Octave 3.8, jcobi/3

Percentage Accurate: 94.4% → 99.8%

Time: 20.7s

Alternatives: 20

Speedup: 1.4×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.4% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.3%

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

3. Simplified 95.3%

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

   1. clear-num 95.2%

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

   2. inv-pow 95.2%

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

6. Applied egg-rr 95.2%

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

   1. unpow-1 95.2%

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

   2. associate-/l* 99.2%

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

   3. +-commutative 99.2%

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

   4. +-commutative 99.2%

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

   5. +-commutative 99.2%

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

8. Simplified 99.2%

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

   1. +-commutative 99.2%

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

   2. +-commutative 99.2%

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

   3. inv-pow 99.2%

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

   4. associate-/r/ 99.2%

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

   5. unpow-prod-down 99.7%

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

   6. inv-pow 99.7%

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

   7. clear-num 99.7%

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

   8. inv-pow 99.7%

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

   9. div-inv 99.8%

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

   10. *-commutative 99.8%

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

   11. associate-*l/ 99.8%

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

10. Applied egg-rr 99.8%

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

11. Final simplification 99.8%

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

Alternative 2: 94.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 3\right)\\ t_1 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 125000000:\\ \;\;\;\;\frac{-1 - \alpha}{t_1} \cdot \frac{-1 - \beta}{t_1 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{t_1} \cdot \left(1 - \frac{1 + \alpha}{\beta}\right)}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 3.0))) (t_1 (+ alpha (+ beta 2.0))))
   (if (<= beta 125000000.0)
     (* (/ (- -1.0 alpha) t_1) (/ (- -1.0 beta) (* t_1 t_0)))
     (/ (* (/ (+ 1.0 alpha) t_1) (- 1.0 (/ (+ 1.0 alpha) beta))) t_0))))

C

double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 3.0);
	double t_1 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 125000000.0) {
		tmp = ((-1.0 - alpha) / t_1) * ((-1.0 - beta) / (t_1 * t_0));
	} else {
		tmp = (((1.0 + alpha) / t_1) * (1.0 - ((1.0 + alpha) / beta))) / t_0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = alpha + (beta + 3.0d0)
    t_1 = alpha + (beta + 2.0d0)
    if (beta <= 125000000.0d0) then
        tmp = (((-1.0d0) - alpha) / t_1) * (((-1.0d0) - beta) / (t_1 * t_0))
    else
        tmp = (((1.0d0 + alpha) / t_1) * (1.0d0 - ((1.0d0 + alpha) / beta))) / t_0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 3.0);
	double t_1 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 125000000.0) {
		tmp = ((-1.0 - alpha) / t_1) * ((-1.0 - beta) / (t_1 * t_0));
	} else {
		tmp = (((1.0 + alpha) / t_1) * (1.0 - ((1.0 + alpha) / beta))) / t_0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = alpha + (beta + 3.0)
	t_1 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 125000000.0:
		tmp = ((-1.0 - alpha) / t_1) * ((-1.0 - beta) / (t_1 * t_0))
	else:
		tmp = (((1.0 + alpha) / t_1) * (1.0 - ((1.0 + alpha) / beta))) / t_0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 3.0))
	t_1 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 125000000.0)
		tmp = Float64(Float64(Float64(-1.0 - alpha) / t_1) * Float64(Float64(-1.0 - beta) / Float64(t_1 * t_0)));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + alpha) / t_1) * Float64(1.0 - Float64(Float64(1.0 + alpha) / beta))) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 3.0);
	t_1 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 125000000.0)
		tmp = ((-1.0 - alpha) / t_1) * ((-1.0 - beta) / (t_1 * t_0));
	else
		tmp = (((1.0 + alpha) / t_1) * (1.0 - ((1.0 + alpha) / beta))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 125000000.0], N[(N[(N[(-1.0 - alpha), $MachinePrecision] / t$95$1), $MachinePrecision] * N[(N[(-1.0 - beta), $MachinePrecision] / N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$1), $MachinePrecision] * N[(1.0 - N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.25e8

   1. Initial program 99.8%

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

   3. Simplified 99.4%

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

   if 1.25e8 < beta

   1. Initial program 76.6%

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

   3. Simplified 84.6%

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

      1. clear-num 84.5%

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

      2. inv-pow 84.5%

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

   6. Applied egg-rr 84.5%

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

      1. unpow-1 84.5%

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

      2. associate-/l* 98.5%

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

      3. +-commutative 98.5%

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

      4. +-commutative 98.5%

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

      5. +-commutative 98.5%

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

   8. Simplified 98.5%

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

      1. +-commutative 98.5%

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

      2. +-commutative 98.5%

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

      3. inv-pow 98.5%

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

      4. associate-/r/ 98.6%

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

      5. unpow-prod-down 99.6%

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

      6. inv-pow 99.6%

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

      7. clear-num 99.5%

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

      8. inv-pow 99.5%

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

      9. div-inv 99.6%

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

      10. *-commutative 99.6%

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

      11. associate-*l/ 99.8%

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

   10. Applied egg-rr 99.8%

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

   11. Taylor expanded in beta around inf 69.3%

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

       1. associate-*r/ 69.3%

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

       2. neg-mul-1 69.3%

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

       3. distribute-neg-in 69.3%

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

       4. metadata-eval 69.3%

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

       5. unsub-neg 69.3%

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

   13. Simplified 69.3%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 125000000:\\ \;\;\;\;\frac{-1 - \alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{-1 - \beta}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \left(1 - \frac{1 + \alpha}{\beta}\right)}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 76000000:\\ \;\;\;\;\frac{1}{\beta + 2} \cdot \frac{1}{\frac{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}{1 + \beta}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \left(1 - \frac{1 + \alpha}{\beta}\right)}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 76000000.0)
   (*
    (/ 1.0 (+ beta 2.0))
    (/ 1.0 (/ (* (+ beta 3.0) (+ beta 2.0)) (+ 1.0 beta))))
   (/
    (* (/ (+ 1.0 alpha) (+ alpha (+ beta 2.0))) (- 1.0 (/ (+ 1.0 alpha) beta)))
    (+ alpha (+ beta 3.0)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 76000000.0) {
		tmp = (1.0 / (beta + 2.0)) * (1.0 / (((beta + 3.0) * (beta + 2.0)) / (1.0 + beta)));
	} else {
		tmp = (((1.0 + alpha) / (alpha + (beta + 2.0))) * (1.0 - ((1.0 + alpha) / beta))) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 76000000.0d0) then
        tmp = (1.0d0 / (beta + 2.0d0)) * (1.0d0 / (((beta + 3.0d0) * (beta + 2.0d0)) / (1.0d0 + beta)))
    else
        tmp = (((1.0d0 + alpha) / (alpha + (beta + 2.0d0))) * (1.0d0 - ((1.0d0 + alpha) / beta))) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 76000000.0) {
		tmp = (1.0 / (beta + 2.0)) * (1.0 / (((beta + 3.0) * (beta + 2.0)) / (1.0 + beta)));
	} else {
		tmp = (((1.0 + alpha) / (alpha + (beta + 2.0))) * (1.0 - ((1.0 + alpha) / beta))) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 76000000.0:
		tmp = (1.0 / (beta + 2.0)) * (1.0 / (((beta + 3.0) * (beta + 2.0)) / (1.0 + beta)))
	else:
		tmp = (((1.0 + alpha) / (alpha + (beta + 2.0))) * (1.0 - ((1.0 + alpha) / beta))) / (alpha + (beta + 3.0))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 76000000.0)
		tmp = Float64(Float64(1.0 / Float64(beta + 2.0)) * Float64(1.0 / Float64(Float64(Float64(beta + 3.0) * Float64(beta + 2.0)) / Float64(1.0 + beta))));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(beta + 2.0))) * Float64(1.0 - Float64(Float64(1.0 + alpha) / beta))) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 76000000.0)
		tmp = (1.0 / (beta + 2.0)) * (1.0 / (((beta + 3.0) * (beta + 2.0)) / (1.0 + beta)));
	else
		tmp = (((1.0 + alpha) / (alpha + (beta + 2.0))) * (1.0 - ((1.0 + alpha) / beta))) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 76000000.0], N[(N[(1.0 / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[(N[(beta + 3.0), $MachinePrecision] * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 7.6e7

   1. Initial program 99.8%

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

   3. Simplified 99.4%

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

      1. clear-num 99.4%

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

      2. inv-pow 99.4%

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

   6. Applied egg-rr 99.4%

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

      1. unpow-1 99.4%

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

      2. associate-/l* 99.4%

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

      3. +-commutative 99.4%

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

      4. +-commutative 99.4%

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

      5. +-commutative 99.4%

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

   8. Simplified 99.4%

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

   9. Taylor expanded in alpha around 0 85.8%

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

   10. Taylor expanded in alpha around 0 66.4%

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

       1. +-commutative 66.4%

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

       2. +-commutative 66.4%

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

   12. Simplified 66.4%

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

   if 7.6e7 < beta

   1. Initial program 76.6%

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

   3. Simplified 84.6%

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

      1. clear-num 84.5%

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

      2. inv-pow 84.5%

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

   6. Applied egg-rr 84.5%

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

      1. unpow-1 84.5%

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

      2. associate-/l* 98.5%

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

      3. +-commutative 98.5%

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

      4. +-commutative 98.5%

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

      5. +-commutative 98.5%

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

   8. Simplified 98.5%

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

      1. +-commutative 98.5%

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

      2. +-commutative 98.5%

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

      3. inv-pow 98.5%

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

      4. associate-/r/ 98.6%

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

      5. unpow-prod-down 99.6%

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

      6. inv-pow 99.6%

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

      7. clear-num 99.5%

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

      8. inv-pow 99.5%

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

      9. div-inv 99.6%

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

      10. *-commutative 99.6%

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

      11. associate-*l/ 99.8%

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

   10. Applied egg-rr 99.8%

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

   11. Taylor expanded in beta around inf 69.3%

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

       1. associate-*r/ 69.3%

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

       2. neg-mul-1 69.3%

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

       3. distribute-neg-in 69.3%

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

       4. metadata-eval 69.3%

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

       5. unsub-neg 69.3%

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

   13. Simplified 69.3%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 76000000:\\ \;\;\;\;\frac{1}{\beta + 2} \cdot \frac{1}{\frac{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}{1 + \beta}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \left(1 - \frac{1 + \alpha}{\beta}\right)}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 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 93.3%

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

3. Simplified 95.3%

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

   1. clear-num 95.3%

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

   2. associate-+r+ 95.3%

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

   3. *-commutative 95.3%

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

   4. frac-times 90.3%

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

   5. *-un-lft-identity 90.3%

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

   6. +-commutative 90.3%

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

   7. *-commutative 90.3%

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

   8. associate-+r+ 90.3%

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

6. Applied egg-rr 90.3%

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

   1. associate-/r* 95.3%

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

   2. associate-/l* 91.1%

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

   3. associate-*l/ 95.3%

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

   4. *-commutative 95.3%

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

   5. times-frac 99.8%

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

   6. associate-/r* 95.3%

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

   7. *-commutative 95.3%

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

   8. associate-/r* 99.8%

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

   9. +-commutative 99.8%

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

   10. +-commutative 99.8%

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

   11. +-commutative 99.8%

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

   12. +-commutative 99.8%

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

8. Simplified 99.8%

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

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 5: 72.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.15 \cdot 10^{+15}:\\ \;\;\;\;\frac{1}{\beta + 2} \cdot \frac{1}{\frac{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}{1 + \beta}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.15e+15)
   (*
    (/ 1.0 (+ beta 2.0))
    (/ 1.0 (/ (* (+ beta 3.0) (+ beta 2.0)) (+ 1.0 beta))))
   (/ (/ (+ 1.0 alpha) (+ alpha (+ beta 2.0))) (+ alpha (+ beta 3.0)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.15e+15) {
		tmp = (1.0 / (beta + 2.0)) * (1.0 / (((beta + 3.0) * (beta + 2.0)) / (1.0 + beta)));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.15d+15) then
        tmp = (1.0d0 / (beta + 2.0d0)) * (1.0d0 / (((beta + 3.0d0) * (beta + 2.0d0)) / (1.0d0 + beta)))
    else
        tmp = ((1.0d0 + alpha) / (alpha + (beta + 2.0d0))) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.15e+15) {
		tmp = (1.0 / (beta + 2.0)) * (1.0 / (((beta + 3.0) * (beta + 2.0)) / (1.0 + beta)));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 1.15e+15:
		tmp = (1.0 / (beta + 2.0)) * (1.0 / (((beta + 3.0) * (beta + 2.0)) / (1.0 + beta)))
	else:
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) / (alpha + (beta + 3.0))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.15e+15)
		tmp = Float64(Float64(1.0 / Float64(beta + 2.0)) * Float64(1.0 / Float64(Float64(Float64(beta + 3.0) * Float64(beta + 2.0)) / Float64(1.0 + beta))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(beta + 2.0))) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.15e+15)
		tmp = (1.0 / (beta + 2.0)) * (1.0 / (((beta + 3.0) * (beta + 2.0)) / (1.0 + beta)));
	else
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 1.15e+15], N[(N[(1.0 / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[(N[(beta + 3.0), $MachinePrecision] * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.15e15

   1. Initial program 99.8%

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

   3. Simplified 99.4%

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

      1. clear-num 99.4%

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

      2. inv-pow 99.4%

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

   6. Applied egg-rr 99.4%

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

      1. unpow-1 99.4%

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

      2. associate-/l* 99.4%

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

      3. +-commutative 99.4%

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

      4. +-commutative 99.4%

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

      5. +-commutative 99.4%

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

   8. Simplified 99.4%

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

   9. Taylor expanded in alpha around 0 85.8%

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

   10. Taylor expanded in alpha around 0 66.4%

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

       1. +-commutative 66.4%

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

       2. +-commutative 66.4%

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

   12. Simplified 66.4%

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

   if 1.15e15 < beta

   1. Initial program 76.6%

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

   3. Simplified 84.6%

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

      1. clear-num 84.5%

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

      2. inv-pow 84.5%

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

   6. Applied egg-rr 84.5%

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

      1. unpow-1 84.5%

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

      2. associate-/l* 98.5%

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

      3. +-commutative 98.5%

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

      4. +-commutative 98.5%

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

      5. +-commutative 98.5%

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

   8. Simplified 98.5%

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

      1. +-commutative 98.5%

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

      2. +-commutative 98.5%

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

      3. inv-pow 98.5%

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

      4. associate-/r/ 98.6%

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

      5. unpow-prod-down 99.6%

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

      6. inv-pow 99.6%

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

      7. clear-num 99.5%

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

      8. inv-pow 99.5%

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

      9. div-inv 99.6%

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

      10. *-commutative 99.6%

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

      11. associate-*l/ 99.8%

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

   10. Applied egg-rr 99.8%

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

   11. Taylor expanded in beta around inf 70.7%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.15 \cdot 10^{+15}:\\ \;\;\;\;\frac{1}{\beta + 2} \cdot \frac{1}{\frac{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}{1 + \beta}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 10^{+39}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 2}}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{1}{\left(\beta + 4\right) + \alpha \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1e+39)
   (/ (/ (+ 1.0 beta) (+ beta 2.0)) (* (+ beta 3.0) (+ beta 2.0)))
   (*
    (/ (+ 1.0 alpha) (+ alpha (+ beta 2.0)))
    (/ 1.0 (+ (+ beta 4.0) (* alpha 2.0))))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1e+39) {
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 3.0) * (beta + 2.0));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) * (1.0 / ((beta + 4.0) + (alpha * 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1d+39) then
        tmp = ((1.0d0 + beta) / (beta + 2.0d0)) / ((beta + 3.0d0) * (beta + 2.0d0))
    else
        tmp = ((1.0d0 + alpha) / (alpha + (beta + 2.0d0))) * (1.0d0 / ((beta + 4.0d0) + (alpha * 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1e+39) {
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 3.0) * (beta + 2.0));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) * (1.0 / ((beta + 4.0) + (alpha * 2.0)));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 1e+39:
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 3.0) * (beta + 2.0))
	else:
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) * (1.0 / ((beta + 4.0) + (alpha * 2.0)))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1e+39)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(beta + 2.0)) / Float64(Float64(beta + 3.0) * Float64(beta + 2.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(beta + 2.0))) * Float64(1.0 / Float64(Float64(beta + 4.0) + Float64(alpha * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1e+39)
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 3.0) * (beta + 2.0));
	else
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) * (1.0 / ((beta + 4.0) + (alpha * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 1e+39], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + 3.0), $MachinePrecision] * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[(beta + 4.0), $MachinePrecision] + N[(alpha * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 9.9999999999999994e38

   1. Initial program 99.8%

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

      1. associate-/l/ 99.4%

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

      2. +-commutative 99.4%

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

      3. +-commutative 99.4%

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

      4. associate-+r+ 99.4%

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

      5. *-commutative 99.4%

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

      6. metadata-eval 99.4%

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

      7. associate-+l+ 99.4%

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

      8. metadata-eval 99.4%

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

      9. associate-+l+ 99.4%

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

      10. metadata-eval 99.4%

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

      11. metadata-eval 99.4%

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

      12. associate-+l+ 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in alpha around 0 85.1%

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

   6. Taylor expanded in alpha around 0 65.6%

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

   if 9.9999999999999994e38 < beta

   1. Initial program 75.2%

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

   3. Simplified 83.8%

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

      1. clear-num 83.7%

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

      2. inv-pow 83.7%

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

   6. Applied egg-rr 83.7%

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

      1. unpow-1 83.7%

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

      2. associate-/l* 98.5%

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

      3. +-commutative 98.5%

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

      4. +-commutative 98.5%

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

      5. +-commutative 98.5%

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

   8. Simplified 98.5%

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

   9. Taylor expanded in beta around inf 73.0%

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

       1. associate-+r+ 73.0%

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

   11. Simplified 73.0%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 10^{+39}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 2}}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{1}{\left(\beta + 4\right) + \alpha \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.4:\\ \;\;\;\;\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(6 + \beta \cdot 5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 - \alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{-1}{\beta + 4}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.4)
   (/ (+ 1.0 beta) (* (+ beta 2.0) (+ 6.0 (* beta 5.0))))
   (* (/ (- -1.0 alpha) (+ alpha (+ beta 2.0))) (/ -1.0 (+ beta 4.0)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.4) {
		tmp = (1.0 + beta) / ((beta + 2.0) * (6.0 + (beta * 5.0)));
	} else {
		tmp = ((-1.0 - alpha) / (alpha + (beta + 2.0))) * (-1.0 / (beta + 4.0));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.4d0) then
        tmp = (1.0d0 + beta) / ((beta + 2.0d0) * (6.0d0 + (beta * 5.0d0)))
    else
        tmp = (((-1.0d0) - alpha) / (alpha + (beta + 2.0d0))) * ((-1.0d0) / (beta + 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.4) {
		tmp = (1.0 + beta) / ((beta + 2.0) * (6.0 + (beta * 5.0)));
	} else {
		tmp = ((-1.0 - alpha) / (alpha + (beta + 2.0))) * (-1.0 / (beta + 4.0));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 1.4:
		tmp = (1.0 + beta) / ((beta + 2.0) * (6.0 + (beta * 5.0)))
	else:
		tmp = ((-1.0 - alpha) / (alpha + (beta + 2.0))) * (-1.0 / (beta + 4.0))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.4)
		tmp = Float64(Float64(1.0 + beta) / Float64(Float64(beta + 2.0) * Float64(6.0 + Float64(beta * 5.0))));
	else
		tmp = Float64(Float64(Float64(-1.0 - alpha) / Float64(alpha + Float64(beta + 2.0))) * Float64(-1.0 / Float64(beta + 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.4)
		tmp = (1.0 + beta) / ((beta + 2.0) * (6.0 + (beta * 5.0)));
	else
		tmp = ((-1.0 - alpha) / (alpha + (beta + 2.0))) * (-1.0 / (beta + 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 1.4], N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(6.0 + N[(beta * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-1.0 - alpha), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[(beta + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.3999999999999999

   1. Initial program 99.8%

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

   3. Simplified 94.9%

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

   5. Taylor expanded in beta around 0 94.6%

      \[\leadsto \frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \color{blue}{\left(\beta \cdot \left(5 + 2 \cdot \alpha\right) + \left(2 + \alpha\right) \cdot \left(3 + \alpha\right)\right)}} \]

   6. Taylor expanded in alpha around 0 65.9%

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

      1. +-commutative 65.9%

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

      2. *-commutative 65.9%

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

   8. Simplified 65.9%

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

   if 1.3999999999999999 < beta

   1. Initial program 76.9%

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

   3. Simplified 84.8%

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

      1. clear-num 84.8%

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

      2. inv-pow 84.8%

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

   6. Applied egg-rr 84.8%

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

      1. unpow-1 84.8%

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

      2. associate-/l* 98.6%

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

      3. +-commutative 98.6%

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

      4. +-commutative 98.6%

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

      5. +-commutative 98.6%

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

   8. Simplified 98.6%

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

   9. Taylor expanded in beta around inf 70.5%

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

       1. associate-+r+ 70.5%

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

   11. Simplified 70.5%

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

   12. Taylor expanded in alpha around 0 69.7%

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

       1. +-commutative 69.7%

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

   14. Simplified 69.7%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.4:\\ \;\;\;\;\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(6 + \beta \cdot 5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 - \alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{-1}{\beta + 4}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.6 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 2}}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.6e+16)
   (/ (/ (+ 1.0 beta) (+ beta 2.0)) (* (+ beta 3.0) (+ beta 2.0)))
   (/ (/ (+ 1.0 alpha) beta) (+ alpha (+ beta 2.0)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.6e+16) {
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 3.0) * (beta + 2.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.6d+16) then
        tmp = ((1.0d0 + beta) / (beta + 2.0d0)) / ((beta + 3.0d0) * (beta + 2.0d0))
    else
        tmp = ((1.0d0 + alpha) / beta) / (alpha + (beta + 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.6e+16) {
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 3.0) * (beta + 2.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 2.0));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 1.6e+16:
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 3.0) * (beta + 2.0))
	else:
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 2.0))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.6e+16)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(beta + 2.0)) / Float64(Float64(beta + 3.0) * Float64(beta + 2.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(alpha + Float64(beta + 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.6e+16)
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 3.0) * (beta + 2.0));
	else
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 1.6e+16], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + 3.0), $MachinePrecision] * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.6e16

   1. Initial program 99.8%

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

      1. associate-/l/ 99.4%

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

      2. +-commutative 99.4%

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

      3. +-commutative 99.4%

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

      4. associate-+r+ 99.4%

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

      5. *-commutative 99.4%

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

      6. metadata-eval 99.4%

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

      7. associate-+l+ 99.4%

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

      8. metadata-eval 99.4%

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

      9. associate-+l+ 99.4%

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

      10. metadata-eval 99.4%

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

      11. metadata-eval 99.4%

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

      12. associate-+l+ 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in alpha around 0 85.8%

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

   6. Taylor expanded in alpha around 0 66.4%

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

   if 1.6e16 < beta

   1. Initial program 76.6%

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

   3. Simplified 84.6%

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

   5. Taylor expanded in beta around inf 69.7%

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

      1. associate-*l/ 69.8%

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

   7. Applied egg-rr 69.8%

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

      1. associate-*r/ 69.8%

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

      2. *-rgt-identity 69.8%

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

      3. +-commutative 69.8%

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

      4. +-commutative 69.8%

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

      5. +-commutative 69.8%

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

      6. +-commutative 69.8%

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

   9. Simplified 69.8%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.6 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 2}}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.15 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 2}}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.15e+15)
   (/ (/ (+ 1.0 beta) (+ beta 2.0)) (* (+ beta 3.0) (+ beta 2.0)))
   (/ (/ (+ 1.0 alpha) (+ alpha (+ beta 2.0))) (+ alpha (+ beta 3.0)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.15e+15) {
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 3.0) * (beta + 2.0));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.15d+15) then
        tmp = ((1.0d0 + beta) / (beta + 2.0d0)) / ((beta + 3.0d0) * (beta + 2.0d0))
    else
        tmp = ((1.0d0 + alpha) / (alpha + (beta + 2.0d0))) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.15e+15) {
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 3.0) * (beta + 2.0));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.15e+15:
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 3.0) * (beta + 2.0))
	else:
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) / (alpha + (beta + 3.0))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.15e+15)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(beta + 2.0)) / Float64(Float64(beta + 3.0) * Float64(beta + 2.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(beta + 2.0))) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.15e+15)
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 3.0) * (beta + 2.0));
	else
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.15e+15], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + 3.0), $MachinePrecision] * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.15e15

   1. Initial program 99.8%

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

      1. associate-/l/ 99.4%

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

      2. +-commutative 99.4%

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

      3. +-commutative 99.4%

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

      4. associate-+r+ 99.4%

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

      5. *-commutative 99.4%

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

      6. metadata-eval 99.4%

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

      7. associate-+l+ 99.4%

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

      8. metadata-eval 99.4%

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

      9. associate-+l+ 99.4%

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

      10. metadata-eval 99.4%

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

      11. metadata-eval 99.4%

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

      12. associate-+l+ 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in alpha around 0 85.8%

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

   6. Taylor expanded in alpha around 0 66.4%

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

   if 2.15e15 < beta

   1. Initial program 76.6%

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

   3. Simplified 84.6%

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

      1. clear-num 84.5%

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

      2. inv-pow 84.5%

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

   6. Applied egg-rr 84.5%

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

      1. unpow-1 84.5%

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

      2. associate-/l* 98.5%

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

      3. +-commutative 98.5%

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

      4. +-commutative 98.5%

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

      5. +-commutative 98.5%

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

   8. Simplified 98.5%

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

      1. +-commutative 98.5%

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

      2. +-commutative 98.5%

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

      3. inv-pow 98.5%

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

      4. associate-/r/ 98.6%

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

      5. unpow-prod-down 99.6%

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

      6. inv-pow 99.6%

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

      7. clear-num 99.5%

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

      8. inv-pow 99.5%

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

      9. div-inv 99.6%

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

      10. *-commutative 99.6%

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

      11. associate-*l/ 99.8%

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

   10. Applied egg-rr 99.8%

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

   11. Taylor expanded in beta around inf 70.7%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.15 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 2}}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 71.2% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if beta < 5

   1. Initial program 99.8%

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

   3. Simplified 94.9%

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

   5. Taylor expanded in beta around 0 94.6%

      \[\leadsto \frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \color{blue}{\left(\beta \cdot \left(5 + 2 \cdot \alpha\right) + \left(2 + \alpha\right) \cdot \left(3 + \alpha\right)\right)}} \]

   6. Taylor expanded in alpha around 0 65.9%

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

      1. +-commutative 65.9%

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

      2. *-commutative 65.9%

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

   8. Simplified 65.9%

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

   if 5 < beta

   1. Initial program 76.9%

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

   3. Simplified 84.8%

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

   5. Taylor expanded in beta around inf 69.3%

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

      1. associate-*l/ 69.4%

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

   7. Applied egg-rr 69.4%

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

      1. associate-*r/ 69.4%

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

      2. *-rgt-identity 69.4%

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

      3. +-commutative 69.4%

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

      4. +-commutative 69.4%

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

      5. +-commutative 69.4%

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

      6. +-commutative 69.4%

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

   9. Simplified 69.4%

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

4. Final simplification 66.9%

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

Alternative 11: 82.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 14.5:\\ \;\;\;\;\frac{0.5}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\beta}{1 + \alpha}} \cdot \frac{1}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 14.5)
   (/ 0.5 (* (+ alpha 2.0) (+ alpha 3.0)))
   (* (/ 1.0 (/ beta (+ 1.0 alpha))) (/ 1.0 beta))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 14.5) {
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	} else {
		tmp = (1.0 / (beta / (1.0 + alpha))) * (1.0 / beta);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 14.5d0) then
        tmp = 0.5d0 / ((alpha + 2.0d0) * (alpha + 3.0d0))
    else
        tmp = (1.0d0 / (beta / (1.0d0 + alpha))) * (1.0d0 / beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 14.5) {
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	} else {
		tmp = (1.0 / (beta / (1.0 + alpha))) * (1.0 / beta);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 14.5:
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0))
	else:
		tmp = (1.0 / (beta / (1.0 + alpha))) * (1.0 / beta)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 14.5)
		tmp = Float64(0.5 / Float64(Float64(alpha + 2.0) * Float64(alpha + 3.0)));
	else
		tmp = Float64(Float64(1.0 / Float64(beta / Float64(1.0 + alpha))) * Float64(1.0 / beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 14.5)
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	else
		tmp = (1.0 / (beta / (1.0 + alpha))) * (1.0 / beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 14.5], N[(0.5 / N[(N[(alpha + 2.0), $MachinePrecision] * N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(beta / N[(1.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 14.5

   1. Initial program 99.8%

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

   3. Simplified 99.4%

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

      1. clear-num 99.4%

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

      2. inv-pow 99.4%

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

   6. Applied egg-rr 99.4%

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

      1. unpow-1 99.4%

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

      2. associate-/l* 99.4%

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

      3. +-commutative 99.4%

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

      4. +-commutative 99.4%

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

      5. +-commutative 99.4%

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

   8. Simplified 99.4%

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

   9. Taylor expanded in alpha around 0 85.7%

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

   10. Taylor expanded in beta around 0 84.6%

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

   if 14.5 < beta

   1. Initial program 76.9%

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

   3. Simplified 84.8%

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

   5. Taylor expanded in beta around inf 69.3%

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

      1. clear-num 69.3%

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

      2. inv-pow 69.3%

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

      3. +-commutative 69.3%

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

   7. Applied egg-rr 69.3%

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

      1. unpow-1 69.3%

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

      2. +-commutative 69.3%

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

   9. Simplified 69.3%

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

   10. Taylor expanded in beta around inf 68.9%

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

       1. +-commutative 68.9%

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

   12. Simplified 68.9%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 14.5:\\ \;\;\;\;\frac{0.5}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\beta}{1 + \alpha}} \cdot \frac{1}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 83.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 23.5:\\ \;\;\;\;\frac{0.5}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 23.5)
   (/ 0.5 (* (+ alpha 2.0) (+ alpha 3.0)))
   (/ (/ (+ 1.0 alpha) beta) (+ alpha (+ beta 2.0)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 23.5) {
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 23.5d0) then
        tmp = 0.5d0 / ((alpha + 2.0d0) * (alpha + 3.0d0))
    else
        tmp = ((1.0d0 + alpha) / beta) / (alpha + (beta + 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 23.5) {
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 2.0));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 23.5:
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0))
	else:
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 2.0))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 23.5)
		tmp = Float64(0.5 / Float64(Float64(alpha + 2.0) * Float64(alpha + 3.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(alpha + Float64(beta + 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 23.5)
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	else
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 23.5], N[(0.5 / N[(N[(alpha + 2.0), $MachinePrecision] * N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 23.5

   1. Initial program 99.8%

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

   3. Simplified 99.4%

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

      1. clear-num 99.4%

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

      2. inv-pow 99.4%

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

   6. Applied egg-rr 99.4%

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

      1. unpow-1 99.4%

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

      2. associate-/l* 99.4%

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

      3. +-commutative 99.4%

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

      4. +-commutative 99.4%

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

      5. +-commutative 99.4%

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

   8. Simplified 99.4%

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

   9. Taylor expanded in alpha around 0 85.7%

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

   10. Taylor expanded in beta around 0 84.6%

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

   if 23.5 < beta

   1. Initial program 76.9%

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

   3. Simplified 84.8%

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

   5. Taylor expanded in beta around inf 69.3%

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

      1. associate-*l/ 69.4%

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

   7. Applied egg-rr 69.4%

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

      1. associate-*r/ 69.4%

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

      2. *-rgt-identity 69.4%

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

      3. +-commutative 69.4%

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

      4. +-commutative 69.4%

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

      5. +-commutative 69.4%

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

      6. +-commutative 69.4%

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

   9. Simplified 69.4%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 23.5:\\ \;\;\;\;\frac{0.5}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 82.9% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.8:\\ \;\;\;\;\frac{0.5}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta} \cdot \frac{1}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5.8)
   (/ 0.5 (* (+ alpha 2.0) (+ alpha 3.0)))
   (* (/ (+ 1.0 alpha) beta) (/ 1.0 beta))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.8) {
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	} else {
		tmp = ((1.0 + alpha) / beta) * (1.0 / beta);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 5.8d0) then
        tmp = 0.5d0 / ((alpha + 2.0d0) * (alpha + 3.0d0))
    else
        tmp = ((1.0d0 + alpha) / beta) * (1.0d0 / beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.8) {
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	} else {
		tmp = ((1.0 + alpha) / beta) * (1.0 / beta);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 5.8:
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0))
	else:
		tmp = ((1.0 + alpha) / beta) * (1.0 / beta)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 5.8)
		tmp = Float64(0.5 / Float64(Float64(alpha + 2.0) * Float64(alpha + 3.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) * Float64(1.0 / beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 5.8)
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	else
		tmp = ((1.0 + alpha) / beta) * (1.0 / beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 5.8], N[(0.5 / N[(N[(alpha + 2.0), $MachinePrecision] * N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] * N[(1.0 / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.79999999999999982

   1. Initial program 99.8%

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

   3. Simplified 99.4%

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

      1. clear-num 99.4%

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

      2. inv-pow 99.4%

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

   6. Applied egg-rr 99.4%

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

      1. unpow-1 99.4%

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

      2. associate-/l* 99.4%

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

      3. +-commutative 99.4%

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

      4. +-commutative 99.4%

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

      5. +-commutative 99.4%

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

   8. Simplified 99.4%

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

   9. Taylor expanded in alpha around 0 85.7%

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

   10. Taylor expanded in beta around 0 84.6%

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

   if 5.79999999999999982 < beta

   1. Initial program 76.9%

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

   3. Simplified 84.8%

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

   5. Taylor expanded in beta around inf 69.3%

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

   6. Taylor expanded in beta around inf 68.9%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.8:\\ \;\;\;\;\frac{0.5}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta} \cdot \frac{1}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 83.0% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 160:\\ \;\;\;\;\frac{0.5}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 160.0)
   (/ 0.5 (* (+ alpha 2.0) (+ alpha 3.0)))
   (/ (/ (+ 1.0 alpha) beta) beta)))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 160.0) {
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / beta;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 160.0d0) then
        tmp = 0.5d0 / ((alpha + 2.0d0) * (alpha + 3.0d0))
    else
        tmp = ((1.0d0 + alpha) / beta) / beta
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 160.0) {
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / beta;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 160.0:
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0))
	else:
		tmp = ((1.0 + alpha) / beta) / beta
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 160.0)
		tmp = Float64(0.5 / Float64(Float64(alpha + 2.0) * Float64(alpha + 3.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / beta);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 160.0)
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	else
		tmp = ((1.0 + alpha) / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 160.0], N[(0.5 / N[(N[(alpha + 2.0), $MachinePrecision] * N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 160

   1. Initial program 99.8%

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

   3. Simplified 99.4%

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

      1. clear-num 99.4%

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

      2. inv-pow 99.4%

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

   6. Applied egg-rr 99.4%

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

      1. unpow-1 99.4%

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

      2. associate-/l* 99.4%

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

      3. +-commutative 99.4%

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

      4. +-commutative 99.4%

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

      5. +-commutative 99.4%

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

   8. Simplified 99.4%

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

   9. Taylor expanded in alpha around 0 85.7%

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

   10. Taylor expanded in beta around 0 84.6%

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

   if 160 < beta

   1. Initial program 76.9%

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

   3. Taylor expanded in beta around -inf 69.4%

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

   4. Taylor expanded in beta around inf 69.1%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 160:\\ \;\;\;\;\frac{0.5}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 28.1% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.16 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha}{\beta \cdot \left(\beta + 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 1.16e-7)
   (/ 1.0 (* beta (+ beta 3.0)))
   (/ alpha (* beta (+ beta 2.0)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1.16e-7) {
		tmp = 1.0 / (beta * (beta + 3.0));
	} else {
		tmp = alpha / (beta * (beta + 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 1.16d-7) then
        tmp = 1.0d0 / (beta * (beta + 3.0d0))
    else
        tmp = alpha / (beta * (beta + 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1.16e-7) {
		tmp = 1.0 / (beta * (beta + 3.0));
	} else {
		tmp = alpha / (beta * (beta + 2.0));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 1.16e-7:
		tmp = 1.0 / (beta * (beta + 3.0))
	else:
		tmp = alpha / (beta * (beta + 2.0))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 1.16e-7)
		tmp = Float64(1.0 / Float64(beta * Float64(beta + 3.0)));
	else
		tmp = Float64(alpha / Float64(beta * Float64(beta + 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 1.16e-7)
		tmp = 1.0 / (beta * (beta + 3.0));
	else
		tmp = alpha / (beta * (beta + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 1.16e-7], N[(1.0 / N[(beta * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(alpha / N[(beta * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.1600000000000001e-7

   1. Initial program 99.8%

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

   3. Taylor expanded in beta around -inf 26.2%

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

   4. Taylor expanded in alpha around 0 26.2%

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

      1. +-commutative 26.2%

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

   6. Simplified 26.2%

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

   if 1.1600000000000001e-7 < alpha

   1. Initial program 82.2%

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

   3. Simplified 87.6%

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

   5. Taylor expanded in beta around inf 14.6%

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

      1. expm1-log1p-u 13.9%

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

      2. expm1-udef 17.4%

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

      3. un-div-inv 17.4%

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

   7. Applied egg-rr 17.4%

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

      1. expm1-def 14.0%

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

      2. expm1-log1p 14.6%

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

      3. associate-/l/ 18.8%

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

      4. +-commutative 18.8%

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

      5. +-commutative 18.8%

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

      6. +-commutative 18.8%

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

      7. +-commutative 18.8%

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

   9. Simplified 18.8%

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

   10. Taylor expanded in alpha around 0 14.2%

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

   11. Taylor expanded in alpha around inf 14.2%

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

4. Final simplification 21.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.16 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha}{\beta \cdot \left(\beta + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 28.4% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 0.0034:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha}{\beta \cdot \left(\beta + 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 0.0034)
   (/ (/ 1.0 beta) (+ beta 2.0))
   (/ alpha (* beta (+ beta 2.0)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 0.0034) {
		tmp = (1.0 / beta) / (beta + 2.0);
	} else {
		tmp = alpha / (beta * (beta + 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 0.0034d0) then
        tmp = (1.0d0 / beta) / (beta + 2.0d0)
    else
        tmp = alpha / (beta * (beta + 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 0.0034) {
		tmp = (1.0 / beta) / (beta + 2.0);
	} else {
		tmp = alpha / (beta * (beta + 2.0));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 0.0034:
		tmp = (1.0 / beta) / (beta + 2.0)
	else:
		tmp = alpha / (beta * (beta + 2.0))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 0.0034)
		tmp = Float64(Float64(1.0 / beta) / Float64(beta + 2.0));
	else
		tmp = Float64(alpha / Float64(beta * Float64(beta + 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 0.0034)
		tmp = (1.0 / beta) / (beta + 2.0);
	else
		tmp = alpha / (beta * (beta + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 0.0034], N[(N[(1.0 / beta), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision], N[(alpha / N[(beta * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 0.00339999999999999981

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in beta around inf 26.0%

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

   6. Taylor expanded in alpha around 0 26.1%

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

      1. associate-/r* 26.1%

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

      2. +-commutative 26.1%

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

   8. Simplified 26.1%

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

   if 0.00339999999999999981 < alpha

   1. Initial program 82.0%

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

   3. Simplified 87.5%

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

   5. Taylor expanded in beta around inf 14.6%

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

      1. expm1-log1p-u 14.0%

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

      2. expm1-udef 17.5%

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

      3. un-div-inv 17.5%

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

   7. Applied egg-rr 17.5%

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

      1. expm1-def 14.0%

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

      2. expm1-log1p 14.6%

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

      3. associate-/l/ 18.9%

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

      4. +-commutative 18.9%

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

      5. +-commutative 18.9%

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

      6. +-commutative 18.9%

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

      7. +-commutative 18.9%

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

   9. Simplified 18.9%

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

   10. Taylor expanded in alpha around 0 14.2%

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

   11. Taylor expanded in alpha around inf 14.2%

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

4. Final simplification 21.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 0.0034:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha}{\beta \cdot \left(\beta + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 29.9% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \frac{1 + \alpha}{\beta} \cdot \frac{1}{\beta} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.3%

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

3. Simplified 95.3%

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

5. Taylor expanded in beta around inf 21.8%

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

6. Taylor expanded in beta around inf 22.2%

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

7. Final simplification 22.2%

   \[\leadsto \frac{1 + \alpha}{\beta} \cdot \frac{1}{\beta} \]
8. Add Preprocessing

Alternative 18: 27.4% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.3%

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

3. Taylor expanded in beta around -inf 21.8%

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

4. Taylor expanded in alpha around 0 21.0%

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

   1. +-commutative 21.0%

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

6. Simplified 21.0%

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

7. Final simplification 21.0%

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

Alternative 19: 4.2% accurate, 11.7× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{\alpha} \end{array} \]

FPCore

(FPCore (alpha beta) :precision binary64 (/ 0.5 alpha))

C

double code(double alpha, double beta) {
	return 0.5 / alpha;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = 0.5d0 / alpha
end function

Java

public static double code(double alpha, double beta) {
	return 0.5 / alpha;
}

Python

def code(alpha, beta):
	return 0.5 / alpha

Julia

function code(alpha, beta)
	return Float64(0.5 / alpha)
end

MATLAB

function tmp = code(alpha, beta)
	tmp = 0.5 / alpha;
end

Wolfram

code[alpha_, beta_] := N[(0.5 / alpha), $MachinePrecision]

Derivation

1. Initial program 93.3%

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

3. Simplified 95.3%

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

   1. clear-num 95.2%

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

   2. inv-pow 95.2%

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

6. Applied egg-rr 95.2%

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

   1. unpow-1 95.2%

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

   2. associate-/l* 99.2%

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

   3. +-commutative 99.2%

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

   4. +-commutative 99.2%

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

   5. +-commutative 99.2%

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

8. Simplified 99.2%

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

9. Taylor expanded in beta around inf 31.2%

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

    1. associate-+r+ 31.2%

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

11. Simplified 31.2%

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

12. Taylor expanded in alpha around inf 4.7%

    \[\leadsto \color{blue}{\frac{0.5}{\alpha}} \]

13. Final simplification 4.7%

    \[\leadsto \frac{0.5}{\alpha} \]
14. Add Preprocessing

Alternative 20: 4.3% accurate, 11.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\beta} \end{array} \]

FPCore

(FPCore (alpha beta) :precision binary64 (/ 1.0 beta))

C

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

Fortran

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

Java

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

Python

def code(alpha, beta):
	return 1.0 / beta

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.3%

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

3. Simplified 95.3%

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

5. Taylor expanded in beta around inf 21.8%

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

6. Taylor expanded in alpha around inf 3.9%

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

7. Final simplification 3.9%

   \[\leadsto \frac{1}{\beta} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024011 
(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)))