Octave 3.8, jcobi/3

Percentage Accurate: 94.5% → 99.6%

Time: 18.5s

Alternatives: 20

Speedup: 2.5×

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: 94.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\alpha + \beta\right)\\ t_1 := \alpha + \left(2 + \beta\right)\\ \mathbf{if}\;\beta \leq 3.7 \cdot 10^{+68}:\\ \;\;\;\;\frac{1 + \alpha}{t\_1} \cdot \frac{1 + \beta}{t\_1 \cdot \left(\alpha + \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{t\_0}}{t\_0} \cdot \left(1 - \frac{\alpha + 2}{\beta}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (+ alpha beta))) (t_1 (+ alpha (+ 2.0 beta))))
   (if (<= beta 3.7e+68)
     (* (/ (+ 1.0 alpha) t_1) (/ (+ 1.0 beta) (* t_1 (+ alpha (+ beta 3.0)))))
     (* (/ (/ (+ 1.0 alpha) t_0) t_0) (- 1.0 (/ (+ alpha 2.0) beta))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (alpha + beta);
	double t_1 = alpha + (2.0 + beta);
	double tmp;
	if (beta <= 3.7e+68) {
		tmp = ((1.0 + alpha) / t_1) * ((1.0 + beta) / (t_1 * (alpha + (beta + 3.0))));
	} else {
		tmp = (((1.0 + alpha) / t_0) / t_0) * (1.0 - ((alpha + 2.0) / 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 2.0d0 + (alpha + beta)
    t_1 = alpha + (2.0d0 + beta)
    if (beta <= 3.7d+68) then
        tmp = ((1.0d0 + alpha) / t_1) * ((1.0d0 + beta) / (t_1 * (alpha + (beta + 3.0d0))))
    else
        tmp = (((1.0d0 + alpha) / t_0) / t_0) * (1.0d0 - ((alpha + 2.0d0) / beta))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = 2.0 + (alpha + beta);
	double t_1 = alpha + (2.0 + beta);
	double tmp;
	if (beta <= 3.7e+68) {
		tmp = ((1.0 + alpha) / t_1) * ((1.0 + beta) / (t_1 * (alpha + (beta + 3.0))));
	} else {
		tmp = (((1.0 + alpha) / t_0) / t_0) * (1.0 - ((alpha + 2.0) / beta));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = 2.0 + (alpha + beta)
	t_1 = alpha + (2.0 + beta)
	tmp = 0
	if beta <= 3.7e+68:
		tmp = ((1.0 + alpha) / t_1) * ((1.0 + beta) / (t_1 * (alpha + (beta + 3.0))))
	else:
		tmp = (((1.0 + alpha) / t_0) / t_0) * (1.0 - ((alpha + 2.0) / beta))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(alpha + beta))
	t_1 = Float64(alpha + Float64(2.0 + beta))
	tmp = 0.0
	if (beta <= 3.7e+68)
		tmp = Float64(Float64(Float64(1.0 + alpha) / t_1) * Float64(Float64(1.0 + beta) / Float64(t_1 * Float64(alpha + Float64(beta + 3.0)))));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + alpha) / t_0) / t_0) * Float64(1.0 - Float64(Float64(alpha + 2.0) / beta)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = 2.0 + (alpha + beta);
	t_1 = alpha + (2.0 + beta);
	tmp = 0.0;
	if (beta <= 3.7e+68)
		tmp = ((1.0 + alpha) / t_1) * ((1.0 + beta) / (t_1 * (alpha + (beta + 3.0))));
	else
		tmp = (((1.0 + alpha) / t_0) / t_0) * (1.0 - ((alpha + 2.0) / beta));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 3.7e+68], N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$1), $MachinePrecision] * N[(N[(1.0 + beta), $MachinePrecision] / N[(t$95$1 * N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(1.0 - N[(N[(alpha + 2.0), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.69999999999999998e68

   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 93.8%

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

      1. times-frac 99.2%

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

      2. +-commutative 99.2%

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

   6. Applied egg-rr 99.2%

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

   if 3.69999999999999998e68 < beta

   1. Initial program 73.9%

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

   3. Simplified 52.2%

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

      1. times-frac 82.3%

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

      2. +-commutative 82.3%

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

   6. Applied egg-rr 82.3%

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

      1. associate-*r/ 82.3%

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

      2. times-frac 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+r+ 99.8%

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

      5. +-commutative 99.8%

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

      6. +-commutative 99.8%

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

      7. associate-+r+ 99.8%

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

      8. +-commutative 99.8%

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

      9. +-commutative 99.8%

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

      10. associate-+r+ 99.8%

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

      11. +-commutative 99.8%

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

      12. +-commutative 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in beta around inf 90.0%

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

       1. associate-*r/ 90.0%

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

       2. mul-1-neg 90.0%

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

   11. Simplified 90.0%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.7 \cdot 10^{+68}:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 + \beta}{\left(\alpha + \left(2 + \beta\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{2 + \left(\alpha + \beta\right)}}{2 + \left(\alpha + \beta\right)} \cdot \left(1 - \frac{\alpha + 2}{\beta}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\alpha + \beta\right)\\ \mathbf{if}\;\beta \leq 3.1 \cdot 10^{+38}:\\ \;\;\;\;\frac{\frac{1 + \beta}{t\_0}}{\left(2 + \beta\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{t\_0}}{t\_0} \cdot \left(1 - \frac{\alpha + 2}{\beta}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (+ alpha beta))))
   (if (<= beta 3.1e+38)
     (/ (/ (+ 1.0 beta) t_0) (* (+ 2.0 beta) (+ beta 3.0)))
     (* (/ (/ (+ 1.0 alpha) t_0) t_0) (- 1.0 (/ (+ alpha 2.0) beta))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (alpha + beta);
	double tmp;
	if (beta <= 3.1e+38) {
		tmp = ((1.0 + beta) / t_0) / ((2.0 + beta) * (beta + 3.0));
	} else {
		tmp = (((1.0 + alpha) / t_0) / t_0) * (1.0 - ((alpha + 2.0) / 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) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 + (alpha + beta)
    if (beta <= 3.1d+38) then
        tmp = ((1.0d0 + beta) / t_0) / ((2.0d0 + beta) * (beta + 3.0d0))
    else
        tmp = (((1.0d0 + alpha) / t_0) / t_0) * (1.0d0 - ((alpha + 2.0d0) / beta))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = 2.0 + (alpha + beta);
	double tmp;
	if (beta <= 3.1e+38) {
		tmp = ((1.0 + beta) / t_0) / ((2.0 + beta) * (beta + 3.0));
	} else {
		tmp = (((1.0 + alpha) / t_0) / t_0) * (1.0 - ((alpha + 2.0) / beta));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = 2.0 + (alpha + beta)
	tmp = 0
	if beta <= 3.1e+38:
		tmp = ((1.0 + beta) / t_0) / ((2.0 + beta) * (beta + 3.0))
	else:
		tmp = (((1.0 + alpha) / t_0) / t_0) * (1.0 - ((alpha + 2.0) / beta))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(alpha + beta))
	tmp = 0.0
	if (beta <= 3.1e+38)
		tmp = Float64(Float64(Float64(1.0 + beta) / t_0) / Float64(Float64(2.0 + beta) * Float64(beta + 3.0)));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + alpha) / t_0) / t_0) * Float64(1.0 - Float64(Float64(alpha + 2.0) / beta)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = 2.0 + (alpha + beta);
	tmp = 0.0;
	if (beta <= 3.1e+38)
		tmp = ((1.0 + beta) / t_0) / ((2.0 + beta) * (beta + 3.0));
	else
		tmp = (((1.0 + alpha) / t_0) / t_0) * (1.0 - ((alpha + 2.0) / beta));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 3.1e+38], N[(N[(N[(1.0 + beta), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(N[(2.0 + beta), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(1.0 - N[(N[(alpha + 2.0), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.10000000000000018e38

   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 93.3%

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

   5. Taylor expanded in alpha around 0 81.3%

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

   6. Taylor expanded in alpha around 0 63.2%

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

      1. +-commutative 63.2%

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

      2. +-commutative 63.2%

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

   8. Simplified 63.2%

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

      1. *-un-lft-identity 63.2%

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

      2. associate-+r+ 63.2%

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

      3. +-commutative 63.2%

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

      4. +-commutative 63.2%

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

      5. +-commutative 63.2%

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

   10. Applied egg-rr 63.2%

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

       1. *-lft-identity 63.2%

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

       2. associate-/r* 62.7%

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

       3. +-commutative 62.7%

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

   12. Simplified 62.7%

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

   if 3.10000000000000018e38 < beta

   1. Initial program 78.5%

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

   3. Simplified 60.7%

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

      1. times-frac 85.4%

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

      2. +-commutative 85.4%

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

   6. Applied egg-rr 85.4%

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

      1. associate-*r/ 85.5%

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

      2. times-frac 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+r+ 99.8%

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

      5. +-commutative 99.8%

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

      6. +-commutative 99.8%

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

      7. associate-+r+ 99.8%

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

      8. +-commutative 99.8%

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

      9. +-commutative 99.8%

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

      10. associate-+r+ 99.8%

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

      11. +-commutative 99.8%

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

      12. +-commutative 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in beta around inf 89.4%

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

       1. associate-*r/ 89.4%

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

       2. mul-1-neg 89.4%

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

   11. Simplified 89.4%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.1 \cdot 10^{+38}:\\ \;\;\;\;\frac{\frac{1 + \beta}{2 + \left(\alpha + \beta\right)}}{\left(2 + \beta\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{2 + \left(\alpha + \beta\right)}}{2 + \left(\alpha + \beta\right)} \cdot \left(1 - \frac{\alpha + 2}{\beta}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\alpha + \beta\right)\\ \mathbf{if}\;\beta \leq 3.1 \cdot 10^{+38}:\\ \;\;\;\;\frac{\frac{1 + \beta}{t\_0}}{\left(2 + \beta\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \beta}{\left(\alpha + \beta\right) + 3} \cdot \frac{\frac{1 + \alpha}{\beta}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (+ alpha beta))))
   (if (<= beta 3.1e+38)
     (/ (/ (+ 1.0 beta) t_0) (* (+ 2.0 beta) (+ beta 3.0)))
     (*
      (/ (+ 1.0 beta) (+ (+ alpha beta) 3.0))
      (/ (/ (+ 1.0 alpha) beta) t_0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (alpha + beta);
	double tmp;
	if (beta <= 3.1e+38) {
		tmp = ((1.0 + beta) / t_0) / ((2.0 + beta) * (beta + 3.0));
	} else {
		tmp = ((1.0 + beta) / ((alpha + beta) + 3.0)) * (((1.0 + alpha) / beta) / t_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) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 + (alpha + beta)
    if (beta <= 3.1d+38) then
        tmp = ((1.0d0 + beta) / t_0) / ((2.0d0 + beta) * (beta + 3.0d0))
    else
        tmp = ((1.0d0 + beta) / ((alpha + beta) + 3.0d0)) * (((1.0d0 + alpha) / beta) / t_0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = 2.0 + (alpha + beta);
	double tmp;
	if (beta <= 3.1e+38) {
		tmp = ((1.0 + beta) / t_0) / ((2.0 + beta) * (beta + 3.0));
	} else {
		tmp = ((1.0 + beta) / ((alpha + beta) + 3.0)) * (((1.0 + alpha) / beta) / t_0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = 2.0 + (alpha + beta)
	tmp = 0
	if beta <= 3.1e+38:
		tmp = ((1.0 + beta) / t_0) / ((2.0 + beta) * (beta + 3.0))
	else:
		tmp = ((1.0 + beta) / ((alpha + beta) + 3.0)) * (((1.0 + alpha) / beta) / t_0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(alpha + beta))
	tmp = 0.0
	if (beta <= 3.1e+38)
		tmp = Float64(Float64(Float64(1.0 + beta) / t_0) / Float64(Float64(2.0 + beta) * Float64(beta + 3.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(Float64(alpha + beta) + 3.0)) * Float64(Float64(Float64(1.0 + alpha) / beta) / t_0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = 2.0 + (alpha + beta);
	tmp = 0.0;
	if (beta <= 3.1e+38)
		tmp = ((1.0 + beta) / t_0) / ((2.0 + beta) * (beta + 3.0));
	else
		tmp = ((1.0 + beta) / ((alpha + beta) + 3.0)) * (((1.0 + alpha) / beta) / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 3.1e+38], N[(N[(N[(1.0 + beta), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(N[(2.0 + beta), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.10000000000000018e38

   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 93.3%

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

   5. Taylor expanded in alpha around 0 81.3%

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

   6. Taylor expanded in alpha around 0 63.2%

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

      1. +-commutative 63.2%

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

      2. +-commutative 63.2%

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

   8. Simplified 63.2%

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

      1. *-un-lft-identity 63.2%

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

      2. associate-+r+ 63.2%

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

      3. +-commutative 63.2%

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

      4. +-commutative 63.2%

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

      5. +-commutative 63.2%

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

   10. Applied egg-rr 63.2%

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

       1. *-lft-identity 63.2%

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

       2. associate-/r* 62.7%

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

       3. +-commutative 62.7%

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

   12. Simplified 62.7%

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

   if 3.10000000000000018e38 < beta

   1. Initial program 78.5%

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

   3. Simplified 60.7%

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

      1. times-frac 85.4%

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

      2. +-commutative 85.4%

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

   6. Applied egg-rr 85.4%

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

      1. associate-*r/ 85.5%

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

      2. times-frac 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+r+ 99.8%

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

      5. +-commutative 99.8%

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

      6. +-commutative 99.8%

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

      7. associate-+r+ 99.8%

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

      8. +-commutative 99.8%

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

      9. +-commutative 99.8%

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

      10. associate-+r+ 99.8%

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

      11. +-commutative 99.8%

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

      12. +-commutative 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in beta around inf 89.2%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.1 \cdot 10^{+38}:\\ \;\;\;\;\frac{\frac{1 + \beta}{2 + \left(\alpha + \beta\right)}}{\left(2 + \beta\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \beta}{\left(\alpha + \beta\right) + 3} \cdot \frac{\frac{1 + \alpha}{\beta}}{2 + \left(\alpha + \beta\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.8% accurate, 1.4× speedup

Math

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

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (+ alpha beta))))
   (* (/ (/ (+ 1.0 alpha) t_0) t_0) (/ (+ 1.0 beta) (+ (+ alpha beta) 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (alpha + beta);
	return (((1.0 + alpha) / t_0) / t_0) * ((1.0 + beta) / ((alpha + 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
    real(8) :: t_0
    t_0 = 2.0d0 + (alpha + beta)
    code = (((1.0d0 + alpha) / t_0) / t_0) * ((1.0d0 + beta) / ((alpha + beta) + 3.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	t_0 = 2.0 + (alpha + beta);
	tmp = (((1.0 + alpha) / t_0) / t_0) * ((1.0 + beta) / ((alpha + beta) + 3.0));
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 92.9%

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

3. Simplified 82.7%

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

   1. times-frac 94.7%

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

   2. +-commutative 94.7%

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

6. Applied egg-rr 94.7%

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

   1. associate-*r/ 94.7%

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

   2. times-frac 99.8%

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

   3. +-commutative 99.8%

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

   4. associate-+r+ 99.8%

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

   5. +-commutative 99.8%

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

   6. +-commutative 99.8%

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

   7. associate-+r+ 99.8%

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

   8. +-commutative 99.8%

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

   9. +-commutative 99.8%

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

   10. associate-+r+ 99.8%

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

   11. +-commutative 99.8%

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

   12. +-commutative 99.8%

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

8. Simplified 99.8%

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

9. Final simplification 99.8%

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

Alternative 5: 98.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.1 \cdot 10^{+24}:\\ \;\;\;\;\frac{1 + \beta}{\left(\alpha + \left(2 + \beta\right)\right) \cdot \left(\left(2 + \beta\right) \cdot \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.1e+24)
   (/ (+ 1.0 beta) (* (+ alpha (+ 2.0 beta)) (* (+ 2.0 beta) (+ beta 3.0))))
   (/ (/ (+ 1.0 alpha) beta) (+ beta (+ alpha 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.1e+24) {
		tmp = (1.0 + beta) / ((alpha + (2.0 + beta)) * ((2.0 + beta) * (beta + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 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 <= 4.1d+24) then
        tmp = (1.0d0 + beta) / ((alpha + (2.0d0 + beta)) * ((2.0d0 + beta) * (beta + 3.0d0)))
    else
        tmp = ((1.0d0 + alpha) / beta) / (beta + (alpha + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.1e+24) {
		tmp = (1.0 + beta) / ((alpha + (2.0 + beta)) * ((2.0 + beta) * (beta + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 4.1e+24:
		tmp = (1.0 + beta) / ((alpha + (2.0 + beta)) * ((2.0 + beta) * (beta + 3.0)))
	else:
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.1e+24)
		tmp = Float64(Float64(1.0 + beta) / Float64(Float64(alpha + Float64(2.0 + beta)) * Float64(Float64(2.0 + beta) * Float64(beta + 3.0))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(beta + Float64(alpha + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4.1e+24)
		tmp = (1.0 + beta) / ((alpha + (2.0 + beta)) * ((2.0 + beta) * (beta + 3.0)));
	else
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 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, 4.1e+24], N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 + beta), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.1000000000000001e24

   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 93.2%

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

   5. Taylor expanded in alpha around 0 81.0%

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

   6. Taylor expanded in alpha around 0 62.6%

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

      1. +-commutative 62.6%

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

      2. +-commutative 62.6%

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

   8. Simplified 62.6%

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

   if 4.1000000000000001e24 < beta

   1. Initial program 79.3%

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

   3. Taylor expanded in beta around inf 88.2%

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

      1. metadata-eval 88.2%

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

      2. associate-+l+ 88.2%

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

      3. metadata-eval 88.2%

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

      4. +-commutative 88.2%

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

      5. associate-+r+ 88.2%

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

      6. *-un-lft-identity 88.2%

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

      7. fma-define 88.2%

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

   5. Applied egg-rr 88.2%

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

      1. fma-undefine 88.2%

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

      2. *-lft-identity 88.2%

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

      3. +-commutative 88.2%

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

   7. Simplified 88.2%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.1 \cdot 10^{+24}:\\ \;\;\;\;\frac{1 + \beta}{\left(\alpha + \left(2 + \beta\right)\right) \cdot \left(\left(2 + \beta\right) \cdot \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.1 \cdot 10^{+24}:\\ \;\;\;\;\frac{1 + \beta}{\left(\alpha + \left(2 + \beta\right)\right) \cdot \left(6 + \beta \cdot \left(\beta + 5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.1e+24)
   (/ (+ 1.0 beta) (* (+ alpha (+ 2.0 beta)) (+ 6.0 (* beta (+ beta 5.0)))))
   (/ (/ (+ 1.0 alpha) beta) (+ beta (+ alpha 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.1e+24) {
		tmp = (1.0 + beta) / ((alpha + (2.0 + beta)) * (6.0 + (beta * (beta + 5.0))));
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 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 <= 4.1d+24) then
        tmp = (1.0d0 + beta) / ((alpha + (2.0d0 + beta)) * (6.0d0 + (beta * (beta + 5.0d0))))
    else
        tmp = ((1.0d0 + alpha) / beta) / (beta + (alpha + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.1e+24) {
		tmp = (1.0 + beta) / ((alpha + (2.0 + beta)) * (6.0 + (beta * (beta + 5.0))));
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 4.1e+24:
		tmp = (1.0 + beta) / ((alpha + (2.0 + beta)) * (6.0 + (beta * (beta + 5.0))))
	else:
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.1e+24)
		tmp = Float64(Float64(1.0 + beta) / Float64(Float64(alpha + Float64(2.0 + beta)) * Float64(6.0 + Float64(beta * Float64(beta + 5.0)))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(beta + Float64(alpha + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4.1e+24)
		tmp = (1.0 + beta) / ((alpha + (2.0 + beta)) * (6.0 + (beta * (beta + 5.0))));
	else
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 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, 4.1e+24], N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] * N[(6.0 + N[(beta * N[(beta + 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.1000000000000001e24

   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 93.2%

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

   5. Taylor expanded in alpha around 0 81.0%

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

   6. Taylor expanded in alpha around 0 62.6%

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

      1. +-commutative 62.6%

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

      2. +-commutative 62.6%

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

   8. Simplified 62.6%

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

   9. Taylor expanded in beta around 0 62.6%

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

       1. +-commutative 62.6%

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

   11. Simplified 62.6%

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

   if 4.1000000000000001e24 < beta

   1. Initial program 79.3%

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

   3. Taylor expanded in beta around inf 88.2%

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

      1. metadata-eval 88.2%

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

      2. associate-+l+ 88.2%

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

      3. metadata-eval 88.2%

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

      4. +-commutative 88.2%

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

      5. associate-+r+ 88.2%

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

      6. *-un-lft-identity 88.2%

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

      7. fma-define 88.2%

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

   5. Applied egg-rr 88.2%

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

      1. fma-undefine 88.2%

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

      2. *-lft-identity 88.2%

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

      3. +-commutative 88.2%

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

   7. Simplified 88.2%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.1 \cdot 10^{+24}:\\ \;\;\;\;\frac{1 + \beta}{\left(\alpha + \left(2 + \beta\right)\right) \cdot \left(6 + \beta \cdot \left(\beta + 5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.35 \cdot 10^{+38}:\\ \;\;\;\;\frac{\frac{1 + \beta}{2 + \left(\alpha + \beta\right)}}{\left(2 + \beta\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.35e+38)
   (/ (/ (+ 1.0 beta) (+ 2.0 (+ alpha beta))) (* (+ 2.0 beta) (+ beta 3.0)))
   (/ (/ (+ 1.0 alpha) beta) (+ beta (+ alpha 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.35e+38) {
		tmp = ((1.0 + beta) / (2.0 + (alpha + beta))) / ((2.0 + beta) * (beta + 3.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 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 <= 3.35d+38) then
        tmp = ((1.0d0 + beta) / (2.0d0 + (alpha + beta))) / ((2.0d0 + beta) * (beta + 3.0d0))
    else
        tmp = ((1.0d0 + alpha) / beta) / (beta + (alpha + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.35e+38) {
		tmp = ((1.0 + beta) / (2.0 + (alpha + beta))) / ((2.0 + beta) * (beta + 3.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 3.35e+38:
		tmp = ((1.0 + beta) / (2.0 + (alpha + beta))) / ((2.0 + beta) * (beta + 3.0))
	else:
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.35e+38)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(2.0 + Float64(alpha + beta))) / Float64(Float64(2.0 + beta) * Float64(beta + 3.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(beta + Float64(alpha + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.35e+38)
		tmp = ((1.0 + beta) / (2.0 + (alpha + beta))) / ((2.0 + beta) * (beta + 3.0));
	else
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 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, 3.35e+38], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 + beta), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.35000000000000012e38

   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 93.3%

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

   5. Taylor expanded in alpha around 0 81.3%

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

   6. Taylor expanded in alpha around 0 63.2%

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

      1. +-commutative 63.2%

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

      2. +-commutative 63.2%

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

   8. Simplified 63.2%

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

      1. *-un-lft-identity 63.2%

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

      2. associate-+r+ 63.2%

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

      3. +-commutative 63.2%

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

      4. +-commutative 63.2%

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

      5. +-commutative 63.2%

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

   10. Applied egg-rr 63.2%

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

       1. *-lft-identity 63.2%

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

       2. associate-/r* 62.7%

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

       3. +-commutative 62.7%

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

   12. Simplified 62.7%

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

   if 3.35000000000000012e38 < beta

   1. Initial program 78.5%

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

   3. Taylor expanded in beta around inf 89.0%

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

      1. metadata-eval 89.0%

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

      2. associate-+l+ 89.0%

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

      3. metadata-eval 89.0%

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

      4. +-commutative 89.0%

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

      5. associate-+r+ 89.0%

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

      6. *-un-lft-identity 89.0%

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

      7. fma-define 89.0%

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

   5. Applied egg-rr 89.0%

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

      1. fma-undefine 89.0%

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

      2. *-lft-identity 89.0%

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

      3. +-commutative 89.0%

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

   7. Simplified 89.0%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.35 \cdot 10^{+38}:\\ \;\;\;\;\frac{\frac{1 + \beta}{2 + \left(\alpha + \beta\right)}}{\left(2 + \beta\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.9:\\ \;\;\;\;\frac{1 + \alpha}{\left(\alpha + 2\right) \cdot \left(\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.9)
   (/ (+ 1.0 alpha) (* (+ alpha 2.0) (* (+ alpha 2.0) (+ alpha 3.0))))
   (/ (/ (+ 1.0 alpha) beta) (+ beta (+ alpha 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.9) {
		tmp = (1.0 + alpha) / ((alpha + 2.0) * ((alpha + 2.0) * (alpha + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 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 <= 2.9d0) then
        tmp = (1.0d0 + alpha) / ((alpha + 2.0d0) * ((alpha + 2.0d0) * (alpha + 3.0d0)))
    else
        tmp = ((1.0d0 + alpha) / beta) / (beta + (alpha + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.9) {
		tmp = (1.0 + alpha) / ((alpha + 2.0) * ((alpha + 2.0) * (alpha + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.9:
		tmp = (1.0 + alpha) / ((alpha + 2.0) * ((alpha + 2.0) * (alpha + 3.0)))
	else:
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.9)
		tmp = Float64(Float64(1.0 + alpha) / Float64(Float64(alpha + 2.0) * Float64(Float64(alpha + 2.0) * Float64(alpha + 3.0))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(beta + Float64(alpha + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.9)
		tmp = (1.0 + alpha) / ((alpha + 2.0) * ((alpha + 2.0) * (alpha + 3.0)));
	else
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 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, 2.9], N[(N[(1.0 + alpha), $MachinePrecision] / N[(N[(alpha + 2.0), $MachinePrecision] * N[(N[(alpha + 2.0), $MachinePrecision] * N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.89999999999999991

   1. Initial program 99.9%

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

      1. associate-/l/ 99.1%

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

      2. +-commutative 99.1%

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

      3. associate-+l+ 99.1%

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

      4. *-commutative 99.1%

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

      5. metadata-eval 99.1%

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

      6. associate-+l+ 99.1%

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

      7. metadata-eval 99.1%

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

      8. associate-+l+ 99.1%

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

      9. metadata-eval 99.1%

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

      10. metadata-eval 99.1%

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

      11. associate-+l+ 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in beta around 0 97.9%

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

      1. *-un-lft-identity 97.9%

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

      2. +-commutative 97.9%

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

      3. associate-+r+ 97.9%

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

      4. *-commutative 97.9%

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

      5. +-commutative 97.9%

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

   7. Applied egg-rr 97.9%

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

      1. *-lft-identity 97.9%

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

      2. associate-/l/ 92.2%

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

      3. +-commutative 92.2%

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

      4. *-commutative 92.2%

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

      5. +-commutative 92.2%

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

      6. associate-+l+ 92.2%

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

   9. Simplified 92.2%

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

   10. Taylor expanded in beta around 0 92.3%

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

       1. +-commutative 92.3%

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

   12. Simplified 92.3%

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

   if 2.89999999999999991 < beta

   1. Initial program 80.3%

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

   3. Taylor expanded in beta around inf 85.6%

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

      1. metadata-eval 85.6%

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

      2. associate-+l+ 85.6%

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

      3. metadata-eval 85.6%

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

      4. +-commutative 85.6%

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

      5. associate-+r+ 85.6%

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

      6. *-un-lft-identity 85.6%

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

      7. fma-define 85.6%

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

   5. Applied egg-rr 85.6%

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

      1. fma-undefine 85.6%

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

      2. *-lft-identity 85.6%

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

      3. +-commutative 85.6%

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

   7. Simplified 85.6%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.9:\\ \;\;\;\;\frac{1 + \alpha}{\left(\alpha + 2\right) \cdot \left(\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.3 \cdot 10^{+20}:\\ \;\;\;\;\frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(\left(2 + \beta\right) \cdot \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.3e+20)
   (/ (+ 1.0 beta) (* (+ 2.0 beta) (* (+ 2.0 beta) (+ beta 3.0))))
   (/ (/ (+ 1.0 alpha) beta) (+ beta (+ alpha 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.3e+20) {
		tmp = (1.0 + beta) / ((2.0 + beta) * ((2.0 + beta) * (beta + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 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 <= 1.3d+20) then
        tmp = (1.0d0 + beta) / ((2.0d0 + beta) * ((2.0d0 + beta) * (beta + 3.0d0)))
    else
        tmp = ((1.0d0 + alpha) / beta) / (beta + (alpha + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.3e+20) {
		tmp = (1.0 + beta) / ((2.0 + beta) * ((2.0 + beta) * (beta + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.3e+20:
		tmp = (1.0 + beta) / ((2.0 + beta) * ((2.0 + beta) * (beta + 3.0)))
	else:
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.3e+20)
		tmp = Float64(Float64(1.0 + beta) / Float64(Float64(2.0 + beta) * Float64(Float64(2.0 + beta) * Float64(beta + 3.0))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(beta + Float64(alpha + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.3e+20)
		tmp = (1.0 + beta) / ((2.0 + beta) * ((2.0 + beta) * (beta + 3.0)));
	else
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 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, 1.3e+20], N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(2.0 + beta), $MachinePrecision] * N[(N[(2.0 + beta), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.3e20

   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 93.2%

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

   5. Taylor expanded in alpha around 0 81.0%

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

   6. Taylor expanded in alpha around 0 62.6%

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

      1. +-commutative 62.6%

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

      2. +-commutative 62.6%

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

   8. Simplified 62.6%

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

   9. Taylor expanded in alpha around 0 61.5%

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

       1. +-commutative 61.5%

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

   11. Simplified 61.5%

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

   if 1.3e20 < beta

   1. Initial program 79.3%

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

   3. Taylor expanded in beta around inf 88.2%

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

      1. metadata-eval 88.2%

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

      2. associate-+l+ 88.2%

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

      3. metadata-eval 88.2%

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

      4. +-commutative 88.2%

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

      5. associate-+r+ 88.2%

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

      6. *-un-lft-identity 88.2%

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

      7. fma-define 88.2%

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

   5. Applied egg-rr 88.2%

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

      1. fma-undefine 88.2%

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

      2. *-lft-identity 88.2%

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

      3. +-commutative 88.2%

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

   7. Simplified 88.2%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.3 \cdot 10^{+20}:\\ \;\;\;\;\frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(\left(2 + \beta\right) \cdot \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 98.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.15 \cdot 10^{+38}:\\ \;\;\;\;\frac{\frac{1 + \beta}{2 + \beta}}{\left(2 + \beta\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.15e+38)
   (/ (/ (+ 1.0 beta) (+ 2.0 beta)) (* (+ 2.0 beta) (+ beta 3.0)))
   (/ (/ (+ 1.0 alpha) beta) (+ beta (+ alpha 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.15e+38) {
		tmp = ((1.0 + beta) / (2.0 + beta)) / ((2.0 + beta) * (beta + 3.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 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 <= 3.15d+38) then
        tmp = ((1.0d0 + beta) / (2.0d0 + beta)) / ((2.0d0 + beta) * (beta + 3.0d0))
    else
        tmp = ((1.0d0 + alpha) / beta) / (beta + (alpha + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.15e+38) {
		tmp = ((1.0 + beta) / (2.0 + beta)) / ((2.0 + beta) * (beta + 3.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 3.15e+38:
		tmp = ((1.0 + beta) / (2.0 + beta)) / ((2.0 + beta) * (beta + 3.0))
	else:
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.15e+38)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(2.0 + beta)) / Float64(Float64(2.0 + beta) * Float64(beta + 3.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(beta + Float64(alpha + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.15e+38)
		tmp = ((1.0 + beta) / (2.0 + beta)) / ((2.0 + beta) * (beta + 3.0));
	else
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 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, 3.15e+38], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 + beta), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.15000000000000001e38

   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 93.3%

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

   5. Taylor expanded in alpha around 0 81.3%

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

   6. Taylor expanded in alpha around 0 63.2%

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

      1. +-commutative 63.2%

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

      2. +-commutative 63.2%

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

   8. Simplified 63.2%

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

      1. *-un-lft-identity 63.2%

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

      2. associate-+r+ 63.2%

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

      3. +-commutative 63.2%

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

      4. +-commutative 63.2%

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

      5. +-commutative 63.2%

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

   10. Applied egg-rr 63.2%

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

       1. *-lft-identity 63.2%

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

       2. associate-/r* 62.7%

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

       3. +-commutative 62.7%

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

   12. Simplified 62.7%

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

   13. Taylor expanded in alpha around 0 61.6%

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

       1. +-commutative 61.6%

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

   15. Simplified 61.6%

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

   if 3.15000000000000001e38 < beta

   1. Initial program 78.5%

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

   3. Taylor expanded in beta around inf 89.0%

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

      1. metadata-eval 89.0%

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

      2. associate-+l+ 89.0%

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

      3. metadata-eval 89.0%

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

      4. +-commutative 89.0%

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

      5. associate-+r+ 89.0%

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

      6. *-un-lft-identity 89.0%

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

      7. fma-define 89.0%

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

   5. Applied egg-rr 89.0%

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

      1. fma-undefine 89.0%

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

      2. *-lft-identity 89.0%

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

      3. +-commutative 89.0%

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

   7. Simplified 89.0%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.15 \cdot 10^{+38}:\\ \;\;\;\;\frac{\frac{1 + \beta}{2 + \beta}}{\left(2 + \beta\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 96.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.95:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot -0.041666666666666664\\ \mathbf{elif}\;\beta \leq 4.5 \cdot 10^{+155}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.95)
   (+ 0.08333333333333333 (* alpha -0.041666666666666664))
   (if (<= beta 4.5e+155)
     (/ (+ 1.0 alpha) (* beta (+ beta 3.0)))
     (/ (/ alpha beta) (+ beta (+ alpha 3.0))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.95) {
		tmp = 0.08333333333333333 + (alpha * -0.041666666666666664);
	} else if (beta <= 4.5e+155) {
		tmp = (1.0 + alpha) / (beta * (beta + 3.0));
	} else {
		tmp = (alpha / beta) / (beta + (alpha + 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 <= 1.95d0) then
        tmp = 0.08333333333333333d0 + (alpha * (-0.041666666666666664d0))
    else if (beta <= 4.5d+155) then
        tmp = (1.0d0 + alpha) / (beta * (beta + 3.0d0))
    else
        tmp = (alpha / beta) / (beta + (alpha + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.95) {
		tmp = 0.08333333333333333 + (alpha * -0.041666666666666664);
	} else if (beta <= 4.5e+155) {
		tmp = (1.0 + alpha) / (beta * (beta + 3.0));
	} else {
		tmp = (alpha / beta) / (beta + (alpha + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.95:
		tmp = 0.08333333333333333 + (alpha * -0.041666666666666664)
	elif beta <= 4.5e+155:
		tmp = (1.0 + alpha) / (beta * (beta + 3.0))
	else:
		tmp = (alpha / beta) / (beta + (alpha + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.95)
		tmp = Float64(0.08333333333333333 + Float64(alpha * -0.041666666666666664));
	elseif (beta <= 4.5e+155)
		tmp = Float64(Float64(1.0 + alpha) / Float64(beta * Float64(beta + 3.0)));
	else
		tmp = Float64(Float64(alpha / beta) / Float64(beta + Float64(alpha + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.95)
		tmp = 0.08333333333333333 + (alpha * -0.041666666666666664);
	elseif (beta <= 4.5e+155)
		tmp = (1.0 + alpha) / (beta * (beta + 3.0));
	else
		tmp = (alpha / beta) / (beta + (alpha + 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, 1.95], N[(0.08333333333333333 + N[(alpha * -0.041666666666666664), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 4.5e+155], N[(N[(1.0 + alpha), $MachinePrecision] / N[(beta * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if beta < 1.94999999999999996

   1. Initial program 99.9%

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

   3. Simplified 93.0%

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

   5. Taylor expanded in alpha around 0 81.0%

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

   6. Taylor expanded in alpha around 0 62.1%

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

      1. +-commutative 62.1%

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

      2. +-commutative 62.1%

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

   8. Simplified 62.1%

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

   9. Taylor expanded in beta around 0 61.3%

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

       1. +-commutative 61.3%

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

   11. Simplified 61.3%

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

   12. Taylor expanded in alpha around 0 59.4%

       \[\leadsto \color{blue}{0.08333333333333333 + -0.041666666666666664 \cdot \alpha} \]
   13. Step-by-step derivation

       1. *-commutative 59.4%

          \[\leadsto 0.08333333333333333 + \color{blue}{\alpha \cdot -0.041666666666666664} \]

   14. Simplified 59.4%

       \[\leadsto \color{blue}{0.08333333333333333 + \alpha \cdot -0.041666666666666664} \]

   if 1.94999999999999996 < beta < 4.49999999999999973e155

   1. Initial program 93.7%

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

   3. Taylor expanded in beta around inf 78.7%

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

      1. metadata-eval 78.7%

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

      2. associate-+l+ 78.7%

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

      3. metadata-eval 78.7%

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

      4. +-commutative 78.7%

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

      5. associate-+r+ 78.7%

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

      6. *-un-lft-identity 78.7%

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

      7. fma-define 78.7%

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

   5. Applied egg-rr 78.7%

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

      1. fma-undefine 78.7%

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

      2. *-lft-identity 78.7%

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

      3. +-commutative 78.7%

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

   7. Simplified 78.7%

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

      1. *-un-lft-identity 78.7%

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

      2. associate-/l/ 80.7%

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

      3. +-commutative 80.7%

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

   9. Applied egg-rr 80.7%

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

       1. *-lft-identity 80.7%

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

       2. *-commutative 80.7%

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

   11. Simplified 80.7%

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

   12. Taylor expanded in alpha around 0 78.5%

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

       1. +-commutative 78.5%

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

   14. Simplified 78.5%

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

   if 4.49999999999999973e155 < beta

   1. Initial program 64.0%

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

   3. Taylor expanded in beta around inf 94.0%

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

      1. metadata-eval 94.0%

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

      2. associate-+l+ 94.0%

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

      3. metadata-eval 94.0%

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

      4. +-commutative 94.0%

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

      5. associate-+r+ 94.0%

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

      6. *-un-lft-identity 94.0%

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

      7. fma-define 94.0%

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

   5. Applied egg-rr 94.0%

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

      1. fma-undefine 94.0%

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

      2. *-lft-identity 94.0%

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

      3. +-commutative 94.0%

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

   7. Simplified 94.0%

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

   8. Taylor expanded in alpha around inf 92.7%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.95:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot -0.041666666666666664\\ \mathbf{elif}\;\beta \leq 4.5 \cdot 10^{+155}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 96.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.0)
   (+ 0.08333333333333333 (* alpha -0.041666666666666664))
   (/ (/ (+ 1.0 alpha) beta) (+ beta (+ alpha 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.0) {
		tmp = 0.08333333333333333 + (alpha * -0.041666666666666664);
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 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 <= 2.0d0) then
        tmp = 0.08333333333333333d0 + (alpha * (-0.041666666666666664d0))
    else
        tmp = ((1.0d0 + alpha) / beta) / (beta + (alpha + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.0) {
		tmp = 0.08333333333333333 + (alpha * -0.041666666666666664);
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.0:
		tmp = 0.08333333333333333 + (alpha * -0.041666666666666664)
	else:
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.0)
		tmp = Float64(0.08333333333333333 + Float64(alpha * -0.041666666666666664));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(beta + Float64(alpha + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.0)
		tmp = 0.08333333333333333 + (alpha * -0.041666666666666664);
	else
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 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, 2.0], N[(0.08333333333333333 + N[(alpha * -0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2

   1. Initial program 99.9%

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

   3. Simplified 93.0%

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

   5. Taylor expanded in alpha around 0 81.0%

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

   6. Taylor expanded in alpha around 0 62.1%

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

      1. +-commutative 62.1%

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

      2. +-commutative 62.1%

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

   8. Simplified 62.1%

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

   9. Taylor expanded in beta around 0 61.3%

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

       1. +-commutative 61.3%

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

   11. Simplified 61.3%

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

   12. Taylor expanded in alpha around 0 59.4%

       \[\leadsto \color{blue}{0.08333333333333333 + -0.041666666666666664 \cdot \alpha} \]
   13. Step-by-step derivation

       1. *-commutative 59.4%

          \[\leadsto 0.08333333333333333 + \color{blue}{\alpha \cdot -0.041666666666666664} \]

   14. Simplified 59.4%

       \[\leadsto \color{blue}{0.08333333333333333 + \alpha \cdot -0.041666666666666664} \]

   if 2 < beta

   1. Initial program 80.3%

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

   3. Taylor expanded in beta around inf 85.6%

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

      1. metadata-eval 85.6%

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

      2. associate-+l+ 85.6%

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

      3. metadata-eval 85.6%

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

      4. +-commutative 85.6%

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

      5. associate-+r+ 85.6%

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

      6. *-un-lft-identity 85.6%

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

      7. fma-define 85.6%

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

   5. Applied egg-rr 85.6%

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

      1. fma-undefine 85.6%

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

      2. *-lft-identity 85.6%

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

      3. +-commutative 85.6%

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

   7. Simplified 85.6%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 94.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.1:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.1)
   (+ 0.08333333333333333 (* alpha -0.041666666666666664))
   (/ (+ 1.0 alpha) (* beta (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.1) {
		tmp = 0.08333333333333333 + (alpha * -0.041666666666666664);
	} else {
		tmp = (1.0 + alpha) / (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 <= 2.1d0) then
        tmp = 0.08333333333333333d0 + (alpha * (-0.041666666666666664d0))
    else
        tmp = (1.0d0 + alpha) / (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 <= 2.1) {
		tmp = 0.08333333333333333 + (alpha * -0.041666666666666664);
	} else {
		tmp = (1.0 + alpha) / (beta * (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.1:
		tmp = 0.08333333333333333 + (alpha * -0.041666666666666664)
	else:
		tmp = (1.0 + alpha) / (beta * (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.1)
		tmp = Float64(0.08333333333333333 + Float64(alpha * -0.041666666666666664));
	else
		tmp = Float64(Float64(1.0 + alpha) / 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 <= 2.1)
		tmp = 0.08333333333333333 + (alpha * -0.041666666666666664);
	else
		tmp = (1.0 + alpha) / (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, 2.1], N[(0.08333333333333333 + N[(alpha * -0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + alpha), $MachinePrecision] / N[(beta * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.10000000000000009

   1. Initial program 99.9%

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

   3. Simplified 93.0%

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

   5. Taylor expanded in alpha around 0 81.0%

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

   6. Taylor expanded in alpha around 0 62.1%

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

      1. +-commutative 62.1%

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

      2. +-commutative 62.1%

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

   8. Simplified 62.1%

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

   9. Taylor expanded in beta around 0 61.3%

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

       1. +-commutative 61.3%

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

   11. Simplified 61.3%

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

   12. Taylor expanded in alpha around 0 59.4%

       \[\leadsto \color{blue}{0.08333333333333333 + -0.041666666666666664 \cdot \alpha} \]
   13. Step-by-step derivation

       1. *-commutative 59.4%

          \[\leadsto 0.08333333333333333 + \color{blue}{\alpha \cdot -0.041666666666666664} \]

   14. Simplified 59.4%

       \[\leadsto \color{blue}{0.08333333333333333 + \alpha \cdot -0.041666666666666664} \]

   if 2.10000000000000009 < beta

   1. Initial program 80.3%

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

   3. Taylor expanded in beta around inf 85.6%

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

      1. metadata-eval 85.6%

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

      2. associate-+l+ 85.6%

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

      3. metadata-eval 85.6%

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

      4. +-commutative 85.6%

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

      5. associate-+r+ 85.6%

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

      6. *-un-lft-identity 85.6%

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

      7. fma-define 85.6%

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

   5. Applied egg-rr 85.6%

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

      1. fma-undefine 85.6%

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

      2. *-lft-identity 85.6%

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

      3. +-commutative 85.6%

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

   7. Simplified 85.6%

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

      1. *-un-lft-identity 85.6%

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

      2. associate-/l/ 79.3%

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

      3. +-commutative 79.3%

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

   9. Applied egg-rr 79.3%

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

       1. *-lft-identity 79.3%

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

       2. *-commutative 79.3%

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

   11. Simplified 79.3%

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

   12. Taylor expanded in alpha around 0 78.1%

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

       1. +-commutative 78.1%

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

   14. Simplified 78.1%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.1:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 96.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.3:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.3)
   (+ 0.08333333333333333 (* alpha -0.041666666666666664))
   (/ (/ (+ 1.0 alpha) beta) (+ beta 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.3) {
		tmp = 0.08333333333333333 + (alpha * -0.041666666666666664);
	} else {
		tmp = ((1.0 + alpha) / 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 <= 2.3d0) then
        tmp = 0.08333333333333333d0 + (alpha * (-0.041666666666666664d0))
    else
        tmp = ((1.0d0 + alpha) / 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 <= 2.3) {
		tmp = 0.08333333333333333 + (alpha * -0.041666666666666664);
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.3:
		tmp = 0.08333333333333333 + (alpha * -0.041666666666666664)
	else:
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.3)
		tmp = Float64(0.08333333333333333 + Float64(alpha * -0.041666666666666664));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / 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 <= 2.3)
		tmp = 0.08333333333333333 + (alpha * -0.041666666666666664);
	else
		tmp = ((1.0 + alpha) / 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, 2.3], N[(0.08333333333333333 + N[(alpha * -0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.2999999999999998

   1. Initial program 99.9%

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

   3. Simplified 93.0%

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

   5. Taylor expanded in alpha around 0 81.0%

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

   6. Taylor expanded in alpha around 0 62.1%

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

      1. +-commutative 62.1%

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

      2. +-commutative 62.1%

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

   8. Simplified 62.1%

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

   9. Taylor expanded in beta around 0 61.3%

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

       1. +-commutative 61.3%

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

   11. Simplified 61.3%

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

   12. Taylor expanded in alpha around 0 59.4%

       \[\leadsto \color{blue}{0.08333333333333333 + -0.041666666666666664 \cdot \alpha} \]
   13. Step-by-step derivation

       1. *-commutative 59.4%

          \[\leadsto 0.08333333333333333 + \color{blue}{\alpha \cdot -0.041666666666666664} \]

   14. Simplified 59.4%

       \[\leadsto \color{blue}{0.08333333333333333 + \alpha \cdot -0.041666666666666664} \]

   if 2.2999999999999998 < beta

   1. Initial program 80.3%

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

   3. Taylor expanded in beta around inf 85.6%

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

   4. Taylor expanded in alpha around 0 85.5%

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

      1. +-commutative 85.5%

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

   6. Simplified 85.5%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.3:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 91.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.2:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.2)
   (+ 0.08333333333333333 (* alpha -0.041666666666666664))
   (/ 1.0 (* beta (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.2) {
		tmp = 0.08333333333333333 + (alpha * -0.041666666666666664);
	} 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 <= 2.2d0) then
        tmp = 0.08333333333333333d0 + (alpha * (-0.041666666666666664d0))
    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 <= 2.2) {
		tmp = 0.08333333333333333 + (alpha * -0.041666666666666664);
	} else {
		tmp = 1.0 / (beta * (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.2:
		tmp = 0.08333333333333333 + (alpha * -0.041666666666666664)
	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 <= 2.2)
		tmp = Float64(0.08333333333333333 + Float64(alpha * -0.041666666666666664));
	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 <= 2.2)
		tmp = 0.08333333333333333 + (alpha * -0.041666666666666664);
	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, 2.2], N[(0.08333333333333333 + N[(alpha * -0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(beta * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.2000000000000002

   1. Initial program 99.9%

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

   3. Simplified 93.0%

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

   5. Taylor expanded in alpha around 0 81.0%

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

   6. Taylor expanded in alpha around 0 62.1%

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

      1. +-commutative 62.1%

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

      2. +-commutative 62.1%

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

   8. Simplified 62.1%

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

   9. Taylor expanded in beta around 0 61.3%

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

       1. +-commutative 61.3%

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

   11. Simplified 61.3%

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

   12. Taylor expanded in alpha around 0 59.4%

       \[\leadsto \color{blue}{0.08333333333333333 + -0.041666666666666664 \cdot \alpha} \]
   13. Step-by-step derivation

       1. *-commutative 59.4%

          \[\leadsto 0.08333333333333333 + \color{blue}{\alpha \cdot -0.041666666666666664} \]

   14. Simplified 59.4%

       \[\leadsto \color{blue}{0.08333333333333333 + \alpha \cdot -0.041666666666666664} \]

   if 2.2000000000000002 < beta

   1. Initial program 80.3%

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

   3. Taylor expanded in beta around inf 85.6%

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

   4. Taylor expanded in alpha around 0 70.5%

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

      1. +-commutative 70.5%

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

   6. Simplified 70.5%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.2:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 91.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.2:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.2)
   (+ 0.08333333333333333 (* alpha -0.041666666666666664))
   (/ (/ 1.0 beta) (+ beta 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.2) {
		tmp = 0.08333333333333333 + (alpha * -0.041666666666666664);
	} 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 <= 2.2d0) then
        tmp = 0.08333333333333333d0 + (alpha * (-0.041666666666666664d0))
    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 <= 2.2) {
		tmp = 0.08333333333333333 + (alpha * -0.041666666666666664);
	} else {
		tmp = (1.0 / beta) / (beta + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.2:
		tmp = 0.08333333333333333 + (alpha * -0.041666666666666664)
	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 <= 2.2)
		tmp = Float64(0.08333333333333333 + Float64(alpha * -0.041666666666666664));
	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 <= 2.2)
		tmp = 0.08333333333333333 + (alpha * -0.041666666666666664);
	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, 2.2], N[(0.08333333333333333 + N[(alpha * -0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.2000000000000002

   1. Initial program 99.9%

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

   3. Simplified 93.0%

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

   5. Taylor expanded in alpha around 0 81.0%

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

   6. Taylor expanded in alpha around 0 62.1%

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

      1. +-commutative 62.1%

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

      2. +-commutative 62.1%

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

   8. Simplified 62.1%

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

   9. Taylor expanded in beta around 0 61.3%

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

       1. +-commutative 61.3%

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

   11. Simplified 61.3%

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

   12. Taylor expanded in alpha around 0 59.4%

       \[\leadsto \color{blue}{0.08333333333333333 + -0.041666666666666664 \cdot \alpha} \]
   13. Step-by-step derivation

       1. *-commutative 59.4%

          \[\leadsto 0.08333333333333333 + \color{blue}{\alpha \cdot -0.041666666666666664} \]

   14. Simplified 59.4%

       \[\leadsto \color{blue}{0.08333333333333333 + \alpha \cdot -0.041666666666666664} \]

   if 2.2000000000000002 < beta

   1. Initial program 80.3%

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

   3. Taylor expanded in beta around inf 85.6%

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

      1. metadata-eval 85.6%

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

      2. associate-+l+ 85.6%

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

      3. metadata-eval 85.6%

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

      4. +-commutative 85.6%

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

      5. associate-+r+ 85.6%

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

      6. *-un-lft-identity 85.6%

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

      7. fma-define 85.6%

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

   5. Applied egg-rr 85.6%

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

      1. fma-undefine 85.6%

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

      2. *-lft-identity 85.6%

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

      3. +-commutative 85.6%

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

   7. Simplified 85.6%

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

   8. Taylor expanded in alpha around 0 70.5%

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

      1. associate-/r* 71.1%

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

      2. +-commutative 71.1%

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

   10. Simplified 71.1%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.2:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 46.2% accurate, 3.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 7.9:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 7.9)
   (+ 0.08333333333333333 (* alpha -0.041666666666666664))
   (/ 1.0 beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 7.9) {
		tmp = 0.08333333333333333 + (alpha * -0.041666666666666664);
	} else {
		tmp = 1.0 / 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 <= 7.9d0) then
        tmp = 0.08333333333333333d0 + (alpha * (-0.041666666666666664d0))
    else
        tmp = 1.0d0 / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 7.9) {
		tmp = 0.08333333333333333 + (alpha * -0.041666666666666664);
	} else {
		tmp = 1.0 / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 7.9:
		tmp = 0.08333333333333333 + (alpha * -0.041666666666666664)
	else:
		tmp = 1.0 / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 7.9)
		tmp = Float64(0.08333333333333333 + Float64(alpha * -0.041666666666666664));
	else
		tmp = Float64(1.0 / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 7.9)
		tmp = 0.08333333333333333 + (alpha * -0.041666666666666664);
	else
		tmp = 1.0 / 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, 7.9], N[(0.08333333333333333 + N[(alpha * -0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(1.0 / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 7.9000000000000004

   1. Initial program 99.9%

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

   3. Simplified 93.0%

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

   5. Taylor expanded in alpha around 0 81.1%

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

   6. Taylor expanded in alpha around 0 62.3%

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

      1. +-commutative 62.3%

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

      2. +-commutative 62.3%

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

   8. Simplified 62.3%

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

   9. Taylor expanded in beta around 0 61.1%

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

       1. +-commutative 61.1%

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

   11. Simplified 61.1%

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

   12. Taylor expanded in alpha around 0 59.1%

       \[\leadsto \color{blue}{0.08333333333333333 + -0.041666666666666664 \cdot \alpha} \]
   13. Step-by-step derivation

       1. *-commutative 59.1%

          \[\leadsto 0.08333333333333333 + \color{blue}{\alpha \cdot -0.041666666666666664} \]

   14. Simplified 59.1%

       \[\leadsto \color{blue}{0.08333333333333333 + \alpha \cdot -0.041666666666666664} \]

   if 7.9000000000000004 < beta

   1. Initial program 80.1%

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

   3. Taylor expanded in beta around inf 86.3%

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

   4. Taylor expanded in alpha around inf 6.9%

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

4. Final simplification 40.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 7.9:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 46.1% accurate, 4.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 12:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 12.0) 0.08333333333333333 (/ 1.0 beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 12.0) {
		tmp = 0.08333333333333333;
	} else {
		tmp = 1.0 / 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 <= 12.0d0) then
        tmp = 0.08333333333333333d0
    else
        tmp = 1.0d0 / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 12.0) {
		tmp = 0.08333333333333333;
	} else {
		tmp = 1.0 / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 12.0:
		tmp = 0.08333333333333333
	else:
		tmp = 1.0 / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 12.0)
		tmp = 0.08333333333333333;
	else
		tmp = Float64(1.0 / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 12.0)
		tmp = 0.08333333333333333;
	else
		tmp = 1.0 / 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, 12.0], 0.08333333333333333, N[(1.0 / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 12

   1. Initial program 99.9%

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

   3. Simplified 93.0%

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

   5. Taylor expanded in alpha around 0 81.1%

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

   6. Taylor expanded in alpha around 0 62.3%

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

      1. +-commutative 62.3%

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

      2. +-commutative 62.3%

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

   8. Simplified 62.3%

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

   9. Taylor expanded in beta around 0 61.1%

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

       1. +-commutative 61.1%

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

   11. Simplified 61.1%

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

   12. Taylor expanded in alpha around 0 59.9%

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

   if 12 < beta

   1. Initial program 80.1%

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

   3. Taylor expanded in beta around inf 86.3%

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

   4. Taylor expanded in alpha around inf 6.9%

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

4. Final simplification 41.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 12:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 44.7% accurate, 7.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \frac{0.16666666666666666}{\alpha + 2} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta) :precision binary64 (/ 0.16666666666666666 (+ alpha 2.0)))

C

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

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 = 0.16666666666666666d0 / (alpha + 2.0d0)
end function

Java

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

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return 0.16666666666666666 / (alpha + 2.0)

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return Float64(0.16666666666666666 / Float64(alpha + 2.0))
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = 0.16666666666666666 / (alpha + 2.0);
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := N[(0.16666666666666666 / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.9%

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

3. Simplified 82.7%

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

5. Taylor expanded in alpha around 0 75.7%

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

6. Taylor expanded in alpha around 0 63.5%

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

   1. +-commutative 63.5%

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

   2. +-commutative 63.5%

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

8. Simplified 63.5%

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

9. Taylor expanded in beta around 0 41.4%

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

    1. +-commutative 41.4%

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

11. Simplified 41.4%

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

12. Final simplification 41.4%

    \[\leadsto \frac{0.16666666666666666}{\alpha + 2} \]
13. Add Preprocessing

Alternative 20: 44.5% 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 92.9%

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

3. Simplified 82.7%

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

5. Taylor expanded in alpha around 0 75.7%

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

6. Taylor expanded in alpha around 0 63.5%

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

   1. +-commutative 63.5%

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

   2. +-commutative 63.5%

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

8. Simplified 63.5%

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

9. Taylor expanded in beta around 0 41.4%

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

    1. +-commutative 41.4%

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

11. Simplified 41.4%

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

12. Taylor expanded in alpha around 0 40.3%

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

13. Final simplification 40.3%

    \[\leadsto 0.08333333333333333 \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024067 
(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)))