Octave 3.8, jcobi/3

Percentage Accurate: 94.4% → 98.1%

Time: 13.4s

Alternatives: 15

Speedup: 2.9×

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: 98.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4 \cdot 10^{+48}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 2}}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\alpha + 1\right) \cdot \frac{1}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4e+48)
   (/ (/ (+ 1.0 beta) (+ beta 2.0)) (* (+ beta 2.0) (+ beta 3.0)))
   (/ (* (+ alpha 1.0) (/ 1.0 beta)) (+ alpha (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4e+48) {
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0));
	} else {
		tmp = ((alpha + 1.0) * (1.0 / beta)) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 4d+48) then
        tmp = ((1.0d0 + beta) / (beta + 2.0d0)) / ((beta + 2.0d0) * (beta + 3.0d0))
    else
        tmp = ((alpha + 1.0d0) * (1.0d0 / beta)) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4e+48) {
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0));
	} else {
		tmp = ((alpha + 1.0) * (1.0 / beta)) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 4e+48:
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0))
	else:
		tmp = ((alpha + 1.0) * (1.0 / beta)) / (alpha + (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4e+48)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(beta + 2.0)) / Float64(Float64(beta + 2.0) * Float64(beta + 3.0)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) * Float64(1.0 / beta)) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4e+48)
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0));
	else
		tmp = ((alpha + 1.0) * (1.0 / beta)) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 4e+48], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] * N[(1.0 / beta), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.00000000000000018e48

   1. Initial program 99.3%

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

   3. Simplified 94.4%

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

      1. times-frac 99.0%

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

      2. +-commutative 99.0%

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

   6. Applied egg-rr 99.0%

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

   7. Taylor expanded in alpha around 0 71.2%

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

      1. associate-/r* 71.2%

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

      2. +-commutative 71.2%

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

      3. +-commutative 71.2%

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

   9. Simplified 71.2%

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

   10. Taylor expanded in alpha around 0 71.3%

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

       1. +-commutative 71.3%

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

   12. Simplified 71.3%

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

       1. frac-times 71.3%

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

       2. *-un-lft-identity 71.3%

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

   14. Applied egg-rr 71.3%

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

   if 4.00000000000000018e48 < beta

   1. Initial program 74.8%

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

   3. Taylor expanded in beta around inf 85.1%

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

      1. div-inv 84.9%

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

      2. +-commutative 84.9%

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

      3. metadata-eval 84.9%

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

      4. associate-+l+ 84.9%

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

      5. metadata-eval 84.9%

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

      6. associate-+r+ 84.9%

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

   5. Applied egg-rr 84.9%

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

      1. associate-*r/ 85.1%

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

      2. *-commutative 85.1%

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

      3. *-lft-identity 85.1%

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

      4. +-commutative 85.1%

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

      5. +-commutative 85.1%

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

      6. +-commutative 85.1%

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

      7. +-commutative 85.1%

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

   7. Simplified 85.1%

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

      1. div-inv 85.0%

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

      2. +-commutative 85.0%

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

   9. Applied egg-rr 85.0%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4 \cdot 10^{+48}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 2}}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\alpha + 1\right) \cdot \frac{1}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.2% accurate, 1.7× speedup

Math

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

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (*
  (/ (+ alpha 1.0) (+ alpha (+ beta 2.0)))
  (/ (/ (+ 1.0 beta) (+ beta 2.0)) (+ beta 3.0))))

C

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

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = ((alpha + 1.0d0) / (alpha + (beta + 2.0d0))) * (((1.0d0 + beta) / (beta + 2.0d0)) / (beta + 3.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = ((alpha + 1.0) / (alpha + (beta + 2.0))) * (((1.0 + beta) / (beta + 2.0)) / (beta + 3.0));
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.5%

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

3. Simplified 81.3%

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

   1. times-frac 95.3%

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

   2. +-commutative 95.3%

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

6. Applied egg-rr 95.3%

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

7. Taylor expanded in alpha around 0 74.1%

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

   1. associate-/r* 75.0%

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

   2. +-commutative 75.0%

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

   3. +-commutative 75.0%

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

9. Simplified 75.0%

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

Alternative 3: 98.3% accurate, 1.7× speedup

Math

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

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2e+16)
   (/ (/ (+ 1.0 beta) (* (+ beta 2.0) (+ beta 3.0))) (+ beta 2.0))
   (/ (* (+ alpha 1.0) (/ 1.0 beta)) (+ alpha (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2e+16) {
		tmp = ((1.0 + beta) / ((beta + 2.0) * (beta + 3.0))) / (beta + 2.0);
	} else {
		tmp = ((alpha + 1.0) * (1.0 / beta)) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2d+16) then
        tmp = ((1.0d0 + beta) / ((beta + 2.0d0) * (beta + 3.0d0))) / (beta + 2.0d0)
    else
        tmp = ((alpha + 1.0d0) * (1.0d0 / beta)) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2e+16) {
		tmp = ((1.0 + beta) / ((beta + 2.0) * (beta + 3.0))) / (beta + 2.0);
	} else {
		tmp = ((alpha + 1.0) * (1.0 / beta)) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2e+16:
		tmp = ((1.0 + beta) / ((beta + 2.0) * (beta + 3.0))) / (beta + 2.0)
	else:
		tmp = ((alpha + 1.0) * (1.0 / beta)) / (alpha + (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2e+16)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(Float64(beta + 2.0) * Float64(beta + 3.0))) / Float64(beta + 2.0));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) * Float64(1.0 / beta)) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2e+16)
		tmp = ((1.0 + beta) / ((beta + 2.0) * (beta + 3.0))) / (beta + 2.0);
	else
		tmp = ((alpha + 1.0) * (1.0 / beta)) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2e+16], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] * N[(1.0 / beta), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2e16

   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 94.7%

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

      1. times-frac 98.9%

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

      2. +-commutative 98.9%

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

   6. Applied egg-rr 98.9%

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

   7. Taylor expanded in alpha around 0 71.9%

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

      1. associate-/r* 71.9%

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

      2. +-commutative 71.9%

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

      3. +-commutative 71.9%

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

   9. Simplified 71.9%

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

   10. Taylor expanded in alpha around 0 72.0%

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

       1. +-commutative 72.0%

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

   12. Simplified 72.0%

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

       1. associate-*l/ 72.0%

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

       2. *-un-lft-identity 72.0%

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

       3. associate-/l/ 71.9%

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

   14. Applied egg-rr 71.9%

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

   if 2e16 < beta

   1. Initial program 76.4%

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

   3. Taylor expanded in beta around inf 82.0%

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

      1. div-inv 81.7%

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

      2. +-commutative 81.7%

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

      3. metadata-eval 81.7%

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

      4. associate-+l+ 81.7%

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

      5. metadata-eval 81.7%

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

      6. associate-+r+ 81.7%

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

   5. Applied egg-rr 81.7%

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

      1. associate-*r/ 82.0%

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

      2. *-commutative 82.0%

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

      3. *-lft-identity 82.0%

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

      4. +-commutative 82.0%

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

      5. +-commutative 82.0%

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

      6. +-commutative 82.0%

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

      7. +-commutative 82.0%

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

   7. Simplified 82.0%

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

      1. div-inv 81.9%

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

      2. +-commutative 81.9%

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

   9. Applied egg-rr 81.9%

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

4. Final simplification 75.1%

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

Alternative 4: 97.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4:\\ \;\;\;\;\frac{0.5 + \alpha \cdot 0.25}{\left(3 + \left(\alpha + \beta\right)\right) \cdot \left(\alpha + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\alpha + 1\right) \cdot \frac{1}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.0)
   (/ (+ 0.5 (* alpha 0.25)) (* (+ 3.0 (+ alpha beta)) (+ alpha 2.0)))
   (/ (* (+ alpha 1.0) (/ 1.0 beta)) (+ alpha (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.0) {
		tmp = (0.5 + (alpha * 0.25)) / ((3.0 + (alpha + beta)) * (alpha + 2.0));
	} else {
		tmp = ((alpha + 1.0) * (1.0 / beta)) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 4.0d0) then
        tmp = (0.5d0 + (alpha * 0.25d0)) / ((3.0d0 + (alpha + beta)) * (alpha + 2.0d0))
    else
        tmp = ((alpha + 1.0d0) * (1.0d0 / beta)) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.0) {
		tmp = (0.5 + (alpha * 0.25)) / ((3.0 + (alpha + beta)) * (alpha + 2.0));
	} else {
		tmp = ((alpha + 1.0) * (1.0 / beta)) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 4.0:
		tmp = (0.5 + (alpha * 0.25)) / ((3.0 + (alpha + beta)) * (alpha + 2.0))
	else:
		tmp = ((alpha + 1.0) * (1.0 / beta)) / (alpha + (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.0)
		tmp = Float64(Float64(0.5 + Float64(alpha * 0.25)) / Float64(Float64(3.0 + Float64(alpha + beta)) * Float64(alpha + 2.0)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) * Float64(1.0 / beta)) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4.0)
		tmp = (0.5 + (alpha * 0.25)) / ((3.0 + (alpha + beta)) * (alpha + 2.0));
	else
		tmp = ((alpha + 1.0) * (1.0 / beta)) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 4.0], N[(N[(0.5 + N[(alpha * 0.25), $MachinePrecision]), $MachinePrecision] / N[(N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision] * N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] * N[(1.0 / beta), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4

   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/ 98.9%

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

      2. +-commutative 98.9%

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

      3. associate-+l+ 98.9%

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

      4. *-commutative 98.9%

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

      5. metadata-eval 98.9%

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

      6. associate-+l+ 98.9%

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

      7. metadata-eval 98.9%

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

      8. +-commutative 98.9%

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

      9. +-commutative 98.9%

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

      10. +-commutative 98.9%

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

      11. metadata-eval 98.9%

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

      12. metadata-eval 98.9%

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

      13. associate-+l+ 98.9%

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

   3. Simplified 98.9%

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

   5. Taylor expanded in beta around 0 95.5%

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

   6. Taylor expanded in beta around 0 96.9%

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

   7. Taylor expanded in alpha around 0 84.1%

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

      1. *-commutative 84.1%

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

   9. Simplified 84.1%

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

   if 4 < beta

   1. Initial program 76.7%

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

   3. Taylor expanded in beta around inf 82.1%

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

      1. div-inv 81.9%

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

      2. +-commutative 81.9%

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

      3. metadata-eval 81.9%

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

      4. associate-+l+ 81.9%

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

      5. metadata-eval 81.9%

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

      6. associate-+r+ 81.9%

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

   5. Applied egg-rr 81.9%

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

      1. associate-*r/ 82.1%

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

      2. *-commutative 82.1%

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

      3. *-lft-identity 82.1%

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

      4. +-commutative 82.1%

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

      5. +-commutative 82.1%

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

      6. +-commutative 82.1%

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

      7. +-commutative 82.1%

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

   7. Simplified 82.1%

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

      1. div-inv 82.1%

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

      2. +-commutative 82.1%

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

   9. Applied egg-rr 82.1%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4:\\ \;\;\;\;\frac{0.5 + \alpha \cdot 0.25}{\left(3 + \left(\alpha + \beta\right)\right) \cdot \left(\alpha + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\alpha + 1\right) \cdot \frac{1}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.7:\\ \;\;\;\;0.08333333333333333 + \beta \cdot \left(\beta \cdot \left(\beta \cdot 0.024691358024691357 - 0.011574074074074073\right) - 0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\alpha + 1\right) \cdot \frac{1}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.7)
   (+
    0.08333333333333333
    (*
     beta
     (-
      (* beta (- (* beta 0.024691358024691357) 0.011574074074074073))
      0.027777777777777776)))
   (/ (* (+ alpha 1.0) (/ 1.0 beta)) (+ alpha (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.7) {
		tmp = 0.08333333333333333 + (beta * ((beta * ((beta * 0.024691358024691357) - 0.011574074074074073)) - 0.027777777777777776));
	} else {
		tmp = ((alpha + 1.0) * (1.0 / beta)) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.7d0) then
        tmp = 0.08333333333333333d0 + (beta * ((beta * ((beta * 0.024691358024691357d0) - 0.011574074074074073d0)) - 0.027777777777777776d0))
    else
        tmp = ((alpha + 1.0d0) * (1.0d0 / beta)) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.7) {
		tmp = 0.08333333333333333 + (beta * ((beta * ((beta * 0.024691358024691357) - 0.011574074074074073)) - 0.027777777777777776));
	} else {
		tmp = ((alpha + 1.0) * (1.0 / beta)) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.7:
		tmp = 0.08333333333333333 + (beta * ((beta * ((beta * 0.024691358024691357) - 0.011574074074074073)) - 0.027777777777777776))
	else:
		tmp = ((alpha + 1.0) * (1.0 / beta)) / (alpha + (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.7)
		tmp = Float64(0.08333333333333333 + Float64(beta * Float64(Float64(beta * Float64(Float64(beta * 0.024691358024691357) - 0.011574074074074073)) - 0.027777777777777776)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) * Float64(1.0 / beta)) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.7)
		tmp = 0.08333333333333333 + (beta * ((beta * ((beta * 0.024691358024691357) - 0.011574074074074073)) - 0.027777777777777776));
	else
		tmp = ((alpha + 1.0) * (1.0 / beta)) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.7], N[(0.08333333333333333 + N[(beta * N[(N[(beta * N[(N[(beta * 0.024691358024691357), $MachinePrecision] - 0.011574074074074073), $MachinePrecision]), $MachinePrecision] - 0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] * N[(1.0 / beta), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.69999999999999996

   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 94.7%

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

      1. times-frac 98.9%

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

      2. +-commutative 98.9%

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

   6. Applied egg-rr 98.9%

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

   7. Taylor expanded in alpha around 0 71.8%

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

      1. associate-/r* 71.8%

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

      2. +-commutative 71.8%

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

      3. +-commutative 71.8%

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

   9. Simplified 71.8%

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

   10. Taylor expanded in alpha around 0 71.8%

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

       1. +-commutative 71.8%

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

   12. Simplified 71.8%

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

   13. Taylor expanded in beta around 0 71.0%

       \[\leadsto \color{blue}{0.08333333333333333 + \beta \cdot \left(\beta \cdot \left(0.024691358024691357 \cdot \beta - 0.011574074074074073\right) - 0.027777777777777776\right)} \]

   if 1.69999999999999996 < beta

   1. Initial program 76.7%

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

   3. Taylor expanded in beta around inf 82.1%

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

      1. div-inv 81.9%

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

      2. +-commutative 81.9%

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

      3. metadata-eval 81.9%

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

      4. associate-+l+ 81.9%

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

      5. metadata-eval 81.9%

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

      6. associate-+r+ 81.9%

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

   5. Applied egg-rr 81.9%

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

      1. associate-*r/ 82.1%

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

      2. *-commutative 82.1%

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

      3. *-lft-identity 82.1%

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

      4. +-commutative 82.1%

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

      5. +-commutative 82.1%

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

      6. +-commutative 82.1%

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

      7. +-commutative 82.1%

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

   7. Simplified 82.1%

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

      1. div-inv 82.1%

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

      2. +-commutative 82.1%

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

   9. Applied egg-rr 82.1%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.7:\\ \;\;\;\;0.08333333333333333 + \beta \cdot \left(\beta \cdot \left(\beta \cdot 0.024691358024691357 - 0.011574074074074073\right) - 0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\alpha + 1\right) \cdot \frac{1}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.7:\\ \;\;\;\;0.08333333333333333 + \beta \cdot \left(\beta \cdot \left(\beta \cdot 0.024691358024691357 - 0.011574074074074073\right) - 0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.7)
   (+
    0.08333333333333333
    (*
     beta
     (-
      (* beta (- (* beta 0.024691358024691357) 0.011574074074074073))
      0.027777777777777776)))
   (/ (/ (+ alpha 1.0) beta) (+ alpha (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.7) {
		tmp = 0.08333333333333333 + (beta * ((beta * ((beta * 0.024691358024691357) - 0.011574074074074073)) - 0.027777777777777776));
	} else {
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.7d0) then
        tmp = 0.08333333333333333d0 + (beta * ((beta * ((beta * 0.024691358024691357d0) - 0.011574074074074073d0)) - 0.027777777777777776d0))
    else
        tmp = ((alpha + 1.0d0) / beta) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.7) {
		tmp = 0.08333333333333333 + (beta * ((beta * ((beta * 0.024691358024691357) - 0.011574074074074073)) - 0.027777777777777776));
	} else {
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.7:
		tmp = 0.08333333333333333 + (beta * ((beta * ((beta * 0.024691358024691357) - 0.011574074074074073)) - 0.027777777777777776))
	else:
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.7)
		tmp = Float64(0.08333333333333333 + Float64(beta * Float64(Float64(beta * Float64(Float64(beta * 0.024691358024691357) - 0.011574074074074073)) - 0.027777777777777776)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.7)
		tmp = 0.08333333333333333 + (beta * ((beta * ((beta * 0.024691358024691357) - 0.011574074074074073)) - 0.027777777777777776));
	else
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.7], N[(0.08333333333333333 + N[(beta * N[(N[(beta * N[(N[(beta * 0.024691358024691357), $MachinePrecision] - 0.011574074074074073), $MachinePrecision]), $MachinePrecision] - 0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.69999999999999996

   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 94.7%

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

      1. times-frac 98.9%

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

      2. +-commutative 98.9%

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

   6. Applied egg-rr 98.9%

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

   7. Taylor expanded in alpha around 0 71.8%

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

      1. associate-/r* 71.8%

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

      2. +-commutative 71.8%

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

      3. +-commutative 71.8%

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

   9. Simplified 71.8%

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

   10. Taylor expanded in alpha around 0 71.8%

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

       1. +-commutative 71.8%

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

   12. Simplified 71.8%

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

   13. Taylor expanded in beta around 0 71.0%

       \[\leadsto \color{blue}{0.08333333333333333 + \beta \cdot \left(\beta \cdot \left(0.024691358024691357 \cdot \beta - 0.011574074074074073\right) - 0.027777777777777776\right)} \]

   if 1.69999999999999996 < beta

   1. Initial program 76.7%

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

   3. Taylor expanded in beta around inf 82.1%

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

      1. div-inv 81.9%

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

      2. +-commutative 81.9%

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

      3. metadata-eval 81.9%

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

      4. associate-+l+ 81.9%

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

      5. metadata-eval 81.9%

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

      6. associate-+r+ 81.9%

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

   5. Applied egg-rr 81.9%

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

      1. associate-*r/ 82.1%

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

      2. *-commutative 82.1%

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

      3. *-lft-identity 82.1%

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

      4. +-commutative 82.1%

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

      5. +-commutative 82.1%

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

      6. +-commutative 82.1%

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

      7. +-commutative 82.1%

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

   7. Simplified 82.1%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.7:\\ \;\;\;\;0.08333333333333333 + \beta \cdot \left(\beta \cdot \left(\beta \cdot 0.024691358024691357 - 0.011574074074074073\right) - 0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.55:\\ \;\;\;\;0.08333333333333333 + \beta \cdot \left(\beta \cdot -0.011574074074074073 - 0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.55)
   (+
    0.08333333333333333
    (* beta (- (* beta -0.011574074074074073) 0.027777777777777776)))
   (/ (/ (+ alpha 1.0) beta) (+ alpha (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.55) {
		tmp = 0.08333333333333333 + (beta * ((beta * -0.011574074074074073) - 0.027777777777777776));
	} else {
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.55d0) then
        tmp = 0.08333333333333333d0 + (beta * ((beta * (-0.011574074074074073d0)) - 0.027777777777777776d0))
    else
        tmp = ((alpha + 1.0d0) / beta) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.55) {
		tmp = 0.08333333333333333 + (beta * ((beta * -0.011574074074074073) - 0.027777777777777776));
	} else {
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.55:
		tmp = 0.08333333333333333 + (beta * ((beta * -0.011574074074074073) - 0.027777777777777776))
	else:
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.55)
		tmp = Float64(0.08333333333333333 + Float64(beta * Float64(Float64(beta * -0.011574074074074073) - 0.027777777777777776)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.55)
		tmp = 0.08333333333333333 + (beta * ((beta * -0.011574074074074073) - 0.027777777777777776));
	else
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.55], N[(0.08333333333333333 + N[(beta * N[(N[(beta * -0.011574074074074073), $MachinePrecision] - 0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.55000000000000004

   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 94.7%

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

      1. times-frac 98.9%

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

      2. +-commutative 98.9%

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

   6. Applied egg-rr 98.9%

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

   7. Taylor expanded in alpha around 0 71.8%

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

      1. associate-/r* 71.8%

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

      2. +-commutative 71.8%

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

      3. +-commutative 71.8%

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

   9. Simplified 71.8%

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

   10. Taylor expanded in alpha around 0 71.8%

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

       1. +-commutative 71.8%

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

   12. Simplified 71.8%

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

   13. Taylor expanded in beta around 0 70.9%

       \[\leadsto \color{blue}{0.08333333333333333 + \beta \cdot \left(-0.011574074074074073 \cdot \beta - 0.027777777777777776\right)} \]

   if 1.55000000000000004 < beta

   1. Initial program 76.7%

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

   3. Taylor expanded in beta around inf 82.1%

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

      1. div-inv 81.9%

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

      2. +-commutative 81.9%

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

      3. metadata-eval 81.9%

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

      4. associate-+l+ 81.9%

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

      5. metadata-eval 81.9%

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

      6. associate-+r+ 81.9%

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

   5. Applied egg-rr 81.9%

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

      1. associate-*r/ 82.1%

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

      2. *-commutative 82.1%

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

      3. *-lft-identity 82.1%

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

      4. +-commutative 82.1%

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

      5. +-commutative 82.1%

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

      6. +-commutative 82.1%

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

      7. +-commutative 82.1%

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

   7. Simplified 82.1%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.55:\\ \;\;\;\;0.08333333333333333 + \beta \cdot \left(\beta \cdot -0.011574074074074073 - 0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.55:\\ \;\;\;\;0.08333333333333333 + \beta \cdot \left(\beta \cdot -0.011574074074074073 - 0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.55)
   (+
    0.08333333333333333
    (* beta (- (* beta -0.011574074074074073) 0.027777777777777776)))
   (/ (/ (+ alpha 1.0) beta) (+ beta 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.55) {
		tmp = 0.08333333333333333 + (beta * ((beta * -0.011574074074074073) - 0.027777777777777776));
	} else {
		tmp = ((alpha + 1.0) / beta) / (beta + 3.0);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.55d0) then
        tmp = 0.08333333333333333d0 + (beta * ((beta * (-0.011574074074074073d0)) - 0.027777777777777776d0))
    else
        tmp = ((alpha + 1.0d0) / beta) / (beta + 3.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.55) {
		tmp = 0.08333333333333333 + (beta * ((beta * -0.011574074074074073) - 0.027777777777777776));
	} else {
		tmp = ((alpha + 1.0) / beta) / (beta + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.55:
		tmp = 0.08333333333333333 + (beta * ((beta * -0.011574074074074073) - 0.027777777777777776))
	else:
		tmp = ((alpha + 1.0) / beta) / (beta + 3.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.55)
		tmp = Float64(0.08333333333333333 + Float64(beta * Float64(Float64(beta * -0.011574074074074073) - 0.027777777777777776)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / Float64(beta + 3.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.55)
		tmp = 0.08333333333333333 + (beta * ((beta * -0.011574074074074073) - 0.027777777777777776));
	else
		tmp = ((alpha + 1.0) / beta) / (beta + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.55], N[(0.08333333333333333 + N[(beta * N[(N[(beta * -0.011574074074074073), $MachinePrecision] - 0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.55000000000000004

   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 94.7%

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

      1. times-frac 98.9%

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

      2. +-commutative 98.9%

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

   6. Applied egg-rr 98.9%

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

   7. Taylor expanded in alpha around 0 71.8%

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

      1. associate-/r* 71.8%

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

      2. +-commutative 71.8%

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

      3. +-commutative 71.8%

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

   9. Simplified 71.8%

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

   10. Taylor expanded in alpha around 0 71.8%

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

       1. +-commutative 71.8%

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

   12. Simplified 71.8%

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

   13. Taylor expanded in beta around 0 70.9%

       \[\leadsto \color{blue}{0.08333333333333333 + \beta \cdot \left(-0.011574074074074073 \cdot \beta - 0.027777777777777776\right)} \]

   if 1.55000000000000004 < beta

   1. Initial program 76.7%

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

   3. Taylor expanded in beta around inf 82.1%

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

   4. Taylor expanded in alpha around 0 81.9%

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

      1. +-commutative 81.9%

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

   6. Simplified 81.9%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.55:\\ \;\;\;\;0.08333333333333333 + \beta \cdot \left(\beta \cdot -0.011574074074074073 - 0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 96.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.66:\\ \;\;\;\;0.08333333333333333 + \beta \cdot \left(\beta \cdot -0.011574074074074073 - 0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.66)
   (+
    0.08333333333333333
    (* beta (- (* beta -0.011574074074074073) 0.027777777777777776)))
   (/ (/ (+ alpha 1.0) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.66) {
		tmp = 0.08333333333333333 + (beta * ((beta * -0.011574074074074073) - 0.027777777777777776));
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.66d0) then
        tmp = 0.08333333333333333d0 + (beta * ((beta * (-0.011574074074074073d0)) - 0.027777777777777776d0))
    else
        tmp = ((alpha + 1.0d0) / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.66) {
		tmp = 0.08333333333333333 + (beta * ((beta * -0.011574074074074073) - 0.027777777777777776));
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.66:
		tmp = 0.08333333333333333 + (beta * ((beta * -0.011574074074074073) - 0.027777777777777776))
	else:
		tmp = ((alpha + 1.0) / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.66)
		tmp = Float64(0.08333333333333333 + Float64(beta * Float64(Float64(beta * -0.011574074074074073) - 0.027777777777777776)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.66)
		tmp = 0.08333333333333333 + (beta * ((beta * -0.011574074074074073) - 0.027777777777777776));
	else
		tmp = ((alpha + 1.0) / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.66], N[(0.08333333333333333 + N[(beta * N[(N[(beta * -0.011574074074074073), $MachinePrecision] - 0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.65999999999999992

   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 94.7%

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

      1. times-frac 98.9%

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

      2. +-commutative 98.9%

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

   6. Applied egg-rr 98.9%

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

   7. Taylor expanded in alpha around 0 71.8%

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

      1. associate-/r* 71.8%

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

      2. +-commutative 71.8%

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

      3. +-commutative 71.8%

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

   9. Simplified 71.8%

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

   10. Taylor expanded in alpha around 0 71.8%

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

       1. +-commutative 71.8%

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

   12. Simplified 71.8%

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

   13. Taylor expanded in beta around 0 70.9%

       \[\leadsto \color{blue}{0.08333333333333333 + \beta \cdot \left(-0.011574074074074073 \cdot \beta - 0.027777777777777776\right)} \]

   if 1.65999999999999992 < beta

   1. Initial program 76.7%

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

   3. Taylor expanded in beta around inf 82.1%

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

      1. div-inv 81.9%

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

      2. +-commutative 81.9%

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

      3. metadata-eval 81.9%

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

      4. associate-+l+ 81.9%

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

      5. metadata-eval 81.9%

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

      6. associate-+r+ 81.9%

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

   5. Applied egg-rr 81.9%

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

      1. associate-*r/ 82.1%

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

      2. *-commutative 82.1%

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

      3. *-lft-identity 82.1%

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

      4. +-commutative 82.1%

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

      5. +-commutative 82.1%

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

      6. +-commutative 82.1%

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

      7. +-commutative 82.1%

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

   7. Simplified 82.1%

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

   8. Taylor expanded in beta around inf 81.9%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.66:\\ \;\;\;\;0.08333333333333333 + \beta \cdot \left(\beta \cdot -0.011574074074074073 - 0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 96.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.8:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.8)
   (+ 0.08333333333333333 (* beta -0.027777777777777776))
   (/ (/ (+ alpha 1.0) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.8) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.8d0) then
        tmp = 0.08333333333333333d0 + (beta * (-0.027777777777777776d0))
    else
        tmp = ((alpha + 1.0d0) / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.8) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.8:
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776)
	else:
		tmp = ((alpha + 1.0) / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.8)
		tmp = Float64(0.08333333333333333 + Float64(beta * -0.027777777777777776));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.8)
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	else
		tmp = ((alpha + 1.0) / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.8], N[(0.08333333333333333 + N[(beta * -0.027777777777777776), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.7999999999999998

   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/ 98.9%

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

      2. +-commutative 98.9%

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

      3. associate-+l+ 98.9%

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

      4. *-commutative 98.9%

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

      5. metadata-eval 98.9%

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

      6. associate-+l+ 98.9%

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

      7. metadata-eval 98.9%

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

      8. +-commutative 98.9%

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

      9. +-commutative 98.9%

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

      10. +-commutative 98.9%

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

      11. metadata-eval 98.9%

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

      12. metadata-eval 98.9%

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

      13. associate-+l+ 98.9%

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

   3. Simplified 98.9%

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

   5. Taylor expanded in beta around 0 95.5%

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

   6. Taylor expanded in beta around 0 96.9%

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

   7. Taylor expanded in alpha around 0 70.5%

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

      1. +-commutative 70.5%

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

   9. Simplified 70.5%

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

   10. Taylor expanded in beta around 0 70.6%

       \[\leadsto \color{blue}{0.08333333333333333 + -0.027777777777777776 \cdot \beta} \]
   11. Step-by-step derivation

       1. *-commutative 70.6%

          \[\leadsto 0.08333333333333333 + \color{blue}{\beta \cdot -0.027777777777777776} \]

   12. Simplified 70.6%

       \[\leadsto \color{blue}{0.08333333333333333 + \beta \cdot -0.027777777777777776} \]

   if 2.7999999999999998 < beta

   1. Initial program 76.7%

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

   3. Taylor expanded in beta around inf 82.1%

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

      1. div-inv 81.9%

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

      2. +-commutative 81.9%

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

      3. metadata-eval 81.9%

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

      4. associate-+l+ 81.9%

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

      5. metadata-eval 81.9%

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

      6. associate-+r+ 81.9%

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

   5. Applied egg-rr 81.9%

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

      1. associate-*r/ 82.1%

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

      2. *-commutative 82.1%

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

      3. *-lft-identity 82.1%

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

      4. +-commutative 82.1%

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

      5. +-commutative 82.1%

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

      6. +-commutative 82.1%

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

      7. +-commutative 82.1%

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

   7. Simplified 82.1%

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

   8. Taylor expanded in beta around inf 81.9%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.8:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 91.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.45:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.45)
   (+ 0.08333333333333333 (* beta -0.027777777777777776))
   (/ (/ 1.0 beta) (+ beta 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.45) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = (1.0 / beta) / (beta + 3.0);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.45d0) then
        tmp = 0.08333333333333333d0 + (beta * (-0.027777777777777776d0))
    else
        tmp = (1.0d0 / beta) / (beta + 3.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.45) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = (1.0 / beta) / (beta + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.45:
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776)
	else:
		tmp = (1.0 / beta) / (beta + 3.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.45)
		tmp = Float64(0.08333333333333333 + Float64(beta * -0.027777777777777776));
	else
		tmp = Float64(Float64(1.0 / beta) / Float64(beta + 3.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.45)
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	else
		tmp = (1.0 / beta) / (beta + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.45], N[(0.08333333333333333 + N[(beta * -0.027777777777777776), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.4500000000000002

   1. Initial program 99.8%

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

      1. associate-/l/ 98.9%

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

      2. +-commutative 98.9%

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

      3. associate-+l+ 98.9%

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

      4. *-commutative 98.9%

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

      5. metadata-eval 98.9%

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

      6. associate-+l+ 98.9%

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

      7. metadata-eval 98.9%

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

      8. +-commutative 98.9%

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

      9. +-commutative 98.9%

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

      10. +-commutative 98.9%

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

      11. metadata-eval 98.9%

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

      12. metadata-eval 98.9%

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

      13. associate-+l+ 98.9%

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

   3. Simplified 98.9%

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

   5. Taylor expanded in beta around 0 95.5%

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

   6. Taylor expanded in beta around 0 96.9%

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

   7. Taylor expanded in alpha around 0 70.5%

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

      1. +-commutative 70.5%

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

   9. Simplified 70.5%

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

   10. Taylor expanded in beta around 0 70.6%

       \[\leadsto \color{blue}{0.08333333333333333 + -0.027777777777777776 \cdot \beta} \]
   11. Step-by-step derivation

       1. *-commutative 70.6%

          \[\leadsto 0.08333333333333333 + \color{blue}{\beta \cdot -0.027777777777777776} \]

   12. Simplified 70.6%

       \[\leadsto \color{blue}{0.08333333333333333 + \beta \cdot -0.027777777777777776} \]

   if 2.4500000000000002 < beta

   1. Initial program 76.7%

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

   3. Taylor expanded in beta around inf 82.1%

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

   4. Taylor expanded in alpha around 0 73.5%

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

      1. associate-/r* 72.7%

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

      2. +-commutative 72.7%

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

   6. Simplified 72.7%

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

Alternative 12: 91.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.45:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.45)
   (+ 0.08333333333333333 (* beta -0.027777777777777776))
   (/ 1.0 (* beta (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.45) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = 1.0 / (beta * (beta + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.45d0) then
        tmp = 0.08333333333333333d0 + (beta * (-0.027777777777777776d0))
    else
        tmp = 1.0d0 / (beta * (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.45) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = 1.0 / (beta * (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.45:
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776)
	else:
		tmp = 1.0 / (beta * (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.45)
		tmp = Float64(0.08333333333333333 + Float64(beta * -0.027777777777777776));
	else
		tmp = Float64(1.0 / Float64(beta * Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.45)
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	else
		tmp = 1.0 / (beta * (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.45], N[(0.08333333333333333 + N[(beta * -0.027777777777777776), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(beta * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.4500000000000002

   1. Initial program 99.8%

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

      1. associate-/l/ 98.9%

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

      2. +-commutative 98.9%

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

      3. associate-+l+ 98.9%

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

      4. *-commutative 98.9%

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

      5. metadata-eval 98.9%

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

      6. associate-+l+ 98.9%

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

      7. metadata-eval 98.9%

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

      8. +-commutative 98.9%

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

      9. +-commutative 98.9%

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

      10. +-commutative 98.9%

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

      11. metadata-eval 98.9%

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

      12. metadata-eval 98.9%

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

      13. associate-+l+ 98.9%

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

   3. Simplified 98.9%

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

   5. Taylor expanded in beta around 0 95.5%

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

   6. Taylor expanded in beta around 0 96.9%

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

   7. Taylor expanded in alpha around 0 70.5%

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

      1. +-commutative 70.5%

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

   9. Simplified 70.5%

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

   10. Taylor expanded in beta around 0 70.6%

       \[\leadsto \color{blue}{0.08333333333333333 + -0.027777777777777776 \cdot \beta} \]
   11. Step-by-step derivation

       1. *-commutative 70.6%

          \[\leadsto 0.08333333333333333 + \color{blue}{\beta \cdot -0.027777777777777776} \]

   12. Simplified 70.6%

       \[\leadsto \color{blue}{0.08333333333333333 + \beta \cdot -0.027777777777777776} \]

   if 2.4500000000000002 < beta

   1. Initial program 76.7%

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

   3. Taylor expanded in beta around inf 82.1%

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

   4. Taylor expanded in alpha around 0 73.5%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.45:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 48.0% accurate, 3.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.65:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.65)
   (+ 0.08333333333333333 (* beta -0.027777777777777776))
   (/ 0.25 beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.65) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = 0.25 / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.65d0) then
        tmp = 0.08333333333333333d0 + (beta * (-0.027777777777777776d0))
    else
        tmp = 0.25d0 / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.65) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = 0.25 / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.65:
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776)
	else:
		tmp = 0.25 / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.65)
		tmp = Float64(0.08333333333333333 + Float64(beta * -0.027777777777777776));
	else
		tmp = Float64(0.25 / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.65)
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	else
		tmp = 0.25 / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.65], N[(0.08333333333333333 + N[(beta * -0.027777777777777776), $MachinePrecision]), $MachinePrecision], N[(0.25 / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.64999999999999991

   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/ 98.9%

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

      2. +-commutative 98.9%

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

      3. associate-+l+ 98.9%

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

      4. *-commutative 98.9%

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

      5. metadata-eval 98.9%

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

      6. associate-+l+ 98.9%

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

      7. metadata-eval 98.9%

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

      8. +-commutative 98.9%

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

      9. +-commutative 98.9%

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

      10. +-commutative 98.9%

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

      11. metadata-eval 98.9%

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

      12. metadata-eval 98.9%

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

      13. associate-+l+ 98.9%

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

   3. Simplified 98.9%

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

   5. Taylor expanded in beta around 0 95.5%

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

   6. Taylor expanded in beta around 0 96.9%

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

   7. Taylor expanded in alpha around 0 70.5%

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

      1. +-commutative 70.5%

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

   9. Simplified 70.5%

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

   10. Taylor expanded in beta around 0 70.6%

       \[\leadsto \color{blue}{0.08333333333333333 + -0.027777777777777776 \cdot \beta} \]
   11. Step-by-step derivation

       1. *-commutative 70.6%

          \[\leadsto 0.08333333333333333 + \color{blue}{\beta \cdot -0.027777777777777776} \]

   12. Simplified 70.6%

       \[\leadsto \color{blue}{0.08333333333333333 + \beta \cdot -0.027777777777777776} \]

   if 2.64999999999999991 < beta

   1. Initial program 76.7%

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

      1. associate-/l/ 72.8%

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

      2. +-commutative 72.8%

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

      3. associate-+l+ 72.8%

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

      4. *-commutative 72.8%

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

      5. metadata-eval 72.8%

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

      6. associate-+l+ 72.8%

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

      7. metadata-eval 72.8%

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

      8. +-commutative 72.8%

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

      9. +-commutative 72.8%

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

      10. +-commutative 72.8%

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

      11. metadata-eval 72.8%

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

      12. metadata-eval 72.8%

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

      13. associate-+l+ 72.8%

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

   3. Simplified 72.8%

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

   5. Taylor expanded in beta around 0 49.7%

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

   6. Taylor expanded in beta around 0 22.2%

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

   7. Taylor expanded in alpha around 0 6.7%

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

      1. +-commutative 6.7%

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

   9. Simplified 6.7%

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

   10. Taylor expanded in beta around inf 6.7%

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

Alternative 14: 48.0% accurate, 7.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \frac{0.25}{\beta + 3} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta) :precision binary64 (/ 0.25 (+ beta 3.0)))

C

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

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = 0.25d0 / (beta + 3.0d0)
end function

Java

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

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return 0.25 / (beta + 3.0)

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return Float64(0.25 / Float64(beta + 3.0))
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = 0.25 / (beta + 3.0);
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := N[(0.25 / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.5%

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

   1. associate-/l/ 90.7%

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

   2. +-commutative 90.7%

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

   3. associate-+l+ 90.7%

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

   4. *-commutative 90.7%

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

   5. metadata-eval 90.7%

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

   6. associate-+l+ 90.7%

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

   7. metadata-eval 90.7%

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

   8. +-commutative 90.7%

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

   9. +-commutative 90.7%

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

   10. +-commutative 90.7%

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

   11. metadata-eval 90.7%

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

   12. metadata-eval 90.7%

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

   13. associate-+l+ 90.7%

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

3. Simplified 90.7%

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

5. Taylor expanded in beta around 0 81.0%

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

6. Taylor expanded in beta around 0 73.3%

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

7. Taylor expanded in alpha around 0 50.3%

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

   1. +-commutative 50.3%

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

9. Simplified 50.3%

   \[\leadsto \color{blue}{\frac{0.25}{\beta + 3}} \]
10. Add Preprocessing

Alternative 15: 45.9% accurate, 35.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ 0.08333333333333333 \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta) :precision binary64 0.08333333333333333)

C

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

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = 0.08333333333333333d0
end function

Java

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

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return 0.08333333333333333

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return 0.08333333333333333
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = 0.08333333333333333;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := 0.08333333333333333

Derivation

1. Initial program 92.5%

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

   1. associate-/l/ 90.7%

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

   2. +-commutative 90.7%

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

   3. associate-+l+ 90.7%

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

   4. *-commutative 90.7%

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

   5. metadata-eval 90.7%

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

   6. associate-+l+ 90.7%

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

   7. metadata-eval 90.7%

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

   8. +-commutative 90.7%

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

   9. +-commutative 90.7%

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

   10. +-commutative 90.7%

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

   11. metadata-eval 90.7%

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

   12. metadata-eval 90.7%

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

   13. associate-+l+ 90.7%

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

3. Simplified 90.7%

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

5. Taylor expanded in beta around 0 81.0%

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

6. Taylor expanded in beta around 0 73.3%

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

7. Taylor expanded in alpha around 0 50.3%

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

   1. +-commutative 50.3%

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

9. Simplified 50.3%

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

10. Taylor expanded in beta around 0 48.5%

    \[\leadsto \color{blue}{0.08333333333333333} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024157 
(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)))