Octave 3.8, jcobi/3

Percentage Accurate: 94.6% → 99.8%

Time: 17.9s

Alternatives: 12

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: 94.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.3%

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

3. Simplified 86.5%

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

   1. times-frac 97.5%

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

   2. +-commutative 97.5%

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

6. Applied egg-rr 97.5%

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

   1. +-commutative 97.5%

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

   2. +-commutative 97.5%

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

   3. +-commutative 97.5%

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

   4. +-commutative 97.5%

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

   5. associate-/r* 99.8%

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

   6. +-commutative 99.8%

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

   7. +-commutative 99.8%

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

   8. +-commutative 99.8%

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

   9. associate-+r+ 99.8%

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

   10. +-commutative 99.8%

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

   11. associate-+r+ 99.8%

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

8. Simplified 99.8%

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

9. Final simplification 99.8%

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

Alternative 2: 71.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.6 \cdot 10^{+18}:\\ \;\;\;\;\frac{1}{\beta + 2} \cdot \frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 - 2 \cdot \frac{\alpha + 2}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.6e+18)
   (* (/ 1.0 (+ beta 2.0)) (/ (+ 1.0 beta) (* (+ beta 2.0) (+ beta 3.0))))
   (*
    (/ (+ 1.0 alpha) (+ alpha (+ beta 2.0)))
    (/ (- 1.0 (* 2.0 (/ (+ alpha 2.0) beta))) beta))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.6e+18) {
		tmp = (1.0 / (beta + 2.0)) * ((1.0 + beta) / ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) * ((1.0 - (2.0 * ((alpha + 2.0) / beta))) / beta);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 4.6d+18) then
        tmp = (1.0d0 / (beta + 2.0d0)) * ((1.0d0 + beta) / ((beta + 2.0d0) * (beta + 3.0d0)))
    else
        tmp = ((1.0d0 + alpha) / (alpha + (beta + 2.0d0))) * ((1.0d0 - (2.0d0 * ((alpha + 2.0d0) / beta))) / beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.6e+18) {
		tmp = (1.0 / (beta + 2.0)) * ((1.0 + beta) / ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) * ((1.0 - (2.0 * ((alpha + 2.0) / beta))) / beta);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 4.6e+18:
		tmp = (1.0 / (beta + 2.0)) * ((1.0 + beta) / ((beta + 2.0) * (beta + 3.0)))
	else:
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) * ((1.0 - (2.0 * ((alpha + 2.0) / beta))) / beta)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.6e+18)
		tmp = Float64(Float64(1.0 / Float64(beta + 2.0)) * Float64(Float64(1.0 + beta) / Float64(Float64(beta + 2.0) * Float64(beta + 3.0))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(beta + 2.0))) * Float64(Float64(1.0 - Float64(2.0 * Float64(Float64(alpha + 2.0) / beta))) / beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4.6e+18)
		tmp = (1.0 / (beta + 2.0)) * ((1.0 + beta) / ((beta + 2.0) * (beta + 3.0)));
	else
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) * ((1.0 - (2.0 * ((alpha + 2.0) / beta))) / beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 4.6e+18], N[(N[(1.0 / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[(2.0 * N[(N[(alpha + 2.0), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.6e18

   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 94.2%

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

      1. times-frac 99.2%

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

      2. +-commutative 99.2%

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

   6. Applied egg-rr 99.2%

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

      1. +-commutative 99.2%

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

      2. +-commutative 99.2%

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

      3. +-commutative 99.2%

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

      4. +-commutative 99.2%

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

      5. associate-/r* 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. +-commutative 99.8%

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

      9. associate-+r+ 99.8%

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

      10. +-commutative 99.8%

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

      11. associate-+r+ 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in alpha around 0 62.5%

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

       1. +-commutative 62.5%

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

       2. +-commutative 62.5%

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

   11. Simplified 62.5%

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

   12. Taylor expanded in alpha around 0 62.6%

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

       1. +-commutative 62.6%

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

   14. Simplified 62.6%

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

   if 4.6e18 < beta

   1. Initial program 87.5%

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

   3. Simplified 66.7%

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

      1. times-frac 93.1%

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

      2. +-commutative 93.1%

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

   6. Applied egg-rr 93.1%

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

      1. +-commutative 93.1%

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

      2. +-commutative 93.1%

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

      3. +-commutative 93.1%

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

      4. +-commutative 93.1%

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

      5. associate-/r* 99.7%

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

      6. +-commutative 99.7%

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

      7. +-commutative 99.7%

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

      8. +-commutative 99.7%

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

      9. associate-+r+ 99.7%

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

      10. +-commutative 99.7%

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

      11. associate-+r+ 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in beta around inf 81.5%

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

       1. mul-1-neg 81.5%

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

       2. metadata-eval 81.5%

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

       3. distribute-lft-in 81.5%

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

       4. associate-*r/ 81.5%

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

   11. Simplified 81.5%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.6 \cdot 10^{+18}:\\ \;\;\;\;\frac{1}{\beta + 2} \cdot \frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 - 2 \cdot \frac{\alpha + 2}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 71.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.8 \cdot 10^{+18}:\\ \;\;\;\;\frac{1}{\beta + 2} \cdot \frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{3 + \left(\alpha + \beta\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.8e+18)
   (* (/ 1.0 (+ beta 2.0)) (/ (+ 1.0 beta) (* (+ beta 2.0) (+ beta 3.0))))
   (/ (/ (+ 1.0 alpha) beta) (+ 3.0 (+ alpha beta)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.8e+18) {
		tmp = (1.0 / (beta + 2.0)) * ((1.0 + beta) / ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / beta) / (3.0 + (alpha + beta));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.8d+18) then
        tmp = (1.0d0 / (beta + 2.0d0)) * ((1.0d0 + beta) / ((beta + 2.0d0) * (beta + 3.0d0)))
    else
        tmp = ((1.0d0 + alpha) / beta) / (3.0d0 + (alpha + beta))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.8e+18) {
		tmp = (1.0 / (beta + 2.0)) * ((1.0 + beta) / ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / beta) / (3.0 + (alpha + beta));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.8e+18:
		tmp = (1.0 / (beta + 2.0)) * ((1.0 + beta) / ((beta + 2.0) * (beta + 3.0)))
	else:
		tmp = ((1.0 + alpha) / beta) / (3.0 + (alpha + beta))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.8e+18)
		tmp = Float64(Float64(1.0 / Float64(beta + 2.0)) * Float64(Float64(1.0 + beta) / Float64(Float64(beta + 2.0) * Float64(beta + 3.0))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(3.0 + Float64(alpha + beta)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.8e+18)
		tmp = (1.0 / (beta + 2.0)) * ((1.0 + beta) / ((beta + 2.0) * (beta + 3.0)));
	else
		tmp = ((1.0 + alpha) / beta) / (3.0 + (alpha + beta));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.8e+18], N[(N[(1.0 / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.8e18

   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 94.2%

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

      1. times-frac 99.2%

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

      2. +-commutative 99.2%

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

   6. Applied egg-rr 99.2%

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

      1. +-commutative 99.2%

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

      2. +-commutative 99.2%

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

      3. +-commutative 99.2%

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

      4. +-commutative 99.2%

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

      5. associate-/r* 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. +-commutative 99.8%

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

      9. associate-+r+ 99.8%

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

      10. +-commutative 99.8%

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

      11. associate-+r+ 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in alpha around 0 62.5%

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

       1. +-commutative 62.5%

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

       2. +-commutative 62.5%

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

   11. Simplified 62.5%

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

   12. Taylor expanded in alpha around 0 62.6%

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

       1. +-commutative 62.6%

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

   14. Simplified 62.6%

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

   if 2.8e18 < beta

   1. Initial program 87.5%

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

   3. Taylor expanded in beta around inf 81.7%

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

   4. Taylor expanded in alpha around 0 81.7%

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

      1. +-commutative 81.7%

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

   6. Simplified 81.7%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.8 \cdot 10^{+18}:\\ \;\;\;\;\frac{1}{\beta + 2} \cdot \frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{3 + \left(\alpha + \beta\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5 \cdot 10^{+18}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 2}}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{3 + \left(\alpha + \beta\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5e+18)
   (/ (/ (+ 1.0 beta) (+ beta 2.0)) (* (+ beta 2.0) (+ beta 3.0)))
   (/ (/ (+ 1.0 alpha) beta) (+ 3.0 (+ alpha beta)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5e+18) {
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / (3.0 + (alpha + beta));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 5d+18) then
        tmp = ((1.0d0 + beta) / (beta + 2.0d0)) / ((beta + 2.0d0) * (beta + 3.0d0))
    else
        tmp = ((1.0d0 + alpha) / beta) / (3.0d0 + (alpha + beta))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5e+18) {
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / (3.0 + (alpha + beta));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 5e+18:
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0))
	else:
		tmp = ((1.0 + alpha) / beta) / (3.0 + (alpha + beta))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 5e+18)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(beta + 2.0)) / Float64(Float64(beta + 2.0) * Float64(beta + 3.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(3.0 + Float64(alpha + beta)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 5e+18)
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0));
	else
		tmp = ((1.0 + alpha) / beta) / (3.0 + (alpha + beta));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 5e+18], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5e18

   1. Initial program 99.8%

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

      1. associate-/l/ 99.2%

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

      2. +-commutative 99.2%

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

      3. associate-+l+ 99.2%

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

      4. *-commutative 99.2%

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

      5. metadata-eval 99.2%

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

      6. associate-+l+ 99.2%

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

      7. metadata-eval 99.2%

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

      8. +-commutative 99.2%

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

      9. +-commutative 99.2%

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

      10. +-commutative 99.2%

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

      11. metadata-eval 99.2%

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

      12. metadata-eval 99.2%

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

      13. associate-+l+ 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in alpha around 0 83.3%

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

      1. +-commutative 83.3%

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

   7. Simplified 83.3%

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

   8. Taylor expanded in alpha around 0 62.6%

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

      1. +-commutative 62.6%

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

      2. +-commutative 62.6%

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

   10. Simplified 62.6%

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

   if 5e18 < beta

   1. Initial program 87.5%

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

   3. Taylor expanded in beta around inf 81.7%

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

   4. Taylor expanded in alpha around 0 81.7%

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

      1. +-commutative 81.7%

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

   6. Simplified 81.7%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5 \cdot 10^{+18}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 2}}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{3 + \left(\alpha + \beta\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.8:\\ \;\;\;\;\frac{0.16666666666666666 + \beta \cdot \left(0.027777777777777776 + \beta \cdot -0.05092592592592592\right)}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{3 + \left(\alpha + \beta\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.8)
   (/
    (+
     0.16666666666666666
     (* beta (+ 0.027777777777777776 (* beta -0.05092592592592592))))
    (+ beta 2.0))
   (/ (/ (+ 1.0 alpha) beta) (+ 3.0 (+ alpha beta)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.8) {
		tmp = (0.16666666666666666 + (beta * (0.027777777777777776 + (beta * -0.05092592592592592)))) / (beta + 2.0);
	} else {
		tmp = ((1.0 + alpha) / beta) / (3.0 + (alpha + beta));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.8d0) then
        tmp = (0.16666666666666666d0 + (beta * (0.027777777777777776d0 + (beta * (-0.05092592592592592d0))))) / (beta + 2.0d0)
    else
        tmp = ((1.0d0 + alpha) / beta) / (3.0d0 + (alpha + beta))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.8) {
		tmp = (0.16666666666666666 + (beta * (0.027777777777777776 + (beta * -0.05092592592592592)))) / (beta + 2.0);
	} else {
		tmp = ((1.0 + alpha) / beta) / (3.0 + (alpha + beta));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 1.8:
		tmp = (0.16666666666666666 + (beta * (0.027777777777777776 + (beta * -0.05092592592592592)))) / (beta + 2.0)
	else:
		tmp = ((1.0 + alpha) / beta) / (3.0 + (alpha + beta))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.8)
		tmp = Float64(Float64(0.16666666666666666 + Float64(beta * Float64(0.027777777777777776 + Float64(beta * -0.05092592592592592)))) / Float64(beta + 2.0));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(3.0 + Float64(alpha + beta)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.8)
		tmp = (0.16666666666666666 + (beta * (0.027777777777777776 + (beta * -0.05092592592592592)))) / (beta + 2.0);
	else
		tmp = ((1.0 + alpha) / beta) / (3.0 + (alpha + beta));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 1.8], N[(N[(0.16666666666666666 + N[(beta * N[(0.027777777777777776 + N[(beta * -0.05092592592592592), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.80000000000000004

   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 94.1%

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

      1. times-frac 99.2%

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

      2. +-commutative 99.2%

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

   6. Applied egg-rr 99.2%

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

      1. +-commutative 99.2%

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

      2. +-commutative 99.2%

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

      3. +-commutative 99.2%

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

      4. +-commutative 99.2%

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

      5. associate-/r* 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. +-commutative 99.8%

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

      9. associate-+r+ 99.8%

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

      10. +-commutative 99.8%

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

      11. associate-+r+ 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in alpha around 0 62.7%

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

       1. +-commutative 62.7%

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

       2. +-commutative 62.7%

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

   11. Simplified 62.7%

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

   12. Taylor expanded in beta around 0 62.5%

       \[\leadsto \frac{1 + \alpha}{\left(\beta + 2\right) + \alpha} \cdot \color{blue}{\left(0.16666666666666666 + \beta \cdot \left(0.027777777777777776 + -0.05092592592592592 \cdot \beta\right)\right)} \]
   13. Step-by-step derivation

       1. *-commutative 62.5%

          \[\leadsto \frac{1 + \alpha}{\left(\beta + 2\right) + \alpha} \cdot \left(0.16666666666666666 + \beta \cdot \left(0.027777777777777776 + \color{blue}{\beta \cdot -0.05092592592592592}\right)\right) \]

   14. Simplified 62.5%

       \[\leadsto \frac{1 + \alpha}{\left(\beta + 2\right) + \alpha} \cdot \color{blue}{\left(0.16666666666666666 + \beta \cdot \left(0.027777777777777776 + \beta \cdot -0.05092592592592592\right)\right)} \]

   15. Taylor expanded in alpha around 0 62.6%

       \[\leadsto \color{blue}{\frac{0.16666666666666666 + \beta \cdot \left(0.027777777777777776 + -0.05092592592592592 \cdot \beta\right)}{2 + \beta}} \]

   if 1.80000000000000004 < beta

   1. Initial program 87.8%

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

   3. Taylor expanded in beta around inf 80.0%

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

   4. Taylor expanded in alpha around 0 80.0%

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

      1. +-commutative 80.0%

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

   6. Simplified 80.0%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.8:\\ \;\;\;\;\frac{0.16666666666666666 + \beta \cdot \left(0.027777777777777776 + \beta \cdot -0.05092592592592592\right)}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{3 + \left(\alpha + \beta\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 70.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.3:\\ \;\;\;\;\frac{0.16666666666666666}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{3 + \left(\alpha + \beta\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5.3)
   (/ 0.16666666666666666 (+ beta 2.0))
   (/ (/ (+ 1.0 alpha) beta) (+ 3.0 (+ alpha beta)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.3) {
		tmp = 0.16666666666666666 / (beta + 2.0);
	} else {
		tmp = ((1.0 + alpha) / beta) / (3.0 + (alpha + beta));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 5.3d0) then
        tmp = 0.16666666666666666d0 / (beta + 2.0d0)
    else
        tmp = ((1.0d0 + alpha) / beta) / (3.0d0 + (alpha + beta))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.3) {
		tmp = 0.16666666666666666 / (beta + 2.0);
	} else {
		tmp = ((1.0 + alpha) / beta) / (3.0 + (alpha + beta));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 5.3:
		tmp = 0.16666666666666666 / (beta + 2.0)
	else:
		tmp = ((1.0 + alpha) / beta) / (3.0 + (alpha + beta))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 5.3)
		tmp = Float64(0.16666666666666666 / Float64(beta + 2.0));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(3.0 + Float64(alpha + beta)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 5.3)
		tmp = 0.16666666666666666 / (beta + 2.0);
	else
		tmp = ((1.0 + alpha) / beta) / (3.0 + (alpha + beta));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 5.3], N[(0.16666666666666666 / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.29999999999999982

   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 94.1%

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

      1. times-frac 99.2%

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

      2. +-commutative 99.2%

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

   6. Applied egg-rr 99.2%

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

      1. +-commutative 99.2%

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

      2. +-commutative 99.2%

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

      3. +-commutative 99.2%

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

      4. +-commutative 99.2%

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

      5. associate-/r* 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. +-commutative 99.8%

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

      9. associate-+r+ 99.8%

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

      10. +-commutative 99.8%

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

      11. associate-+r+ 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in alpha around 0 62.7%

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

       1. +-commutative 62.7%

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

       2. +-commutative 62.7%

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

   11. Simplified 62.7%

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

   12. Taylor expanded in beta around 0 61.0%

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

   13. Taylor expanded in alpha around 0 61.1%

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

   if 5.29999999999999982 < beta

   1. Initial program 87.8%

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

   3. Taylor expanded in beta around inf 80.0%

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

   4. Taylor expanded in alpha around 0 80.0%

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

      1. +-commutative 80.0%

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

   6. Simplified 80.0%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.3:\\ \;\;\;\;\frac{0.16666666666666666}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{3 + \left(\alpha + \beta\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.3:\\ \;\;\;\;\frac{0.16666666666666666}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5.3)
   (/ 0.16666666666666666 (+ beta 2.0))
   (/ 1.0 (* beta (+ beta 3.0)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.3) {
		tmp = 0.16666666666666666 / (beta + 2.0);
	} else {
		tmp = 1.0 / (beta * (beta + 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 5.3d0) then
        tmp = 0.16666666666666666d0 / (beta + 2.0d0)
    else
        tmp = 1.0d0 / (beta * (beta + 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.3) {
		tmp = 0.16666666666666666 / (beta + 2.0);
	} else {
		tmp = 1.0 / (beta * (beta + 3.0));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 5.3:
		tmp = 0.16666666666666666 / (beta + 2.0)
	else:
		tmp = 1.0 / (beta * (beta + 3.0))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 5.3)
		tmp = Float64(0.16666666666666666 / Float64(beta + 2.0));
	else
		tmp = Float64(1.0 / Float64(beta * Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 5.3)
		tmp = 0.16666666666666666 / (beta + 2.0);
	else
		tmp = 1.0 / (beta * (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 5.3], N[(0.16666666666666666 / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(beta * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.29999999999999982

   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 94.1%

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

      1. times-frac 99.2%

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

      2. +-commutative 99.2%

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

   6. Applied egg-rr 99.2%

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

      1. +-commutative 99.2%

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

      2. +-commutative 99.2%

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

      3. +-commutative 99.2%

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

      4. +-commutative 99.2%

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

      5. associate-/r* 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. +-commutative 99.8%

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

      9. associate-+r+ 99.8%

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

      10. +-commutative 99.8%

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

      11. associate-+r+ 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in alpha around 0 62.7%

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

       1. +-commutative 62.7%

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

       2. +-commutative 62.7%

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

   11. Simplified 62.7%

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

   12. Taylor expanded in beta around 0 61.0%

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

   13. Taylor expanded in alpha around 0 61.1%

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

   if 5.29999999999999982 < beta

   1. Initial program 87.8%

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

   3. Taylor expanded in beta around inf 80.0%

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

   4. Taylor expanded in alpha around 0 71.6%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.3:\\ \;\;\;\;\frac{0.16666666666666666}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.3:\\ \;\;\;\;\frac{0.16666666666666666}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta + 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5.3)
   (/ 0.16666666666666666 (+ beta 2.0))
   (/ (/ 1.0 beta) (+ beta 3.0))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.3) {
		tmp = 0.16666666666666666 / (beta + 2.0);
	} else {
		tmp = (1.0 / beta) / (beta + 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 5.3d0) then
        tmp = 0.16666666666666666d0 / (beta + 2.0d0)
    else
        tmp = (1.0d0 / beta) / (beta + 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.3) {
		tmp = 0.16666666666666666 / (beta + 2.0);
	} else {
		tmp = (1.0 / beta) / (beta + 3.0);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 5.3:
		tmp = 0.16666666666666666 / (beta + 2.0)
	else:
		tmp = (1.0 / beta) / (beta + 3.0)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 5.3)
		tmp = Float64(0.16666666666666666 / Float64(beta + 2.0));
	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 <= 5.3)
		tmp = 0.16666666666666666 / (beta + 2.0);
	else
		tmp = (1.0 / beta) / (beta + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 5.3], N[(0.16666666666666666 / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.29999999999999982

   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 94.1%

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

      1. times-frac 99.2%

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

      2. +-commutative 99.2%

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

   6. Applied egg-rr 99.2%

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

      1. +-commutative 99.2%

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

      2. +-commutative 99.2%

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

      3. +-commutative 99.2%

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

      4. +-commutative 99.2%

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

      5. associate-/r* 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. +-commutative 99.8%

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

      9. associate-+r+ 99.8%

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

      10. +-commutative 99.8%

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

      11. associate-+r+ 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in alpha around 0 62.7%

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

       1. +-commutative 62.7%

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

       2. +-commutative 62.7%

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

   11. Simplified 62.7%

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

   12. Taylor expanded in beta around 0 61.0%

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

   13. Taylor expanded in alpha around 0 61.1%

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

   if 5.29999999999999982 < beta

   1. Initial program 87.8%

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

   3. Taylor expanded in beta around inf 80.0%

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

   4. Taylor expanded in alpha around 0 71.6%

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

      1. associate-/r* 72.6%

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

      2. +-commutative 72.6%

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

   6. Simplified 72.6%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.3:\\ \;\;\;\;\frac{0.16666666666666666}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 7.8:\\ \;\;\;\;\frac{0.16666666666666666}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 7.8)
   (/ 0.16666666666666666 (+ beta 2.0))
   (/ (/ (+ 1.0 alpha) beta) beta)))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 7.8) {
		tmp = 0.16666666666666666 / (beta + 2.0);
	} 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 <= 7.8d0) then
        tmp = 0.16666666666666666d0 / (beta + 2.0d0)
    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 <= 7.8) {
		tmp = 0.16666666666666666 / (beta + 2.0);
	} else {
		tmp = ((1.0 + alpha) / beta) / beta;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 7.8:
		tmp = 0.16666666666666666 / (beta + 2.0)
	else:
		tmp = ((1.0 + alpha) / beta) / beta
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 7.8)
		tmp = Float64(0.16666666666666666 / Float64(beta + 2.0));
	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 <= 7.8)
		tmp = 0.16666666666666666 / (beta + 2.0);
	else
		tmp = ((1.0 + alpha) / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 7.8], N[(0.16666666666666666 / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 7.79999999999999982

   1. Initial program 99.8%

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

   3. Simplified 94.1%

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

      1. times-frac 99.2%

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

      2. +-commutative 99.2%

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

   6. Applied egg-rr 99.2%

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

      1. +-commutative 99.2%

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

      2. +-commutative 99.2%

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

      3. +-commutative 99.2%

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

      4. +-commutative 99.2%

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

      5. associate-/r* 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. +-commutative 99.8%

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

      9. associate-+r+ 99.8%

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

      10. +-commutative 99.8%

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

      11. associate-+r+ 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in alpha around 0 62.7%

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

       1. +-commutative 62.7%

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

       2. +-commutative 62.7%

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

   11. Simplified 62.7%

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

   12. Taylor expanded in beta around 0 61.0%

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

   13. Taylor expanded in alpha around 0 61.1%

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

   if 7.79999999999999982 < beta

   1. Initial program 87.8%

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

   3. Taylor expanded in beta around inf 80.0%

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

   4. Taylor expanded in beta around inf 79.8%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 7.8:\\ \;\;\;\;\frac{0.16666666666666666}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 11.5% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2:\\ \;\;\;\;0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.0) 0.16666666666666666 (/ 0.3333333333333333 beta)))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.0) {
		tmp = 0.16666666666666666;
	} else {
		tmp = 0.3333333333333333 / beta;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.0d0) then
        tmp = 0.16666666666666666d0
    else
        tmp = 0.3333333333333333d0 / beta
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.0) {
		tmp = 0.16666666666666666;
	} else {
		tmp = 0.3333333333333333 / beta;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.0:
		tmp = 0.16666666666666666
	else:
		tmp = 0.3333333333333333 / beta
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.0)
		tmp = 0.16666666666666666;
	else
		tmp = Float64(0.3333333333333333 / beta);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.0)
		tmp = 0.16666666666666666;
	else
		tmp = 0.3333333333333333 / beta;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.0], 0.16666666666666666, N[(0.3333333333333333 / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2

   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 94.1%

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

      1. times-frac 99.2%

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

      2. +-commutative 99.2%

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

   6. Applied egg-rr 99.2%

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

      1. +-commutative 99.2%

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

      2. +-commutative 99.2%

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

      3. +-commutative 99.2%

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

      4. +-commutative 99.2%

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

      5. associate-/r* 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. +-commutative 99.8%

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

      9. associate-+r+ 99.8%

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

      10. +-commutative 99.8%

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

      11. associate-+r+ 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in alpha around 0 62.7%

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

       1. +-commutative 62.7%

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

       2. +-commutative 62.7%

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

   11. Simplified 62.7%

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

   12. Taylor expanded in beta around 0 61.0%

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

   13. Taylor expanded in alpha around inf 13.4%

       \[\leadsto \color{blue}{0.16666666666666666} \]

   if 2 < beta

   1. Initial program 87.8%

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

   3. Taylor expanded in beta around inf 80.0%

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

   4. Taylor expanded in alpha around 0 71.6%

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

   5. Taylor expanded in beta around 0 6.6%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{\beta}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 11.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2:\\ \;\;\;\;0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 45.5% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.3%

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

3. Simplified 86.5%

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

   1. times-frac 97.5%

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

   2. +-commutative 97.5%

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

6. Applied egg-rr 97.5%

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

   1. +-commutative 97.5%

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

   2. +-commutative 97.5%

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

   3. +-commutative 97.5%

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

   4. +-commutative 97.5%

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

   5. associate-/r* 99.8%

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

   6. +-commutative 99.8%

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

   7. +-commutative 99.8%

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

   8. +-commutative 99.8%

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

   9. associate-+r+ 99.8%

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

   10. +-commutative 99.8%

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

   11. associate-+r+ 99.8%

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

8. Simplified 99.8%

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

9. Taylor expanded in alpha around 0 67.6%

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

    1. +-commutative 67.6%

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

    2. +-commutative 67.6%

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

11. Simplified 67.6%

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

12. Taylor expanded in beta around 0 45.1%

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

13. Taylor expanded in alpha around 0 45.3%

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

14. Final simplification 45.3%

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

Alternative 12: 10.6% accurate, 35.0× speedup

Math

\[\begin{array}{l} \\ 0.16666666666666666 \end{array} \]

FPCore

(FPCore (alpha beta) :precision binary64 0.16666666666666666)

C

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

Fortran

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

Java

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

Python

def code(alpha, beta):
	return 0.16666666666666666

Julia

function code(alpha, beta)
	return 0.16666666666666666
end

MATLAB

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

Wolfram

code[alpha_, beta_] := 0.16666666666666666

Derivation

1. Initial program 96.3%

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

3. Simplified 86.5%

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

   1. times-frac 97.5%

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

   2. +-commutative 97.5%

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

6. Applied egg-rr 97.5%

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

   1. +-commutative 97.5%

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

   2. +-commutative 97.5%

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

   3. +-commutative 97.5%

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

   4. +-commutative 97.5%

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

   5. associate-/r* 99.8%

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

   6. +-commutative 99.8%

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

   7. +-commutative 99.8%

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

   8. +-commutative 99.8%

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

   9. associate-+r+ 99.8%

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

   10. +-commutative 99.8%

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

   11. associate-+r+ 99.8%

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

8. Simplified 99.8%

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

9. Taylor expanded in alpha around 0 67.6%

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

    1. +-commutative 67.6%

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

    2. +-commutative 67.6%

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

11. Simplified 67.6%

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

12. Taylor expanded in beta around 0 45.1%

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

13. Taylor expanded in alpha around inf 10.6%

    \[\leadsto \color{blue}{0.16666666666666666} \]

14. Final simplification 10.6%

    \[\leadsto 0.16666666666666666 \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024095 
(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)))