Octave 3.8, jcobi/3

Percentage Accurate: 94.0% → 99.8%

Time: 18.3s

Alternatives: 18

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: 94.0% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = ((1.0 + alpha) * (((1.0 + beta) / t_0) / (alpha + (beta + 3.0)))) / t_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] * N[(N[(N[(1.0 + beta), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]

Derivation

1. Initial program 96.7%

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

3. Simplified 87.5%

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

   1. times-frac 96.9%

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

   2. +-commutative 96.9%

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

6. Applied egg-rr 96.9%

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

   1. +-commutative 96.9%

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

   2. +-commutative 96.9%

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

   3. associate-/r* 99.8%

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

   4. +-commutative 99.8%

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

   5. +-commutative 99.8%

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

   6. +-commutative 99.8%

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

   7. +-commutative 99.8%

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

8. Simplified 99.8%

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

   1. associate-*l/ 99.8%

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

   2. +-commutative 99.8%

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

   3. +-commutative 99.8%

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

   4. +-commutative 99.8%

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

10. Applied egg-rr 99.8%

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

11. Final simplification 99.8%

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

Alternative 2: 92.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 6.7:\\ \;\;\;\;\frac{1}{t\_0 \cdot \left(\left(\alpha + 2\right) \cdot \frac{\alpha + 3}{1 + \alpha}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{t\_0} \cdot \frac{\frac{\beta - \left(4 + \alpha \cdot 2\right)}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 6.7)
     (/ 1.0 (* t_0 (* (+ alpha 2.0) (/ (+ alpha 3.0) (+ 1.0 alpha)))))
     (*
      (/ (+ 1.0 alpha) t_0)
      (/ (/ (- beta (+ 4.0 (* alpha 2.0))) beta) beta)))))

C

double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 6.7) {
		tmp = 1.0 / (t_0 * ((alpha + 2.0) * ((alpha + 3.0) / (1.0 + alpha))));
	} else {
		tmp = ((1.0 + alpha) / t_0) * (((beta - (4.0 + (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) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    if (beta <= 6.7d0) then
        tmp = 1.0d0 / (t_0 * ((alpha + 2.0d0) * ((alpha + 3.0d0) / (1.0d0 + alpha))))
    else
        tmp = ((1.0d0 + alpha) / t_0) * (((beta - (4.0d0 + (alpha * 2.0d0))) / beta) / beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 6.7) {
		tmp = 1.0 / (t_0 * ((alpha + 2.0) * ((alpha + 3.0) / (1.0 + alpha))));
	} else {
		tmp = ((1.0 + alpha) / t_0) * (((beta - (4.0 + (alpha * 2.0))) / beta) / beta);
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 6.7:
		tmp = 1.0 / (t_0 * ((alpha + 2.0) * ((alpha + 3.0) / (1.0 + alpha))))
	else:
		tmp = ((1.0 + alpha) / t_0) * (((beta - (4.0 + (alpha * 2.0))) / beta) / beta)
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 6.7)
		tmp = Float64(1.0 / Float64(t_0 * Float64(Float64(alpha + 2.0) * Float64(Float64(alpha + 3.0) / Float64(1.0 + alpha)))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / t_0) * Float64(Float64(Float64(beta - Float64(4.0 + Float64(alpha * 2.0))) / beta) / beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 6.7)
		tmp = 1.0 / (t_0 * ((alpha + 2.0) * ((alpha + 3.0) / (1.0 + alpha))));
	else
		tmp = ((1.0 + alpha) / t_0) * (((beta - (4.0 + (alpha * 2.0))) / beta) / beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 6.7], N[(1.0 / N[(t$95$0 * N[(N[(alpha + 2.0), $MachinePrecision] * N[(N[(alpha + 3.0), $MachinePrecision] / N[(1.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(N[(N[(beta - N[(4.0 + N[(alpha * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.70000000000000018

   1. Initial program 99.9%

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

   3. Simplified 93.0%

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

      1. associate-+r+ 93.0%

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

      2. fma-undefine 93.0%

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

      3. *-commutative 93.0%

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

      4. associate-+l+ 93.0%

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

      5. +-commutative 93.0%

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

      6. associate-+l+ 93.0%

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

      7. *-commutative 93.0%

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

      8. associate-*r* 92.9%

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

      9. associate-+r+ 92.9%

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

      10. +-commutative 92.9%

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

      11. associate-/l/ 99.6%

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

      12. clear-num 99.6%

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

      13. inv-pow 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. unpow-1 99.6%

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

      2. associate-/l* 99.6%

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

      3. +-commutative 99.6%

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

      4. +-commutative 99.6%

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

      5. +-commutative 99.6%

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

      6. +-commutative 99.6%

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

      7. fma-undefine 99.6%

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

      8. +-commutative 99.6%

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

      9. *-commutative 99.6%

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

      10. +-commutative 99.6%

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

      11. associate-+r+ 99.6%

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

      12. distribute-rgt1-in 99.6%

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

      13. +-commutative 99.6%

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

      14. +-commutative 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in beta around 0 99.0%

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

       1. associate-/l* 99.0%

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

   11. Simplified 99.0%

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

   if 6.70000000000000018 < beta

   1. Initial program 90.8%

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

   3. Simplified 77.3%

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

      1. times-frac 91.9%

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

      2. +-commutative 91.9%

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

   6. Applied egg-rr 91.9%

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

      1. +-commutative 91.9%

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

      2. +-commutative 91.9%

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

      3. associate-/r* 99.6%

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

      4. +-commutative 99.6%

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

      5. +-commutative 99.6%

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

      6. +-commutative 99.6%

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

      7. +-commutative 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in beta around inf 89.9%

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

       1. mul-1-neg 89.9%

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

   11. Simplified 89.9%

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

   12. Taylor expanded in beta around 0 89.9%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.7:\\ \;\;\;\;\frac{1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\alpha + 2\right) \cdot \frac{\alpha + 3}{1 + \alpha}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{\frac{\beta - \left(4 + \alpha \cdot 2\right)}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.3:\\ \;\;\;\;\frac{\frac{1 + \alpha}{4 + \alpha \cdot \left(\alpha + 4\right)}}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{\frac{\beta - \left(4 + \alpha \cdot 2\right)}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.3)
   (/
    (/ (+ 1.0 alpha) (+ 4.0 (* alpha (+ alpha 4.0))))
    (+ 1.0 (+ 2.0 (+ alpha beta))))
   (*
    (/ (+ 1.0 alpha) (+ alpha (+ beta 2.0)))
    (/ (/ (- beta (+ 4.0 (* alpha 2.0))) beta) beta))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.3) {
		tmp = ((1.0 + alpha) / (4.0 + (alpha * (alpha + 4.0)))) / (1.0 + (2.0 + (alpha + beta)));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) * (((beta - (4.0 + (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 <= 6.3d0) then
        tmp = ((1.0d0 + alpha) / (4.0d0 + (alpha * (alpha + 4.0d0)))) / (1.0d0 + (2.0d0 + (alpha + beta)))
    else
        tmp = ((1.0d0 + alpha) / (alpha + (beta + 2.0d0))) * (((beta - (4.0d0 + (alpha * 2.0d0))) / beta) / beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.3) {
		tmp = ((1.0 + alpha) / (4.0 + (alpha * (alpha + 4.0)))) / (1.0 + (2.0 + (alpha + beta)));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) * (((beta - (4.0 + (alpha * 2.0))) / beta) / beta);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 6.3:
		tmp = ((1.0 + alpha) / (4.0 + (alpha * (alpha + 4.0)))) / (1.0 + (2.0 + (alpha + beta)))
	else:
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) * (((beta - (4.0 + (alpha * 2.0))) / beta) / beta)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.3)
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(4.0 + Float64(alpha * Float64(alpha + 4.0)))) / Float64(1.0 + Float64(2.0 + Float64(alpha + beta))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(beta + 2.0))) * Float64(Float64(Float64(beta - Float64(4.0 + Float64(alpha * 2.0))) / beta) / beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 6.3)
		tmp = ((1.0 + alpha) / (4.0 + (alpha * (alpha + 4.0)))) / (1.0 + (2.0 + (alpha + beta)));
	else
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) * (((beta - (4.0 + (alpha * 2.0))) / beta) / beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 6.3], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(4.0 + N[(alpha * N[(alpha + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(beta - N[(4.0 + N[(alpha * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.29999999999999982

   1. Initial program 99.9%

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

   3. Taylor expanded in beta around 0 99.0%

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

   4. Taylor expanded in alpha around 0 99.0%

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

      1. +-commutative 99.0%

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

   6. Simplified 99.0%

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

   if 6.29999999999999982 < beta

   1. Initial program 90.8%

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

   3. Simplified 77.3%

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

      1. times-frac 91.9%

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

      2. +-commutative 91.9%

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

   6. Applied egg-rr 91.9%

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

      1. +-commutative 91.9%

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

      2. +-commutative 91.9%

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

      3. associate-/r* 99.6%

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

      4. +-commutative 99.6%

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

      5. +-commutative 99.6%

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

      6. +-commutative 99.6%

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

      7. +-commutative 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in beta around inf 89.9%

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

       1. mul-1-neg 89.9%

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

   11. Simplified 89.9%

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

   12. Taylor expanded in beta around 0 89.9%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.3:\\ \;\;\;\;\frac{\frac{1 + \alpha}{4 + \alpha \cdot \left(\alpha + 4\right)}}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{\frac{\beta - \left(4 + \alpha \cdot 2\right)}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.4:\\ \;\;\;\;\frac{\frac{1 + \alpha}{4 + \alpha \cdot \left(\alpha + 4\right)}}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 - \frac{4 + \alpha \cdot 2}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.4)
   (/
    (/ (+ 1.0 alpha) (+ 4.0 (* alpha (+ alpha 4.0))))
    (+ 1.0 (+ 2.0 (+ alpha beta))))
   (*
    (/ (+ 1.0 alpha) (+ alpha (+ beta 2.0)))
    (/ (- 1.0 (/ (+ 4.0 (* alpha 2.0)) beta)) beta))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.4) {
		tmp = ((1.0 + alpha) / (4.0 + (alpha * (alpha + 4.0)))) / (1.0 + (2.0 + (alpha + beta)));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) * ((1.0 - ((4.0 + (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 <= 6.4d0) then
        tmp = ((1.0d0 + alpha) / (4.0d0 + (alpha * (alpha + 4.0d0)))) / (1.0d0 + (2.0d0 + (alpha + beta)))
    else
        tmp = ((1.0d0 + alpha) / (alpha + (beta + 2.0d0))) * ((1.0d0 - ((4.0d0 + (alpha * 2.0d0)) / beta)) / beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.4) {
		tmp = ((1.0 + alpha) / (4.0 + (alpha * (alpha + 4.0)))) / (1.0 + (2.0 + (alpha + beta)));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) * ((1.0 - ((4.0 + (alpha * 2.0)) / beta)) / beta);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 6.4:
		tmp = ((1.0 + alpha) / (4.0 + (alpha * (alpha + 4.0)))) / (1.0 + (2.0 + (alpha + beta)))
	else:
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) * ((1.0 - ((4.0 + (alpha * 2.0)) / beta)) / beta)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.4)
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(4.0 + Float64(alpha * Float64(alpha + 4.0)))) / Float64(1.0 + Float64(2.0 + Float64(alpha + beta))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(beta + 2.0))) * Float64(Float64(1.0 - Float64(Float64(4.0 + Float64(alpha * 2.0)) / beta)) / beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 6.4)
		tmp = ((1.0 + alpha) / (4.0 + (alpha * (alpha + 4.0)))) / (1.0 + (2.0 + (alpha + beta)));
	else
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) * ((1.0 - ((4.0 + (alpha * 2.0)) / beta)) / beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 6.4], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(4.0 + N[(alpha * N[(alpha + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[(N[(4.0 + N[(alpha * 2.0), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.4000000000000004

   1. Initial program 99.9%

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

   3. Taylor expanded in beta around 0 99.0%

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

   4. Taylor expanded in alpha around 0 99.0%

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

      1. +-commutative 99.0%

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

   6. Simplified 99.0%

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

   if 6.4000000000000004 < beta

   1. Initial program 90.8%

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

   3. Simplified 77.3%

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

      1. times-frac 91.9%

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

      2. +-commutative 91.9%

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

   6. Applied egg-rr 91.9%

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

      1. +-commutative 91.9%

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

      2. +-commutative 91.9%

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

      3. associate-/r* 99.6%

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

      4. +-commutative 99.6%

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

      5. +-commutative 99.6%

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

      6. +-commutative 99.6%

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

      7. +-commutative 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in beta around inf 89.9%

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

       1. mul-1-neg 89.9%

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

   11. Simplified 89.9%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.4:\\ \;\;\;\;\frac{\frac{1 + \alpha}{4 + \alpha \cdot \left(\alpha + 4\right)}}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 - \frac{4 + \alpha \cdot 2}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.8% accurate, 1.4× speedup

Math

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

FPCore

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.7%

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

3. Simplified 87.5%

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

   1. times-frac 96.9%

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

   2. +-commutative 96.9%

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

6. Applied egg-rr 96.9%

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

   1. +-commutative 96.9%

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

   2. +-commutative 96.9%

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

   3. associate-/r* 99.8%

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

   4. +-commutative 99.8%

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

   5. +-commutative 99.8%

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

   6. +-commutative 99.8%

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

   7. +-commutative 99.8%

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

8. Simplified 99.8%

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

9. Final simplification 99.8%

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

Alternative 6: 73.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.75 \cdot 10^{+14}:\\ \;\;\;\;\frac{1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \frac{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}{1 + \beta}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \alpha\right) \cdot \frac{1}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.75e+14)
   (/
    1.0
    (* (+ alpha (+ beta 2.0)) (/ (* (+ beta 2.0) (+ beta 3.0)) (+ 1.0 beta))))
   (/ (* (+ 1.0 alpha) (/ 1.0 beta)) (+ alpha (+ beta 3.0)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.75e+14) {
		tmp = 1.0 / ((alpha + (beta + 2.0)) * (((beta + 2.0) * (beta + 3.0)) / (1.0 + beta)));
	} else {
		tmp = ((1.0 + alpha) * (1.0 / beta)) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.75d+14) then
        tmp = 1.0d0 / ((alpha + (beta + 2.0d0)) * (((beta + 2.0d0) * (beta + 3.0d0)) / (1.0d0 + beta)))
    else
        tmp = ((1.0d0 + alpha) * (1.0d0 / beta)) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.75e+14) {
		tmp = 1.0 / ((alpha + (beta + 2.0)) * (((beta + 2.0) * (beta + 3.0)) / (1.0 + beta)));
	} else {
		tmp = ((1.0 + alpha) * (1.0 / beta)) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.75e+14:
		tmp = 1.0 / ((alpha + (beta + 2.0)) * (((beta + 2.0) * (beta + 3.0)) / (1.0 + beta)))
	else:
		tmp = ((1.0 + alpha) * (1.0 / beta)) / (alpha + (beta + 3.0))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.75e+14)
		tmp = Float64(1.0 / Float64(Float64(alpha + Float64(beta + 2.0)) * Float64(Float64(Float64(beta + 2.0) * Float64(beta + 3.0)) / Float64(1.0 + beta))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) * Float64(1.0 / beta)) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.75e+14)
		tmp = 1.0 / ((alpha + (beta + 2.0)) * (((beta + 2.0) * (beta + 3.0)) / (1.0 + beta)));
	else
		tmp = ((1.0 + alpha) * (1.0 / beta)) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.75e+14], N[(1.0 / N[(N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] * N[(1.0 / beta), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.75e14

   1. Initial program 99.9%

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

   3. Simplified 93.0%

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

      1. associate-+r+ 93.0%

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

      2. fma-undefine 93.0%

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

      3. *-commutative 93.0%

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

      4. associate-+l+ 93.0%

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

      5. +-commutative 93.0%

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

      6. associate-+l+ 93.0%

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

      7. *-commutative 93.0%

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

      8. associate-*r* 92.9%

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

      9. associate-+r+ 92.9%

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

      10. +-commutative 92.9%

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

      11. associate-/l/ 99.6%

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

      12. clear-num 99.6%

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

      13. inv-pow 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. unpow-1 99.6%

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

      2. associate-/l* 99.6%

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

      3. +-commutative 99.6%

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

      4. +-commutative 99.6%

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

      5. +-commutative 99.6%

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

      6. +-commutative 99.6%

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

      7. fma-undefine 99.6%

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

      8. +-commutative 99.6%

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

      9. *-commutative 99.6%

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

      10. +-commutative 99.6%

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

      11. associate-+r+ 99.6%

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

      12. distribute-rgt1-in 99.6%

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

      13. +-commutative 99.6%

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

      14. +-commutative 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in alpha around 0 66.3%

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

   if 2.75e14 < beta

   1. Initial program 90.8%

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

   3. Taylor expanded in beta around inf 90.0%

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

      1. *-un-lft-identity 90.0%

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

      2. metadata-eval 90.0%

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

      3. associate-+l+ 90.0%

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

      4. metadata-eval 90.0%

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

      5. associate-+l+ 90.0%

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

   5. Applied egg-rr 90.0%

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

      1. *-lft-identity 90.0%

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

      2. +-commutative 90.0%

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

      3. +-commutative 90.0%

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

      4. +-commutative 90.0%

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

   7. Simplified 90.0%

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

      1. div-inv 90.0%

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

   9. Applied egg-rr 90.0%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.75 \cdot 10^{+14}:\\ \;\;\;\;\frac{1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \frac{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}{1 + \beta}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \alpha\right) \cdot \frac{1}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 3\right)\\ \mathbf{if}\;\beta \leq 4.4:\\ \;\;\;\;\frac{0.25}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \mathbf{elif}\;\beta \leq 3 \cdot 10^{+154}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 3.0))))
   (if (<= beta 4.4)
     (/ 0.25 (+ 1.0 (+ 2.0 (+ alpha beta))))
     (if (<= beta 3e+154)
       (/ (+ 1.0 alpha) (* beta t_0))
       (/ (/ alpha beta) t_0)))))

C

double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 3.0);
	double tmp;
	if (beta <= 4.4) {
		tmp = 0.25 / (1.0 + (2.0 + (alpha + beta)));
	} else if (beta <= 3e+154) {
		tmp = (1.0 + alpha) / (beta * t_0);
	} else {
		tmp = (alpha / beta) / t_0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 3.0d0)
    if (beta <= 4.4d0) then
        tmp = 0.25d0 / (1.0d0 + (2.0d0 + (alpha + beta)))
    else if (beta <= 3d+154) then
        tmp = (1.0d0 + alpha) / (beta * t_0)
    else
        tmp = (alpha / beta) / t_0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 3.0);
	double tmp;
	if (beta <= 4.4) {
		tmp = 0.25 / (1.0 + (2.0 + (alpha + beta)));
	} else if (beta <= 3e+154) {
		tmp = (1.0 + alpha) / (beta * t_0);
	} else {
		tmp = (alpha / beta) / t_0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = alpha + (beta + 3.0)
	tmp = 0
	if beta <= 4.4:
		tmp = 0.25 / (1.0 + (2.0 + (alpha + beta)))
	elif beta <= 3e+154:
		tmp = (1.0 + alpha) / (beta * t_0)
	else:
		tmp = (alpha / beta) / t_0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 3.0))
	tmp = 0.0
	if (beta <= 4.4)
		tmp = Float64(0.25 / Float64(1.0 + Float64(2.0 + Float64(alpha + beta))));
	elseif (beta <= 3e+154)
		tmp = Float64(Float64(1.0 + alpha) / Float64(beta * t_0));
	else
		tmp = Float64(Float64(alpha / beta) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 3.0);
	tmp = 0.0;
	if (beta <= 4.4)
		tmp = 0.25 / (1.0 + (2.0 + (alpha + beta)));
	elseif (beta <= 3e+154)
		tmp = (1.0 + alpha) / (beta * t_0);
	else
		tmp = (alpha / beta) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 4.4], N[(0.25 / N[(1.0 + N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 3e+154], N[(N[(1.0 + alpha), $MachinePrecision] / N[(beta * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if beta < 4.4000000000000004

   1. Initial program 99.9%

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

   3. Taylor expanded in beta around 0 99.0%

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

   4. Taylor expanded in alpha around 0 66.2%

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

   if 4.4000000000000004 < beta < 3.00000000000000026e154

   1. Initial program 97.2%

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

   3. Taylor expanded in beta around inf 83.7%

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

      1. *-un-lft-identity 83.7%

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

      2. metadata-eval 83.7%

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

      3. associate-+l+ 83.7%

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

      4. metadata-eval 83.7%

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

      5. associate-+l+ 83.7%

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

   5. Applied egg-rr 83.7%

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

      1. *-lft-identity 83.7%

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

      2. +-commutative 83.7%

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

      3. +-commutative 83.7%

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

      4. +-commutative 83.7%

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

   7. Simplified 83.7%

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

      1. *-un-lft-identity 83.7%

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

      2. associate-/l/ 86.0%

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

      3. +-commutative 86.0%

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

   9. Applied egg-rr 86.0%

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

       1. *-lft-identity 86.0%

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

       2. *-commutative 86.0%

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

       3. +-commutative 86.0%

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

       4. +-commutative 86.0%

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

       5. +-commutative 86.0%

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

   11. Simplified 86.0%

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

   if 3.00000000000000026e154 < beta

   1. Initial program 84.7%

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

   3. Taylor expanded in beta around inf 96.1%

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

      1. *-un-lft-identity 96.1%

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

      2. metadata-eval 96.1%

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

      3. associate-+l+ 96.1%

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

      4. metadata-eval 96.1%

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

      5. associate-+l+ 96.1%

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

   5. Applied egg-rr 96.1%

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

      1. *-lft-identity 96.1%

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

      2. +-commutative 96.1%

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

      3. +-commutative 96.1%

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

      4. +-commutative 96.1%

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

   7. Simplified 96.1%

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

   8. Taylor expanded in alpha around inf 96.1%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.4:\\ \;\;\;\;\frac{0.25}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \mathbf{elif}\;\beta \leq 3 \cdot 10^{+154}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \left(\alpha + \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.2 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 2}}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \alpha\right) \cdot \frac{1}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.2e+15)
   (/ (/ (+ 1.0 beta) (+ beta 2.0)) (* (+ beta 2.0) (+ beta 3.0)))
   (/ (* (+ 1.0 alpha) (/ 1.0 beta)) (+ alpha (+ beta 3.0)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.2e+15) {
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0));
	} else {
		tmp = ((1.0 + alpha) * (1.0 / beta)) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.2d+15) then
        tmp = ((1.0d0 + beta) / (beta + 2.0d0)) / ((beta + 2.0d0) * (beta + 3.0d0))
    else
        tmp = ((1.0d0 + alpha) * (1.0d0 / beta)) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.2e+15) {
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0));
	} else {
		tmp = ((1.0 + alpha) * (1.0 / beta)) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 1.2e+15:
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0))
	else:
		tmp = ((1.0 + alpha) * (1.0 / beta)) / (alpha + (beta + 3.0))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.2e+15)
		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) * Float64(1.0 / beta)) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.2e+15)
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0));
	else
		tmp = ((1.0 + alpha) * (1.0 / beta)) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 1.2e+15], 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] * N[(1.0 / beta), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.2e15

   1. Initial program 99.9%

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

      1. associate-/l/ 99.6%

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

      2. +-commutative 99.6%

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

      3. associate-+l+ 99.6%

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

      4. *-commutative 99.6%

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

      5. metadata-eval 99.6%

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

      6. associate-+l+ 99.6%

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

      7. metadata-eval 99.6%

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

      8. associate-+l+ 99.6%

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

      9. metadata-eval 99.6%

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

      10. metadata-eval 99.6%

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

      11. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in alpha around 0 84.9%

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

      1. +-commutative 84.9%

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

   7. Simplified 84.9%

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

   8. Taylor expanded in alpha around 0 65.4%

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

      1. +-commutative 65.4%

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

      2. +-commutative 65.4%

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

   10. Simplified 65.4%

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

   if 1.2e15 < beta

   1. Initial program 90.8%

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

   3. Taylor expanded in beta around inf 90.0%

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

      1. *-un-lft-identity 90.0%

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

      2. metadata-eval 90.0%

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

      3. associate-+l+ 90.0%

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

      4. metadata-eval 90.0%

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

      5. associate-+l+ 90.0%

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

   5. Applied egg-rr 90.0%

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

      1. *-lft-identity 90.0%

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

      2. +-commutative 90.0%

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

      3. +-commutative 90.0%

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

      4. +-commutative 90.0%

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

   7. Simplified 90.0%

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

      1. div-inv 90.0%

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

   9. Applied egg-rr 90.0%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.2 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 2}}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \alpha\right) \cdot \frac{1}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.4:\\ \;\;\;\;\frac{0.25}{\alpha + 3}\\ \mathbf{elif}\;\beta \leq 7 \cdot 10^{+159}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.4)
   (/ 0.25 (+ alpha 3.0))
   (if (<= beta 7e+159)
     (/ (/ 1.0 beta) (+ beta 3.0))
     (/ (/ alpha beta) (+ alpha (+ beta 3.0))))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.4) {
		tmp = 0.25 / (alpha + 3.0);
	} else if (beta <= 7e+159) {
		tmp = (1.0 / beta) / (beta + 3.0);
	} else {
		tmp = (alpha / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.4d0) then
        tmp = 0.25d0 / (alpha + 3.0d0)
    else if (beta <= 7d+159) then
        tmp = (1.0d0 / beta) / (beta + 3.0d0)
    else
        tmp = (alpha / beta) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.4) {
		tmp = 0.25 / (alpha + 3.0);
	} else if (beta <= 7e+159) {
		tmp = (1.0 / beta) / (beta + 3.0);
	} else {
		tmp = (alpha / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.4:
		tmp = 0.25 / (alpha + 3.0)
	elif beta <= 7e+159:
		tmp = (1.0 / beta) / (beta + 3.0)
	else:
		tmp = (alpha / beta) / (alpha + (beta + 3.0))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.4)
		tmp = Float64(0.25 / Float64(alpha + 3.0));
	elseif (beta <= 7e+159)
		tmp = Float64(Float64(1.0 / beta) / Float64(beta + 3.0));
	else
		tmp = Float64(Float64(alpha / beta) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.4)
		tmp = 0.25 / (alpha + 3.0);
	elseif (beta <= 7e+159)
		tmp = (1.0 / beta) / (beta + 3.0);
	else
		tmp = (alpha / beta) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.4], N[(0.25 / N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 7e+159], N[(N[(1.0 / beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if beta < 2.39999999999999991

   1. Initial program 99.9%

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

   3. Taylor expanded in beta around 0 99.0%

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

   4. Taylor expanded in alpha around 0 66.2%

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

   5. Taylor expanded in beta around 0 66.2%

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

   if 2.39999999999999991 < beta < 6.9999999999999999e159

   1. Initial program 97.2%

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

   3. Taylor expanded in beta around inf 83.7%

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

   4. Taylor expanded in alpha around 0 82.5%

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

      1. associate-/r* 82.3%

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

      2. +-commutative 82.3%

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

   6. Simplified 82.3%

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

   if 6.9999999999999999e159 < beta

   1. Initial program 84.7%

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

   3. Taylor expanded in beta around inf 96.1%

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

      1. *-un-lft-identity 96.1%

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

      2. metadata-eval 96.1%

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

      3. associate-+l+ 96.1%

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

      4. metadata-eval 96.1%

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

      5. associate-+l+ 96.1%

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

   5. Applied egg-rr 96.1%

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

      1. *-lft-identity 96.1%

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

      2. +-commutative 96.1%

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

      3. +-commutative 96.1%

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

      4. +-commutative 96.1%

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

   7. Simplified 96.1%

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

   8. Taylor expanded in alpha around inf 96.1%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.4:\\ \;\;\;\;\frac{0.25}{\alpha + 3}\\ \mathbf{elif}\;\beta \leq 7 \cdot 10^{+159}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 71.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.2:\\ \;\;\;\;\frac{0.25}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \mathbf{elif}\;\beta \leq 7 \cdot 10^{+159}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.2)
   (/ 0.25 (+ 1.0 (+ 2.0 (+ alpha beta))))
   (if (<= beta 7e+159)
     (/ (/ 1.0 beta) (+ beta 3.0))
     (/ (/ alpha beta) (+ alpha (+ beta 3.0))))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.2) {
		tmp = 0.25 / (1.0 + (2.0 + (alpha + beta)));
	} else if (beta <= 7e+159) {
		tmp = (1.0 / beta) / (beta + 3.0);
	} else {
		tmp = (alpha / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 4.2d0) then
        tmp = 0.25d0 / (1.0d0 + (2.0d0 + (alpha + beta)))
    else if (beta <= 7d+159) then
        tmp = (1.0d0 / beta) / (beta + 3.0d0)
    else
        tmp = (alpha / beta) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.2) {
		tmp = 0.25 / (1.0 + (2.0 + (alpha + beta)));
	} else if (beta <= 7e+159) {
		tmp = (1.0 / beta) / (beta + 3.0);
	} else {
		tmp = (alpha / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 4.2:
		tmp = 0.25 / (1.0 + (2.0 + (alpha + beta)))
	elif beta <= 7e+159:
		tmp = (1.0 / beta) / (beta + 3.0)
	else:
		tmp = (alpha / beta) / (alpha + (beta + 3.0))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.2)
		tmp = Float64(0.25 / Float64(1.0 + Float64(2.0 + Float64(alpha + beta))));
	elseif (beta <= 7e+159)
		tmp = Float64(Float64(1.0 / beta) / Float64(beta + 3.0));
	else
		tmp = Float64(Float64(alpha / beta) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4.2)
		tmp = 0.25 / (1.0 + (2.0 + (alpha + beta)));
	elseif (beta <= 7e+159)
		tmp = (1.0 / beta) / (beta + 3.0);
	else
		tmp = (alpha / beta) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 4.2], N[(0.25 / N[(1.0 + N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 7e+159], N[(N[(1.0 / beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if beta < 4.20000000000000018

   1. Initial program 99.9%

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

   3. Taylor expanded in beta around 0 99.0%

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

   4. Taylor expanded in alpha around 0 66.2%

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

   if 4.20000000000000018 < beta < 6.9999999999999999e159

   1. Initial program 97.2%

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

   3. Taylor expanded in beta around inf 83.7%

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

   4. Taylor expanded in alpha around 0 82.5%

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

      1. associate-/r* 82.3%

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

      2. +-commutative 82.3%

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

   6. Simplified 82.3%

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

   if 6.9999999999999999e159 < beta

   1. Initial program 84.7%

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

   3. Taylor expanded in beta around inf 96.1%

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

      1. *-un-lft-identity 96.1%

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

      2. metadata-eval 96.1%

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

      3. associate-+l+ 96.1%

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

      4. metadata-eval 96.1%

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

      5. associate-+l+ 96.1%

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

   5. Applied egg-rr 96.1%

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

      1. *-lft-identity 96.1%

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

      2. +-commutative 96.1%

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

      3. +-commutative 96.1%

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

      4. +-commutative 96.1%

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

   7. Simplified 96.1%

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

   8. Taylor expanded in alpha around inf 96.1%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.2:\\ \;\;\;\;\frac{0.25}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \mathbf{elif}\;\beta \leq 7 \cdot 10^{+159}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 72.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if beta < 4.20000000000000018

   1. Initial program 99.9%

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

   3. Taylor expanded in beta around 0 99.0%

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

   4. Taylor expanded in alpha around 0 66.2%

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

   if 4.20000000000000018 < beta

   1. Initial program 90.8%

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

   3. Taylor expanded in beta around inf 90.0%

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

      1. *-un-lft-identity 90.0%

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

      2. metadata-eval 90.0%

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

      3. associate-+l+ 90.0%

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

      4. metadata-eval 90.0%

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

      5. associate-+l+ 90.0%

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

   5. Applied egg-rr 90.0%

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

      1. *-lft-identity 90.0%

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

      2. +-commutative 90.0%

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

      3. +-commutative 90.0%

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

      4. +-commutative 90.0%

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

   7. Simplified 90.0%

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

      1. div-inv 90.0%

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

   9. Applied egg-rr 90.0%

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

4. Final simplification 74.5%

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

Alternative 12: 72.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.5:\\ \;\;\;\;\frac{0.25}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.5)
   (/ 0.25 (+ 1.0 (+ 2.0 (+ alpha beta))))
   (/ (/ (+ 1.0 alpha) beta) (+ alpha (+ beta 3.0)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.5) {
		tmp = 0.25 / (1.0 + (2.0 + (alpha + beta)));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 4.5d0) then
        tmp = 0.25d0 / (1.0d0 + (2.0d0 + (alpha + beta)))
    else
        tmp = ((1.0d0 + alpha) / beta) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.5) {
		tmp = 0.25 / (1.0 + (2.0 + (alpha + beta)));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 4.5:
		tmp = 0.25 / (1.0 + (2.0 + (alpha + beta)))
	else:
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.5)
		tmp = Float64(0.25 / Float64(1.0 + Float64(2.0 + Float64(alpha + beta))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4.5)
		tmp = 0.25 / (1.0 + (2.0 + (alpha + beta)));
	else
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 4.5], N[(0.25 / N[(1.0 + N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.5

   1. Initial program 99.9%

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

   3. Taylor expanded in beta around 0 99.0%

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

   4. Taylor expanded in alpha around 0 66.2%

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

   if 4.5 < beta

   1. Initial program 90.8%

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

   3. Taylor expanded in beta around inf 90.0%

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

      1. *-un-lft-identity 90.0%

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

      2. metadata-eval 90.0%

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

      3. associate-+l+ 90.0%

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

      4. metadata-eval 90.0%

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

      5. associate-+l+ 90.0%

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

   5. Applied egg-rr 90.0%

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

      1. *-lft-identity 90.0%

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

      2. +-commutative 90.0%

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

      3. +-commutative 90.0%

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

      4. +-commutative 90.0%

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

   7. Simplified 90.0%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.5:\\ \;\;\;\;\frac{0.25}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 70.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.4:\\ \;\;\;\;\frac{0.25}{\alpha + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.4) (/ 0.25 (+ alpha 3.0)) (/ 1.0 (* beta (+ beta 3.0)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.4) {
		tmp = 0.25 / (alpha + 3.0);
	} 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 <= 2.4d0) then
        tmp = 0.25d0 / (alpha + 3.0d0)
    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 <= 2.4) {
		tmp = 0.25 / (alpha + 3.0);
	} else {
		tmp = 1.0 / (beta * (beta + 3.0));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.4:
		tmp = 0.25 / (alpha + 3.0)
	else:
		tmp = 1.0 / (beta * (beta + 3.0))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.4)
		tmp = Float64(0.25 / Float64(alpha + 3.0));
	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 <= 2.4)
		tmp = 0.25 / (alpha + 3.0);
	else
		tmp = 1.0 / (beta * (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.4], N[(0.25 / N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(beta * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.39999999999999991

   1. Initial program 99.9%

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

   3. Taylor expanded in beta around 0 99.0%

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

   4. Taylor expanded in alpha around 0 66.2%

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

   5. Taylor expanded in beta around 0 66.2%

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

   if 2.39999999999999991 < beta

   1. Initial program 90.8%

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

   3. Taylor expanded in beta around inf 90.0%

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

   4. Taylor expanded in alpha around 0 84.6%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.4:\\ \;\;\;\;\frac{0.25}{\alpha + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 70.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.3:\\ \;\;\;\;\frac{0.25}{\alpha + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta + 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.3) (/ 0.25 (+ alpha 3.0)) (/ (/ 1.0 beta) (+ beta 3.0))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.3) {
		tmp = 0.25 / (alpha + 3.0);
	} 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 <= 2.3d0) then
        tmp = 0.25d0 / (alpha + 3.0d0)
    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 <= 2.3) {
		tmp = 0.25 / (alpha + 3.0);
	} else {
		tmp = (1.0 / beta) / (beta + 3.0);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.3:
		tmp = 0.25 / (alpha + 3.0)
	else:
		tmp = (1.0 / beta) / (beta + 3.0)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.3)
		tmp = Float64(0.25 / Float64(alpha + 3.0));
	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 <= 2.3)
		tmp = 0.25 / (alpha + 3.0);
	else
		tmp = (1.0 / beta) / (beta + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.3], N[(0.25 / N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.2999999999999998

   1. Initial program 99.9%

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

   3. Taylor expanded in beta around 0 99.0%

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

   4. Taylor expanded in alpha around 0 66.2%

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

   5. Taylor expanded in beta around 0 66.2%

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

   if 2.2999999999999998 < beta

   1. Initial program 90.8%

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

   3. Taylor expanded in beta around inf 90.0%

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

   4. Taylor expanded in alpha around 0 84.6%

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

      1. associate-/r* 84.5%

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

      2. +-commutative 84.5%

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

   6. Simplified 84.5%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.3:\\ \;\;\;\;\frac{0.25}{\alpha + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 46.1% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{0.16666666666666666}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.0) 0.08333333333333333 (/ 0.16666666666666666 beta)))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.0) {
		tmp = 0.08333333333333333;
	} else {
		tmp = 0.16666666666666666 / 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 = 0.08333333333333333d0
    else
        tmp = 0.16666666666666666d0 / beta
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.0) {
		tmp = 0.08333333333333333;
	} else {
		tmp = 0.16666666666666666 / beta;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.0:
		tmp = 0.08333333333333333
	else:
		tmp = 0.16666666666666666 / beta
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.0)
		tmp = 0.08333333333333333;
	else
		tmp = Float64(0.16666666666666666 / beta);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.0)
		tmp = 0.08333333333333333;
	else
		tmp = 0.16666666666666666 / beta;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.0], 0.08333333333333333, N[(0.16666666666666666 / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2

   1. Initial program 99.9%

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

   3. Simplified 93.0%

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

      1. associate-+r+ 93.0%

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

      2. fma-undefine 93.0%

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

      3. *-commutative 93.0%

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

      4. associate-+l+ 93.0%

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

      5. +-commutative 93.0%

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

      6. associate-+l+ 93.0%

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

      7. *-commutative 93.0%

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

      8. associate-*r* 92.9%

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

      9. associate-+r+ 92.9%

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

      10. +-commutative 92.9%

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

      11. associate-/l/ 99.6%

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

      12. clear-num 99.6%

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

      13. inv-pow 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. unpow-1 99.6%

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

      2. associate-/l* 99.6%

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

      3. +-commutative 99.6%

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

      4. +-commutative 99.6%

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

      5. +-commutative 99.6%

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

      6. +-commutative 99.6%

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

      7. fma-undefine 99.6%

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

      8. +-commutative 99.6%

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

      9. *-commutative 99.6%

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

      10. +-commutative 99.6%

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

      11. associate-+r+ 99.6%

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

      12. distribute-rgt1-in 99.6%

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

      13. +-commutative 99.6%

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

      14. +-commutative 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in beta around 0 99.0%

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

       1. associate-/l* 99.0%

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

   11. Simplified 99.0%

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

   12. Taylor expanded in alpha around 0 64.9%

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

       1. +-commutative 64.9%

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

   14. Simplified 64.9%

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

   15. Taylor expanded in beta around 0 64.9%

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

   if 2 < beta

   1. Initial program 90.8%

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

   3. Simplified 77.3%

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

      1. associate-+r+ 77.3%

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

      2. fma-undefine 77.3%

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

      3. *-commutative 77.3%

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

      4. associate-+l+ 77.3%

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

      5. +-commutative 77.3%

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

      6. associate-+l+ 77.3%

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

      7. *-commutative 77.3%

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

      8. associate-*r* 77.3%

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

      9. associate-+r+ 77.3%

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

      10. +-commutative 77.3%

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

      11. associate-/l/ 86.7%

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

      12. clear-num 86.7%

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

      13. inv-pow 86.7%

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

   6. Applied egg-rr 86.7%

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

      1. unpow-1 86.7%

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

      2. associate-/l* 90.9%

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

      3. +-commutative 90.9%

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

      4. +-commutative 90.9%

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

      5. +-commutative 90.9%

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

      6. +-commutative 90.9%

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

      7. fma-undefine 90.9%

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

      8. +-commutative 90.9%

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

      9. *-commutative 90.9%

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

      10. +-commutative 90.9%

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

      11. associate-+r+ 90.9%

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

      12. distribute-rgt1-in 90.9%

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

      13. +-commutative 90.9%

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

      14. +-commutative 90.9%

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

   8. Simplified 90.9%

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

   9. Taylor expanded in beta around 0 14.1%

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

       1. associate-/l* 14.1%

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

   11. Simplified 14.1%

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

   12. Taylor expanded in alpha around 0 7.2%

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

       1. +-commutative 7.2%

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

   14. Simplified 7.2%

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

   15. Taylor expanded in beta around inf 7.2%

       \[\leadsto \color{blue}{\frac{0.16666666666666666}{\beta}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{0.16666666666666666}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 46.2% 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 96.7%

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

3. Simplified 87.5%

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

   1. associate-+r+ 87.5%

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

   2. fma-undefine 87.5%

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

   3. *-commutative 87.5%

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

   4. associate-+l+ 87.5%

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

   5. +-commutative 87.5%

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

   6. associate-+l+ 87.5%

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

   7. *-commutative 87.5%

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

   8. associate-*r* 87.5%

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

   9. associate-+r+ 87.5%

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

   10. +-commutative 87.5%

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

   11. associate-/l/ 95.1%

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

   12. clear-num 95.1%

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

   13. inv-pow 95.1%

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

6. Applied egg-rr 95.1%

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

   1. unpow-1 95.1%

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

   2. associate-/l* 96.6%

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

   3. +-commutative 96.6%

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

   4. +-commutative 96.6%

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

   5. +-commutative 96.6%

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

   6. +-commutative 96.6%

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

   7. fma-undefine 96.6%

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

   8. +-commutative 96.6%

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

   9. *-commutative 96.6%

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

   10. +-commutative 96.6%

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

   11. associate-+r+ 96.6%

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

   12. distribute-rgt1-in 96.6%

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

   13. +-commutative 96.6%

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

   14. +-commutative 96.6%

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

8. Simplified 96.6%

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

9. Taylor expanded in beta around 0 69.5%

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

    1. associate-/l* 69.5%

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

11. Simplified 69.5%

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

12. Taylor expanded in alpha around 0 44.9%

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

    1. +-commutative 44.9%

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

14. Simplified 44.9%

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

15. Final simplification 44.9%

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

Alternative 17: 46.5% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.25}{\beta + 3} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.7%

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

3. Taylor expanded in beta around 0 69.4%

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

4. Taylor expanded in alpha around 0 44.8%

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

   1. +-commutative 44.8%

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

6. Simplified 44.8%

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

7. Final simplification 44.8%

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

Alternative 18: 45.2% accurate, 35.0× speedup

Math

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

FPCore

(FPCore (alpha beta) :precision binary64 0.08333333333333333)

C

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

Fortran

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

Java

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

Python

def code(alpha, beta):
	return 0.08333333333333333

Julia

function code(alpha, beta)
	return 0.08333333333333333
end

MATLAB

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

Wolfram

code[alpha_, beta_] := 0.08333333333333333

Derivation

1. Initial program 96.7%

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

3. Simplified 87.5%

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

   1. associate-+r+ 87.5%

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

   2. fma-undefine 87.5%

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

   3. *-commutative 87.5%

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

   4. associate-+l+ 87.5%

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

   5. +-commutative 87.5%

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

   6. associate-+l+ 87.5%

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

   7. *-commutative 87.5%

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

   8. associate-*r* 87.5%

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

   9. associate-+r+ 87.5%

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

   10. +-commutative 87.5%

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

   11. associate-/l/ 95.1%

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

   12. clear-num 95.1%

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

   13. inv-pow 95.1%

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

6. Applied egg-rr 95.1%

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

   1. unpow-1 95.1%

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

   2. associate-/l* 96.6%

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

   3. +-commutative 96.6%

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

   4. +-commutative 96.6%

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

   5. +-commutative 96.6%

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

   6. +-commutative 96.6%

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

   7. fma-undefine 96.6%

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

   8. +-commutative 96.6%

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

   9. *-commutative 96.6%

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

   10. +-commutative 96.6%

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

   11. associate-+r+ 96.6%

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

   12. distribute-rgt1-in 96.6%

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

   13. +-commutative 96.6%

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

   14. +-commutative 96.6%

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

8. Simplified 96.6%

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

9. Taylor expanded in beta around 0 69.5%

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

    1. associate-/l* 69.5%

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

11. Simplified 69.5%

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

12. Taylor expanded in alpha around 0 44.9%

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

    1. +-commutative 44.9%

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

14. Simplified 44.9%

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

15. Taylor expanded in beta around 0 43.8%

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

16. Final simplification 43.8%

    \[\leadsto 0.08333333333333333 \]
17. Add Preprocessing

Reproduce

herbie shell --seed 2024059 
(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)))