

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

   10. Simplified 99.7%

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

       1. associate-*l/ 99.8%

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

       2. +-commutative 99.8%

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

       3. +-commutative 99.8%

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

       4. +-commutative 99.8%

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

       5. +-commutative 99.8%

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

       6. +-commutative 99.8%

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

       7. +-commutative 99.8%

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

       8. +-commutative 99.8%

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

   12. Applied egg-rr 99.8%

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

   13. Taylor expanded in beta around inf 93.8%

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

       1. mul-1-neg 93.8%

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

       2. unsub-neg 93.8%

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

   15. Simplified 93.8%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2 \cdot 10^{+92}:\\ \;\;\;\;\left(1 + \alpha\right) \cdot \frac{\frac{\frac{1 + \beta}{\beta + \left(2 + \alpha\right)}}{\alpha + \left(\beta + 3\right)}}{\beta + \left(2 + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta + \left(2 + \alpha\right)} \cdot \left(1 + \frac{-1 - \alpha}{\beta}\right)}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 70.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.85 \cdot 10^{-20}:\\ \;\;\;\;\frac{1 + \beta}{\beta + 2} \cdot \frac{\frac{1}{\beta + 2}}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta + \left(2 + \alpha\right)} \cdot \frac{1 + \frac{-1 - \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 1.85e-20)
   (* (/ (+ 1.0 beta) (+ beta 2.0)) (/ (/ 1.0 (+ beta 2.0)) (+ beta 3.0)))
   (*
    (/ (+ 1.0 alpha) (+ beta (+ 2.0 alpha)))
    (/ (+ 1.0 (/ (- -1.0 alpha) beta)) (+ alpha (+ beta 3.0))))))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1.85e-20) {
		tmp = ((1.0 + beta) / (beta + 2.0)) * ((1.0 / (beta + 2.0)) / (beta + 3.0));
	} else {
		tmp = ((1.0 + alpha) / (beta + (2.0 + alpha))) * ((1.0 + ((-1.0 - alpha) / beta)) / (alpha + (beta + 3.0)));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 1.85d-20) then
        tmp = ((1.0d0 + beta) / (beta + 2.0d0)) * ((1.0d0 / (beta + 2.0d0)) / (beta + 3.0d0))
    else
        tmp = ((1.0d0 + alpha) / (beta + (2.0d0 + alpha))) * ((1.0d0 + (((-1.0d0) - alpha) / beta)) / (alpha + (beta + 3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1.85e-20) {
		tmp = ((1.0 + beta) / (beta + 2.0)) * ((1.0 / (beta + 2.0)) / (beta + 3.0));
	} else {
		tmp = ((1.0 + alpha) / (beta + (2.0 + alpha))) * ((1.0 + ((-1.0 - alpha) / beta)) / (alpha + (beta + 3.0)));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 1.85e-20:
		tmp = ((1.0 + beta) / (beta + 2.0)) * ((1.0 / (beta + 2.0)) / (beta + 3.0))
	else:
		tmp = ((1.0 + alpha) / (beta + (2.0 + alpha))) * ((1.0 + ((-1.0 - alpha) / beta)) / (alpha + (beta + 3.0)))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 1.85e-20)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(beta + 2.0)) * Float64(Float64(1.0 / Float64(beta + 2.0)) / Float64(beta + 3.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(beta + Float64(2.0 + alpha))) * Float64(Float64(1.0 + Float64(Float64(-1.0 - alpha) / beta)) / Float64(alpha + Float64(beta + 3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 1.85e-20)
		tmp = ((1.0 + beta) / (beta + 2.0)) * ((1.0 / (beta + 2.0)) / (beta + 3.0));
	else
		tmp = ((1.0 + alpha) / (beta + (2.0 + alpha))) * ((1.0 + ((-1.0 - alpha) / beta)) / (alpha + (beta + 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 1.85e-20], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(beta + N[(2.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(N[(-1.0 - alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.85e-20

   1. Initial program 99.9%

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

      1. associate-/l/ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+l+ 99.8%

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

      4. *-commutative 99.8%

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

      5. metadata-eval 99.8%

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

      6. associate-+l+ 99.8%

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

      7. metadata-eval 99.8%

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

      8. associate-+l+ 99.8%

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

      9. metadata-eval 99.8%

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

      10. metadata-eval 99.8%

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

      11. associate-+l+ 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in alpha around 0 99.0%

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

      1. +-commutative 99.0%

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

   7. Simplified 99.0%

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

      1. div-inv 99.0%

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

      2. *-commutative 99.0%

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

      3. associate-+r+ 99.0%

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

      4. +-commutative 99.0%

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

      5. associate-+l+ 99.0%

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

      6. +-commutative 99.0%

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

   9. Applied egg-rr 99.0%

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

   10. Taylor expanded in alpha around 0 99.0%

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

       1. associate-/r* 99.1%

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

   12. Simplified 99.1%

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

   if 1.85e-20 < alpha

   1. Initial program 90.9%

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

   3. Simplified 77.3%

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

      1. times-frac 93.4%

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

      2. +-commutative 93.4%

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

   6. Applied egg-rr 93.4%

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

      1. add-log-exp 58.8%

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

      2. exp-prod 58.8%

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

      3. +-commutative 58.8%

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

      4. +-commutative 58.8%

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

      5. associate-+l+ 58.8%

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

      6. +-commutative 58.8%

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

      7. +-commutative 58.8%

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

      8. associate-+l+ 58.8%

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

      9. +-commutative 58.8%

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

   8. Applied egg-rr 58.8%

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

      1. log-pow 90.4%

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

      2. rem-log-exp 93.4%

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

      3. associate-/r* 99.6%

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

      4. +-commutative 99.6%

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

      5. +-commutative 99.6%

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

      6. +-commutative 99.6%

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

      7. +-commutative 99.6%

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

      8. +-commutative 99.6%

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

   10. Simplified 99.6%

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

   11. Taylor expanded in beta around inf 22.3%

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

       1. mul-1-neg 22.3%

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

       2. unsub-neg 22.3%

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

   13. Simplified 22.3%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.85 \cdot 10^{-20}:\\ \;\;\;\;\frac{1 + \beta}{\beta + 2} \cdot \frac{\frac{1}{\beta + 2}}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta + \left(2 + \alpha\right)} \cdot \frac{1 + \frac{-1 - \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 14200000000:\\ \;\;\;\;\frac{1 + \beta}{\beta + 2} \cdot \frac{\frac{1}{\beta + 2}}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta + \left(2 + \alpha\right)} \cdot \left(1 + \frac{-1 - \alpha}{\beta}\right)}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 14200000000.0)
   (* (/ (+ 1.0 beta) (+ beta 2.0)) (/ (/ 1.0 (+ beta 2.0)) (+ beta 3.0)))
   (/
    (*
     (/ (+ 1.0 alpha) (+ beta (+ 2.0 alpha)))
     (+ 1.0 (/ (- -1.0 alpha) beta)))
    (+ alpha (+ beta 3.0)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 14200000000.0) {
		tmp = ((1.0 + beta) / (beta + 2.0)) * ((1.0 / (beta + 2.0)) / (beta + 3.0));
	} else {
		tmp = (((1.0 + alpha) / (beta + (2.0 + alpha))) * (1.0 + ((-1.0 - alpha) / beta))) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 14200000000.0d0) then
        tmp = ((1.0d0 + beta) / (beta + 2.0d0)) * ((1.0d0 / (beta + 2.0d0)) / (beta + 3.0d0))
    else
        tmp = (((1.0d0 + alpha) / (beta + (2.0d0 + alpha))) * (1.0d0 + (((-1.0d0) - alpha) / beta))) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 14200000000.0) {
		tmp = ((1.0 + beta) / (beta + 2.0)) * ((1.0 / (beta + 2.0)) / (beta + 3.0));
	} else {
		tmp = (((1.0 + alpha) / (beta + (2.0 + alpha))) * (1.0 + ((-1.0 - alpha) / beta))) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 14200000000.0:
		tmp = ((1.0 + beta) / (beta + 2.0)) * ((1.0 / (beta + 2.0)) / (beta + 3.0))
	else:
		tmp = (((1.0 + alpha) / (beta + (2.0 + alpha))) * (1.0 + ((-1.0 - alpha) / beta))) / (alpha + (beta + 3.0))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 14200000000.0)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(beta + 2.0)) * Float64(Float64(1.0 / Float64(beta + 2.0)) / Float64(beta + 3.0)));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + alpha) / Float64(beta + Float64(2.0 + alpha))) * Float64(1.0 + Float64(Float64(-1.0 - alpha) / beta))) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 14200000000.0)
		tmp = ((1.0 + beta) / (beta + 2.0)) * ((1.0 / (beta + 2.0)) / (beta + 3.0));
	else
		tmp = (((1.0 + alpha) / (beta + (2.0 + alpha))) * (1.0 + ((-1.0 - alpha) / beta))) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 14200000000.0], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(beta + N[(2.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(-1.0 - alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.42e10

   1. Initial program 99.9%

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

      1. associate-/l/ 99.0%

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

      2. +-commutative 99.0%

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

      3. associate-+l+ 99.0%

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

      4. *-commutative 99.0%

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

      5. metadata-eval 99.0%

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

      6. associate-+l+ 99.0%

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

      7. metadata-eval 99.0%

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

      8. associate-+l+ 99.1%

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

      9. metadata-eval 99.1%

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

      10. metadata-eval 99.1%

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

      11. associate-+l+ 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in alpha around 0 84.0%

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

      1. +-commutative 84.0%

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

   7. Simplified 84.0%

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

      1. div-inv 84.0%

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

      2. *-commutative 84.0%

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

      3. associate-+r+ 84.0%

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

      4. +-commutative 84.0%

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

      5. associate-+l+ 84.0%

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

      6. +-commutative 84.0%

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

   9. Applied egg-rr 84.0%

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

   10. Taylor expanded in alpha around 0 63.0%

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

       1. associate-/r* 63.0%

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

   12. Simplified 63.0%

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

   if 1.42e10 < beta

   1. Initial program 89.8%

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

   3. Simplified 65.6%

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

      1. times-frac 93.9%

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

      2. +-commutative 93.9%

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

   6. Applied egg-rr 93.9%

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

      1. add-log-exp 50.1%

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

      2. exp-prod 50.1%

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

      3. +-commutative 50.1%

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

      4. +-commutative 50.1%

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

      5. associate-+l+ 50.1%

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

      6. +-commutative 50.1%

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

      7. +-commutative 50.1%

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

      8. associate-+l+ 50.1%

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

      9. +-commutative 50.1%

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

   8. Applied egg-rr 50.1%

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

      1. log-pow 58.3%

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

      2. rem-log-exp 93.9%

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

      3. associate-/r* 99.5%

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

      4. +-commutative 99.5%

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

      5. +-commutative 99.5%

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

      6. +-commutative 99.5%

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

      7. +-commutative 99.5%

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

      8. +-commutative 99.5%

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

   10. Simplified 99.5%

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

       1. associate-*l/ 99.7%

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

       2. +-commutative 99.7%

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

       3. +-commutative 99.7%

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

       4. +-commutative 99.7%

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

       5. +-commutative 99.7%

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

       6. +-commutative 99.7%

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

       7. +-commutative 99.7%

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

       8. +-commutative 99.7%

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

   12. Applied egg-rr 99.7%

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

   13. Taylor expanded in beta around inf 85.4%

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

       1. mul-1-neg 85.4%

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

       2. unsub-neg 85.4%

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

   15. Simplified 85.4%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 14200000000:\\ \;\;\;\;\frac{1 + \beta}{\beta + 2} \cdot \frac{\frac{1}{\beta + 2}}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta + \left(2 + \alpha\right)} \cdot \left(1 + \frac{-1 - \alpha}{\beta}\right)}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.4%

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

3. Simplified 85.3%

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

   1. times-frac 97.3%

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

   2. +-commutative 97.3%

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

6. Applied egg-rr 97.3%

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

   1. add-log-exp 70.6%

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

   2. exp-prod 70.6%

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

   3. +-commutative 70.6%

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

   4. +-commutative 70.6%

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

   5. associate-+l+ 70.6%

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

   6. +-commutative 70.6%

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

   7. +-commutative 70.6%

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

   8. associate-+l+ 70.6%

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

   9. +-commutative 70.6%

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

8. Applied egg-rr 70.6%

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

   1. log-pow 84.8%

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

   2. rem-log-exp 97.3%

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

   3. associate-/r* 99.7%

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

   4. +-commutative 99.7%

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

   5. +-commutative 99.7%

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

   6. +-commutative 99.7%

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

   7. +-commutative 99.7%

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

   8. +-commutative 99.7%

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

10. Simplified 99.7%

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

11. Final simplification 99.7%

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

Alternative 6: 71.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.4 \cdot 10^{+15}:\\ \;\;\;\;\frac{1 + \beta}{\beta + 2} \cdot \frac{\frac{1}{\beta + 2}}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.4e+15)
   (* (/ (+ 1.0 beta) (+ beta 2.0)) (/ (/ 1.0 (+ beta 2.0)) (+ beta 3.0)))
   (/ (/ (+ 1.0 alpha) beta) (+ alpha (+ beta 3.0)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.4e+15) {
		tmp = ((1.0 + beta) / (beta + 2.0)) * ((1.0 / (beta + 2.0)) / (beta + 3.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.4d+15) then
        tmp = ((1.0d0 + beta) / (beta + 2.0d0)) * ((1.0d0 / (beta + 2.0d0)) / (beta + 3.0d0))
    else
        tmp = ((1.0d0 + alpha) / beta) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.4e+15) {
		tmp = ((1.0 + beta) / (beta + 2.0)) * ((1.0 / (beta + 2.0)) / (beta + 3.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.4e+15:
		tmp = ((1.0 + beta) / (beta + 2.0)) * ((1.0 / (beta + 2.0)) / (beta + 3.0))
	else:
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.4e+15)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(beta + 2.0)) * Float64(Float64(1.0 / Float64(beta + 2.0)) / Float64(beta + 3.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.4e+15)
		tmp = ((1.0 + beta) / (beta + 2.0)) * ((1.0 / (beta + 2.0)) / (beta + 3.0));
	else
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.4e+15], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.4e15

   1. Initial program 99.9%

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

      1. associate-/l/ 99.0%

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

      2. +-commutative 99.0%

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

      3. associate-+l+ 99.0%

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

      4. *-commutative 99.0%

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

      5. metadata-eval 99.0%

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

      6. associate-+l+ 99.0%

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

      7. metadata-eval 99.0%

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

      8. associate-+l+ 99.1%

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

      9. metadata-eval 99.1%

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

      10. metadata-eval 99.1%

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

      11. associate-+l+ 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in alpha around 0 84.0%

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

      1. +-commutative 84.0%

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

   7. Simplified 84.0%

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

      1. div-inv 84.0%

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

      2. *-commutative 84.0%

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

      3. associate-+r+ 84.0%

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

      4. +-commutative 84.0%

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

      5. associate-+l+ 84.0%

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

      6. +-commutative 84.0%

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

   9. Applied egg-rr 84.0%

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

   10. Taylor expanded in alpha around 0 63.0%

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

       1. associate-/r* 63.0%

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

   12. Simplified 63.0%

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

   if 2.4e15 < beta

   1. Initial program 89.8%

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

   3. Taylor expanded in beta around inf 85.2%

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

      1. div-inv 85.0%

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

      2. metadata-eval 85.0%

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

      3. associate-+l+ 85.0%

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

      4. metadata-eval 85.0%

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

      5. associate-+r+ 85.0%

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

   5. Applied egg-rr 85.0%

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

      1. associate-*r/ 85.2%

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

      2. *-commutative 85.2%

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

      3. *-lft-identity 85.2%

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

      4. +-commutative 85.2%

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

   7. Simplified 85.2%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.4 \cdot 10^{+15}:\\ \;\;\;\;\frac{1 + \beta}{\beta + 2} \cdot \frac{\frac{1}{\beta + 2}}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 8.2 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 2}}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 8.2e+15)
   (/ (/ (+ 1.0 beta) (+ beta 2.0)) (* (+ beta 3.0) (+ beta 2.0)))
   (/ (/ (+ 1.0 alpha) beta) (+ alpha (+ beta 3.0)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 8.2e+15) {
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 3.0) * (beta + 2.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 8.2d+15) then
        tmp = ((1.0d0 + beta) / (beta + 2.0d0)) / ((beta + 3.0d0) * (beta + 2.0d0))
    else
        tmp = ((1.0d0 + alpha) / beta) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 8.2e+15) {
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 3.0) * (beta + 2.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 8.2e+15:
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 3.0) * (beta + 2.0))
	else:
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 8.2e+15)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(beta + 2.0)) / Float64(Float64(beta + 3.0) * Float64(beta + 2.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 8.2e+15)
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 3.0) * (beta + 2.0));
	else
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 8.2e+15], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + 3.0), $MachinePrecision] * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 8.2e15

   1. Initial program 99.9%

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

      1. associate-/l/ 99.0%

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

      2. +-commutative 99.0%

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

      3. associate-+l+ 99.0%

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

      4. *-commutative 99.0%

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

      5. metadata-eval 99.0%

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

      6. associate-+l+ 99.0%

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

      7. metadata-eval 99.0%

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

      8. associate-+l+ 99.1%

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

      9. metadata-eval 99.1%

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

      10. metadata-eval 99.1%

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

      11. associate-+l+ 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in alpha around 0 84.0%

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

      1. +-commutative 84.0%

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

   7. Simplified 84.0%

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

   8. Taylor expanded in alpha around 0 63.0%

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

   if 8.2e15 < beta

   1. Initial program 89.8%

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

   3. Taylor expanded in beta around inf 85.2%

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

      1. div-inv 85.0%

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

      2. metadata-eval 85.0%

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

      3. associate-+l+ 85.0%

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

      4. metadata-eval 85.0%

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

      5. associate-+r+ 85.0%

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

   5. Applied egg-rr 85.0%

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

      1. associate-*r/ 85.2%

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

      2. *-commutative 85.2%

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

      3. *-lft-identity 85.2%

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

      4. +-commutative 85.2%

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

   7. Simplified 85.2%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 8.2 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 2}}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.2:\\ \;\;\;\;\frac{0.5 + \beta \cdot 0.25}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.2)
   (/ (+ 0.5 (* beta 0.25)) (* (+ beta 3.0) (+ beta 2.0)))
   (/ (/ (+ 1.0 alpha) beta) (+ alpha (+ beta 3.0)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.2) {
		tmp = (0.5 + (beta * 0.25)) / ((beta + 3.0) * (beta + 2.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 4.2d0) then
        tmp = (0.5d0 + (beta * 0.25d0)) / ((beta + 3.0d0) * (beta + 2.0d0))
    else
        tmp = ((1.0d0 + alpha) / beta) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.2) {
		tmp = (0.5 + (beta * 0.25)) / ((beta + 3.0) * (beta + 2.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 4.2:
		tmp = (0.5 + (beta * 0.25)) / ((beta + 3.0) * (beta + 2.0))
	else:
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.2)
		tmp = Float64(Float64(0.5 + Float64(beta * 0.25)) / Float64(Float64(beta + 3.0) * Float64(beta + 2.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4.2)
		tmp = (0.5 + (beta * 0.25)) / ((beta + 3.0) * (beta + 2.0));
	else
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 4.2], N[(N[(0.5 + N[(beta * 0.25), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + 3.0), $MachinePrecision] * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.20000000000000018

   1. Initial program 99.9%

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

      1. associate-/l/ 99.0%

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

      2. +-commutative 99.0%

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

      3. associate-+l+ 99.0%

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

      4. *-commutative 99.0%

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

      5. metadata-eval 99.0%

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

      6. associate-+l+ 99.0%

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

      7. metadata-eval 99.0%

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

      8. associate-+l+ 99.0%

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

      9. metadata-eval 99.0%

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

      10. metadata-eval 99.0%

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

      11. associate-+l+ 99.0%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in alpha around 0 85.2%

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

      1. +-commutative 85.2%

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

   7. Simplified 85.2%

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

   8. Taylor expanded in beta around 0 84.8%

      \[\leadsto \frac{\color{blue}{0.5 + 0.25 \cdot \beta}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative 84.8%

         \[\leadsto \frac{0.5 + \color{blue}{\beta \cdot 0.25}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]

   10. Simplified 84.8%

       \[\leadsto \frac{\color{blue}{0.5 + \beta \cdot 0.25}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]

   11. Taylor expanded in alpha around 0 63.5%

       \[\leadsto \frac{0.5 + \beta \cdot 0.25}{\color{blue}{\left(2 + \beta\right) \cdot \left(3 + \beta\right)}} \]

   if 4.20000000000000018 < beta

   1. Initial program 90.2%

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

   3. Taylor expanded in beta around inf 81.9%

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

      1. div-inv 81.8%

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

      2. metadata-eval 81.8%

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

      3. associate-+l+ 81.8%

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

      4. metadata-eval 81.8%

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

      5. associate-+r+ 81.8%

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

   5. Applied egg-rr 81.8%

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

      1. associate-*r/ 81.9%

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

      2. *-commutative 81.9%

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

      3. *-lft-identity 81.9%

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

      4. +-commutative 81.9%

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

   7. Simplified 81.9%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.2:\\ \;\;\;\;\frac{0.5 + \beta \cdot 0.25}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.5% accurate, 2.2× speedup

Math

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

FPCore

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

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.3) {
		tmp = 0.08333333333333333;
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.3d0) then
        tmp = 0.08333333333333333d0
    else
        tmp = ((1.0d0 + alpha) / beta) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if beta < 2.2999999999999998

   1. Initial program 99.9%

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

      1. associate-/l/ 99.0%

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

      2. +-commutative 99.0%

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

      3. associate-+l+ 99.0%

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

      4. *-commutative 99.0%

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

      5. metadata-eval 99.0%

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

      6. associate-+l+ 99.0%

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

      7. metadata-eval 99.0%

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

      8. associate-+l+ 99.0%

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

      9. metadata-eval 99.0%

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

      10. metadata-eval 99.0%

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

      11. associate-+l+ 99.0%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in alpha around 0 85.2%

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

      1. +-commutative 85.2%

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

   7. Simplified 85.2%

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

      1. div-inv 85.2%

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

      2. *-commutative 85.2%

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

      3. associate-+r+ 85.2%

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

      4. +-commutative 85.2%

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

      5. associate-+l+ 85.2%

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

      6. +-commutative 85.2%

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

   9. Applied egg-rr 85.2%

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

   10. Taylor expanded in beta around 0 84.3%

       \[\leadsto \color{blue}{\frac{0.5}{\left(2 + \alpha\right) \cdot \left(3 + \alpha\right)}} \]
   11. Step-by-step derivation

       1. +-commutative 84.3%

          \[\leadsto \frac{0.5}{\color{blue}{\left(\alpha + 2\right)} \cdot \left(3 + \alpha\right)} \]

       2. +-commutative 84.3%

          \[\leadsto \frac{0.5}{\left(\alpha + 2\right) \cdot \color{blue}{\left(\alpha + 3\right)}} \]

   12. Simplified 84.3%

       \[\leadsto \color{blue}{\frac{0.5}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}} \]

   13. Taylor expanded in alpha around 0 63.0%

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

   if 2.2999999999999998 < beta

   1. Initial program 90.2%

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

   3. Taylor expanded in beta around inf 81.9%

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

      1. div-inv 81.8%

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

      2. metadata-eval 81.8%

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

      3. associate-+l+ 81.8%

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

      4. metadata-eval 81.8%

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

      5. associate-+r+ 81.8%

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

   5. Applied egg-rr 81.8%

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

      1. associate-*r/ 81.9%

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

      2. *-commutative 81.9%

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

      3. *-lft-identity 81.9%

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

      4. +-commutative 81.9%

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

   7. Simplified 81.9%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.3:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 68.3% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.3d0) then
        tmp = 0.08333333333333333d0
    else
        tmp = 1.0d0 / (beta * (beta + 3.0d0))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if beta < 2.2999999999999998

   1. Initial program 99.9%

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

      1. associate-/l/ 99.0%

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

      2. +-commutative 99.0%

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

      3. associate-+l+ 99.0%

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

      4. *-commutative 99.0%

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

      5. metadata-eval 99.0%

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

      6. associate-+l+ 99.0%

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

      7. metadata-eval 99.0%

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

      8. associate-+l+ 99.0%

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

      9. metadata-eval 99.0%

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

      10. metadata-eval 99.0%

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

      11. associate-+l+ 99.0%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in alpha around 0 85.2%

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

      1. +-commutative 85.2%

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

   7. Simplified 85.2%

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

      1. div-inv 85.2%

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

      2. *-commutative 85.2%

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

      3. associate-+r+ 85.2%

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

      4. +-commutative 85.2%

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

      5. associate-+l+ 85.2%

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

      6. +-commutative 85.2%

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

   9. Applied egg-rr 85.2%

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

   10. Taylor expanded in beta around 0 84.3%

       \[\leadsto \color{blue}{\frac{0.5}{\left(2 + \alpha\right) \cdot \left(3 + \alpha\right)}} \]
   11. Step-by-step derivation

       1. +-commutative 84.3%

          \[\leadsto \frac{0.5}{\color{blue}{\left(\alpha + 2\right)} \cdot \left(3 + \alpha\right)} \]

       2. +-commutative 84.3%

          \[\leadsto \frac{0.5}{\left(\alpha + 2\right) \cdot \color{blue}{\left(\alpha + 3\right)}} \]

   12. Simplified 84.3%

       \[\leadsto \color{blue}{\frac{0.5}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}} \]

   13. Taylor expanded in alpha around 0 63.0%

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

   if 2.2999999999999998 < beta

   1. Initial program 90.2%

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

   3. Taylor expanded in beta around inf 81.9%

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

   4. Taylor expanded in alpha around 0 80.0%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.3:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 68.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.3:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta + 3}\\ \end{array} \end{array} \]

FPCore

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

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.3) {
		tmp = 0.08333333333333333;
	} else {
		tmp = (1.0 / beta) / (beta + 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.3d0) then
        tmp = 0.08333333333333333d0
    else
        tmp = (1.0d0 / beta) / (beta + 3.0d0)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if beta < 2.2999999999999998

   1. Initial program 99.9%

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

      1. associate-/l/ 99.0%

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

      2. +-commutative 99.0%

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

      3. associate-+l+ 99.0%

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

      4. *-commutative 99.0%

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

      5. metadata-eval 99.0%

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

      6. associate-+l+ 99.0%

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

      7. metadata-eval 99.0%

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

      8. associate-+l+ 99.0%

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

      9. metadata-eval 99.0%

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

      10. metadata-eval 99.0%

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

      11. associate-+l+ 99.0%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in alpha around 0 85.2%

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

      1. +-commutative 85.2%

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

   7. Simplified 85.2%

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

      1. div-inv 85.2%

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

      2. *-commutative 85.2%

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

      3. associate-+r+ 85.2%

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

      4. +-commutative 85.2%

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

      5. associate-+l+ 85.2%

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

      6. +-commutative 85.2%

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

   9. Applied egg-rr 85.2%

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

   10. Taylor expanded in beta around 0 84.3%

       \[\leadsto \color{blue}{\frac{0.5}{\left(2 + \alpha\right) \cdot \left(3 + \alpha\right)}} \]
   11. Step-by-step derivation

       1. +-commutative 84.3%

          \[\leadsto \frac{0.5}{\color{blue}{\left(\alpha + 2\right)} \cdot \left(3 + \alpha\right)} \]

       2. +-commutative 84.3%

          \[\leadsto \frac{0.5}{\left(\alpha + 2\right) \cdot \color{blue}{\left(\alpha + 3\right)}} \]

   12. Simplified 84.3%

       \[\leadsto \color{blue}{\frac{0.5}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}} \]

   13. Taylor expanded in alpha around 0 63.0%

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

   if 2.2999999999999998 < beta

   1. Initial program 90.2%

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

   3. Taylor expanded in beta around inf 81.9%

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

   4. Taylor expanded in alpha around 0 80.0%

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

      1. associate-/r* 80.1%

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

   6. Simplified 80.1%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.3:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 70.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.7:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.7) 0.08333333333333333 (/ (/ (+ 1.0 alpha) beta) beta)))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.7) {
		tmp = 0.08333333333333333;
	} else {
		tmp = ((1.0 + alpha) / beta) / beta;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 3.7d0) then
        tmp = 0.08333333333333333d0
    else
        tmp = ((1.0d0 + alpha) / beta) / beta
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.7) {
		tmp = 0.08333333333333333;
	} else {
		tmp = ((1.0 + alpha) / beta) / beta;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 3.7:
		tmp = 0.08333333333333333
	else:
		tmp = ((1.0 + alpha) / beta) / beta
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.7)
		tmp = 0.08333333333333333;
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / beta);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.7)
		tmp = 0.08333333333333333;
	else
		tmp = ((1.0 + alpha) / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 3.7], 0.08333333333333333, N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.7000000000000002

   1. Initial program 99.9%

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

      1. associate-/l/ 99.0%

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

      2. +-commutative 99.0%

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

      3. associate-+l+ 99.0%

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

      4. *-commutative 99.0%

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

      5. metadata-eval 99.0%

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

      6. associate-+l+ 99.0%

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

      7. metadata-eval 99.0%

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

      8. associate-+l+ 99.0%

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

      9. metadata-eval 99.0%

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

      10. metadata-eval 99.0%

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

      11. associate-+l+ 99.0%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in alpha around 0 85.2%

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

      1. +-commutative 85.2%

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

   7. Simplified 85.2%

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

      1. div-inv 85.2%

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

      2. *-commutative 85.2%

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

      3. associate-+r+ 85.2%

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

      4. +-commutative 85.2%

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

      5. associate-+l+ 85.2%

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

      6. +-commutative 85.2%

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

   9. Applied egg-rr 85.2%

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

   10. Taylor expanded in beta around 0 84.3%

       \[\leadsto \color{blue}{\frac{0.5}{\left(2 + \alpha\right) \cdot \left(3 + \alpha\right)}} \]
   11. Step-by-step derivation

       1. +-commutative 84.3%

          \[\leadsto \frac{0.5}{\color{blue}{\left(\alpha + 2\right)} \cdot \left(3 + \alpha\right)} \]

       2. +-commutative 84.3%

          \[\leadsto \frac{0.5}{\left(\alpha + 2\right) \cdot \color{blue}{\left(\alpha + 3\right)}} \]

   12. Simplified 84.3%

       \[\leadsto \color{blue}{\frac{0.5}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}} \]

   13. Taylor expanded in alpha around 0 63.0%

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

   if 3.7000000000000002 < beta

   1. Initial program 90.2%

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

   3. Taylor expanded in beta around inf 81.9%

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

   4. Taylor expanded in beta around inf 81.7%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.7:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 45.6% accurate, 4.4× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 3.0d0) then
        tmp = 0.08333333333333333d0
    else
        tmp = 0.25d0 / beta
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if beta < 3

   1. Initial program 99.9%

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

      1. associate-/l/ 99.0%

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

      2. +-commutative 99.0%

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

      3. associate-+l+ 99.0%

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

      4. *-commutative 99.0%

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

      5. metadata-eval 99.0%

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

      6. associate-+l+ 99.0%

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

      7. metadata-eval 99.0%

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

      8. associate-+l+ 99.0%

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

      9. metadata-eval 99.0%

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

      10. metadata-eval 99.0%

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

      11. associate-+l+ 99.0%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in alpha around 0 85.2%

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

      1. +-commutative 85.2%

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

   7. Simplified 85.2%

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

      1. div-inv 85.2%

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

      2. *-commutative 85.2%

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

      3. associate-+r+ 85.2%

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

      4. +-commutative 85.2%

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

      5. associate-+l+ 85.2%

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

      6. +-commutative 85.2%

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

   9. Applied egg-rr 85.2%

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

   10. Taylor expanded in beta around 0 84.3%

       \[\leadsto \color{blue}{\frac{0.5}{\left(2 + \alpha\right) \cdot \left(3 + \alpha\right)}} \]
   11. Step-by-step derivation

       1. +-commutative 84.3%

          \[\leadsto \frac{0.5}{\color{blue}{\left(\alpha + 2\right)} \cdot \left(3 + \alpha\right)} \]

       2. +-commutative 84.3%

          \[\leadsto \frac{0.5}{\left(\alpha + 2\right) \cdot \color{blue}{\left(\alpha + 3\right)}} \]

   12. Simplified 84.3%

       \[\leadsto \color{blue}{\frac{0.5}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}} \]

   13. Taylor expanded in alpha around 0 63.0%

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

   if 3 < beta

   1. Initial program 90.2%

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

      1. associate-/l/ 86.6%

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

      2. +-commutative 86.6%

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

      3. associate-+l+ 86.6%

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

      4. *-commutative 86.6%

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

      5. metadata-eval 86.6%

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

      6. associate-+l+ 86.6%

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

      7. metadata-eval 86.6%

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

      8. associate-+l+ 86.6%

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

      9. metadata-eval 86.6%

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

      10. metadata-eval 86.6%

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

      11. associate-+l+ 86.6%

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

   3. Simplified 86.6%

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

   5. Taylor expanded in alpha around 0 83.1%

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

      1. +-commutative 83.1%

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

   7. Simplified 83.1%

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

   8. Taylor expanded in beta around 0 49.4%

      \[\leadsto \frac{\color{blue}{0.5 + 0.25 \cdot \beta}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative 49.4%

         \[\leadsto \frac{0.5 + \color{blue}{\beta \cdot 0.25}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]

   10. Simplified 49.4%

       \[\leadsto \frac{\color{blue}{0.5 + \beta \cdot 0.25}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]

   11. Taylor expanded in beta around inf 6.6%

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

4. Final simplification 42.5%

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

Alternative 14: 44.7% 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 96.4%

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

   1. associate-/l/ 94.5%

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

   2. +-commutative 94.5%

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

   3. associate-+l+ 94.5%

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

   4. *-commutative 94.5%

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

   5. metadata-eval 94.5%

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

   6. associate-+l+ 94.5%

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

   7. metadata-eval 94.5%

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

   8. associate-+l+ 94.5%

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

   9. metadata-eval 94.5%

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

   10. metadata-eval 94.5%

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

   11. associate-+l+ 94.5%

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

3. Simplified 94.5%

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

5. Taylor expanded in alpha around 0 84.4%

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

   1. +-commutative 84.4%

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

7. Simplified 84.4%

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

   1. div-inv 84.4%

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

   2. *-commutative 84.4%

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

   3. associate-+r+ 84.4%

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

   4. +-commutative 84.4%

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

   5. associate-+l+ 84.4%

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

   6. +-commutative 84.4%

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

9. Applied egg-rr 84.4%

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

10. Taylor expanded in beta around 0 57.3%

    \[\leadsto \color{blue}{\frac{0.5}{\left(2 + \alpha\right) \cdot \left(3 + \alpha\right)}} \]
11. Step-by-step derivation

    1. +-commutative 57.3%

       \[\leadsto \frac{0.5}{\color{blue}{\left(\alpha + 2\right)} \cdot \left(3 + \alpha\right)} \]

    2. +-commutative 57.3%

       \[\leadsto \frac{0.5}{\left(\alpha + 2\right) \cdot \color{blue}{\left(\alpha + 3\right)}} \]

12. Simplified 57.3%

    \[\leadsto \color{blue}{\frac{0.5}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}} \]

13. Taylor expanded in alpha around 0 41.5%

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

14. Final simplification 41.5%

    \[\leadsto 0.08333333333333333 \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024046 
(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)))