Octave 3.8, jcobi/3

Percentage Accurate: 94.4% → 99.8%

Time: 16.2s

Alternatives: 15

Speedup: 3.2×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\beta + \alpha\right)\\ t_1 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 5 \cdot 10^{+45}:\\ \;\;\;\;\frac{1 + \alpha}{t_1} \cdot \frac{1 + \beta}{t_1 \cdot \left(\alpha + \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta}{t_0} \cdot \frac{\frac{1 + \alpha}{3 + \left(\beta + \alpha\right)}}{t_0}\\ \end{array} \end{array} \]

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 (+ beta alpha))) (t_1 (+ alpha (+ beta 2.0))))
   (if (<= beta 5e+45)
     (* (/ (+ 1.0 alpha) t_1) (/ (+ 1.0 beta) (* t_1 (+ alpha (+ beta 3.0)))))
     (* (/ beta t_0) (/ (/ (+ 1.0 alpha) (+ 3.0 (+ beta alpha))) t_0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta + alpha);
	double t_1 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 5e+45) {
		tmp = ((1.0 + alpha) / t_1) * ((1.0 + beta) / (t_1 * (alpha + (beta + 3.0))));
	} else {
		tmp = (beta / t_0) * (((1.0 + alpha) / (3.0 + (beta + alpha))) / 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) :: t_1
    real(8) :: tmp
    t_0 = 2.0d0 + (beta + alpha)
    t_1 = alpha + (beta + 2.0d0)
    if (beta <= 5d+45) then
        tmp = ((1.0d0 + alpha) / t_1) * ((1.0d0 + beta) / (t_1 * (alpha + (beta + 3.0d0))))
    else
        tmp = (beta / t_0) * (((1.0d0 + alpha) / (3.0d0 + (beta + alpha))) / 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 + (beta + alpha);
	double t_1 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 5e+45) {
		tmp = ((1.0 + alpha) / t_1) * ((1.0 + beta) / (t_1 * (alpha + (beta + 3.0))));
	} else {
		tmp = (beta / t_0) * (((1.0 + alpha) / (3.0 + (beta + alpha))) / t_0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = 2.0 + (beta + alpha)
	t_1 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 5e+45:
		tmp = ((1.0 + alpha) / t_1) * ((1.0 + beta) / (t_1 * (alpha + (beta + 3.0))))
	else:
		tmp = (beta / t_0) * (((1.0 + alpha) / (3.0 + (beta + alpha))) / t_0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(beta + alpha))
	t_1 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 5e+45)
		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(beta / t_0) * Float64(Float64(Float64(1.0 + alpha) / Float64(3.0 + Float64(beta + alpha))) / t_0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = 2.0 + (beta + alpha);
	t_1 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 5e+45)
		tmp = ((1.0 + alpha) / t_1) * ((1.0 + beta) / (t_1 * (alpha + (beta + 3.0))));
	else
		tmp = (beta / t_0) * (((1.0 + alpha) / (3.0 + (beta + alpha))) / 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[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 5e+45], 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[(beta / t$95$0), $MachinePrecision] * N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 5e45

   1. Initial program 99.8%

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

   3. Simplified 99.0%

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

   if 5e45 < beta

   1. Initial program 85.4%

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

   3. Simplified 63.5%

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

   4. Taylor expanded in beta around inf 63.5%

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

      1. times-frac 86.9%

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

      2. associate-+r+ 86.9%

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

      3. +-commutative 86.9%

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

      4. +-commutative 86.9%

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

      5. *-commutative 86.9%

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

      6. associate-+r+ 86.9%

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

      7. +-commutative 86.9%

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

      8. associate-+r+ 86.9%

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

      9. +-commutative 86.9%

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

      10. +-commutative 86.9%

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

   6. Applied egg-rr 86.9%

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

      1. +-commutative 86.9%

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

      2. associate-/r* 99.6%

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

      3. +-commutative 99.6%

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

      4. +-commutative 99.6%

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

      5. +-commutative 99.6%

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

   8. Simplified 99.6%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5 \cdot 10^{+45}:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta}{2 + \left(\beta + \alpha\right)} \cdot \frac{\frac{1 + \alpha}{3 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}\\ \end{array} \]

Alternative 2: 98.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 8200000000000:\\ \;\;\;\;\frac{1 + \beta}{\left(\beta + 3\right) \cdot \left(4 + \beta \cdot \left(\beta + 4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta}{t_0} \cdot \frac{\frac{1 + \alpha}{3 + \left(\beta + \alpha\right)}}{t_0}\\ \end{array} \end{array} \]

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 (+ beta alpha))))
   (if (<= beta 8200000000000.0)
     (/ (+ 1.0 beta) (* (+ beta 3.0) (+ 4.0 (* beta (+ beta 4.0)))))
     (* (/ beta t_0) (/ (/ (+ 1.0 alpha) (+ 3.0 (+ beta alpha))) t_0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta + alpha);
	double tmp;
	if (beta <= 8200000000000.0) {
		tmp = (1.0 + beta) / ((beta + 3.0) * (4.0 + (beta * (beta + 4.0))));
	} else {
		tmp = (beta / t_0) * (((1.0 + alpha) / (3.0 + (beta + alpha))) / 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 + (beta + alpha)
    if (beta <= 8200000000000.0d0) then
        tmp = (1.0d0 + beta) / ((beta + 3.0d0) * (4.0d0 + (beta * (beta + 4.0d0))))
    else
        tmp = (beta / t_0) * (((1.0d0 + alpha) / (3.0d0 + (beta + alpha))) / 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 + (beta + alpha);
	double tmp;
	if (beta <= 8200000000000.0) {
		tmp = (1.0 + beta) / ((beta + 3.0) * (4.0 + (beta * (beta + 4.0))));
	} else {
		tmp = (beta / t_0) * (((1.0 + alpha) / (3.0 + (beta + alpha))) / t_0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = 2.0 + (beta + alpha)
	tmp = 0
	if beta <= 8200000000000.0:
		tmp = (1.0 + beta) / ((beta + 3.0) * (4.0 + (beta * (beta + 4.0))))
	else:
		tmp = (beta / t_0) * (((1.0 + alpha) / (3.0 + (beta + alpha))) / t_0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(beta + alpha))
	tmp = 0.0
	if (beta <= 8200000000000.0)
		tmp = Float64(Float64(1.0 + beta) / Float64(Float64(beta + 3.0) * Float64(4.0 + Float64(beta * Float64(beta + 4.0)))));
	else
		tmp = Float64(Float64(beta / t_0) * Float64(Float64(Float64(1.0 + alpha) / Float64(3.0 + Float64(beta + alpha))) / t_0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = 2.0 + (beta + alpha);
	tmp = 0.0;
	if (beta <= 8200000000000.0)
		tmp = (1.0 + beta) / ((beta + 3.0) * (4.0 + (beta * (beta + 4.0))));
	else
		tmp = (beta / t_0) * (((1.0 + alpha) / (3.0 + (beta + alpha))) / 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[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 8200000000000.0], N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(beta + 3.0), $MachinePrecision] * N[(4.0 + N[(beta * N[(beta + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(beta / t$95$0), $MachinePrecision] * N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 8.2e12

   1. Initial program 99.8%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in alpha around 0 61.3%

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

   5. Taylor expanded in beta around 0 61.3%

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

      1. +-commutative 61.3%

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

      2. unpow2 61.3%

         \[\leadsto \frac{1 + \beta}{\left(4 + \left(\color{blue}{\beta \cdot \beta} + 4 \cdot \beta\right)\right) \cdot \left(3 + \beta\right)} \]

      3. distribute-rgt-out 61.3%

         \[\leadsto \frac{1 + \beta}{\left(4 + \color{blue}{\beta \cdot \left(\beta + 4\right)}\right) \cdot \left(3 + \beta\right)} \]

   7. Simplified 61.3%

      \[\leadsto \frac{1 + \beta}{\color{blue}{\left(4 + \beta \cdot \left(\beta + 4\right)\right)} \cdot \left(3 + \beta\right)} \]

   if 8.2e12 < beta

   1. Initial program 86.5%

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

   3. Simplified 66.0%

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

   4. Taylor expanded in beta around inf 66.0%

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

      1. times-frac 87.6%

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

      2. associate-+r+ 87.6%

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

      3. +-commutative 87.6%

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

      4. +-commutative 87.6%

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

      5. *-commutative 87.6%

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

      6. associate-+r+ 87.6%

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

      7. +-commutative 87.6%

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

      8. associate-+r+ 87.6%

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

      9. +-commutative 87.6%

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

      10. +-commutative 87.6%

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

   6. Applied egg-rr 87.6%

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

      1. +-commutative 87.6%

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

      2. associate-/r* 99.7%

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

      3. +-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. +-commutative 99.7%

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

   8. Simplified 99.7%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 8200000000000:\\ \;\;\;\;\frac{1 + \beta}{\left(\beta + 3\right) \cdot \left(4 + \beta \cdot \left(\beta + 4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta}{2 + \left(\beta + \alpha\right)} \cdot \frac{\frac{1 + \alpha}{3 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}\\ \end{array} \]

Alternative 3: 99.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = (((1.0 + beta) / t_0) / (alpha + (beta + 3.0))) * ((1.0 + alpha) / t_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[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(1.0 + beta), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 95.2%

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

3. Simplified 95.1%

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

   1. clear-num 95.1%

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

   2. associate-+r+ 95.1%

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

   3. *-commutative 95.1%

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

   4. frac-times 90.8%

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

   5. *-un-lft-identity 90.8%

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

   6. +-commutative 90.8%

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

   7. *-commutative 90.8%

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

   8. associate-+r+ 90.8%

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

5. Applied egg-rr 90.8%

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

   1. associate-/r* 95.2%

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

   2. associate-/l* 93.4%

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

   3. associate-*l/ 95.1%

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

   4. *-commutative 95.1%

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

   5. times-frac 99.7%

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

   6. associate-/r* 95.1%

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

   7. *-commutative 95.1%

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

   8. associate-/r* 99.7%

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

   9. +-commutative 99.7%

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

   10. +-commutative 99.7%

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

   11. +-commutative 99.7%

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

   12. +-commutative 99.7%

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

7. Simplified 99.7%

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

8. Final simplification 99.7%

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

Alternative 4: 97.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.8:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \left(0.16666666666666666 + \alpha \cdot -0.1388888888888889\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{3 + \left(\beta + \alpha\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.8)
   (*
    (/ (+ 1.0 alpha) (+ alpha (+ beta 2.0)))
    (+ 0.16666666666666666 (* alpha -0.1388888888888889)))
   (/ (/ (+ 1.0 alpha) beta) (+ 3.0 (+ beta alpha)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.8) {
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) * (0.16666666666666666 + (alpha * -0.1388888888888889));
	} else {
		tmp = ((1.0 + alpha) / beta) / (3.0 + (beta + alpha));
	}
	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 <= 4.8d0) then
        tmp = ((1.0d0 + alpha) / (alpha + (beta + 2.0d0))) * (0.16666666666666666d0 + (alpha * (-0.1388888888888889d0)))
    else
        tmp = ((1.0d0 + alpha) / beta) / (3.0d0 + (beta + alpha))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.8) {
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) * (0.16666666666666666 + (alpha * -0.1388888888888889));
	} else {
		tmp = ((1.0 + alpha) / beta) / (3.0 + (beta + alpha));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 4.8:
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) * (0.16666666666666666 + (alpha * -0.1388888888888889))
	else:
		tmp = ((1.0 + alpha) / beta) / (3.0 + (beta + alpha))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.8)
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(beta + 2.0))) * Float64(0.16666666666666666 + Float64(alpha * -0.1388888888888889)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(3.0 + Float64(beta + alpha)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4.8)
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) * (0.16666666666666666 + (alpha * -0.1388888888888889));
	else
		tmp = ((1.0 + alpha) / beta) / (3.0 + (beta + alpha));
	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, 4.8], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.16666666666666666 + N[(alpha * -0.1388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.79999999999999982

   1. Initial program 99.8%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in beta around 0 97.0%

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

      1. +-commutative 97.0%

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

   6. Simplified 97.0%

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

   7. Taylor expanded in alpha around 0 59.4%

      \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \color{blue}{\left(0.16666666666666666 + -0.1388888888888889 \cdot \alpha\right)} \]
   8. Step-by-step derivation

      1. *-commutative 59.4%

         \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \left(0.16666666666666666 + \color{blue}{\alpha \cdot -0.1388888888888889}\right) \]

   9. Simplified 59.4%

      \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \color{blue}{\left(0.16666666666666666 + \alpha \cdot -0.1388888888888889\right)} \]

   if 4.79999999999999982 < beta

   1. Initial program 86.9%

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

   2. Taylor expanded in beta around -inf 81.8%

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

      1. expm1-log1p-u 81.8%

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

      2. expm1-udef 50.6%

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

      3. mul-1-neg 50.6%

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

      4. *-commutative 50.6%

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

      5. fma-neg 50.6%

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

      6. metadata-eval 50.6%

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

      7. metadata-eval 50.6%

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

      8. associate-+l+ 50.6%

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

      9. metadata-eval 50.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{-\frac{\mathsf{fma}\left(\alpha, -1, -1\right)}{\beta}}{\left(\alpha + \beta\right) + \color{blue}{3}}\right)} - 1 \]

      10. +-commutative 50.6%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{-\frac{\mathsf{fma}\left(\alpha, -1, -1\right)}{\beta}}{\color{blue}{\left(\beta + \alpha\right)} + 3}\right)} - 1 \]

   4. Applied egg-rr 50.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-\frac{\mathsf{fma}\left(\alpha, -1, -1\right)}{\beta}}{\left(\beta + \alpha\right) + 3}\right)} - 1} \]
   5. Step-by-step derivation

      1. expm1-def 81.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-\frac{\mathsf{fma}\left(\alpha, -1, -1\right)}{\beta}}{\left(\beta + \alpha\right) + 3}\right)\right)} \]

      2. expm1-log1p 81.8%

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

      3. distribute-neg-frac 81.8%

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

      4. fma-udef 81.8%

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

      5. distribute-lft1-in 81.8%

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

      6. +-commutative 81.8%

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

      7. *-commutative 81.8%

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

      8. distribute-lft-in 81.8%

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

      9. metadata-eval 81.8%

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

      10. neg-mul-1 81.8%

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

      11. unsub-neg 81.8%

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

      12. +-commutative 81.8%

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

      13. +-commutative 81.8%

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

   6. Simplified 81.8%

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

      1. expm1-log1p-u 81.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{-\left(-1 - \alpha\right)}{\beta}}{3 + \left(\alpha + \beta\right)}\right)\right)} \]

      2. expm1-udef 50.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{-\left(-1 - \alpha\right)}{\beta}}{3 + \left(\alpha + \beta\right)}\right)} - 1} \]

      3. associate-/l/ 50.6%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{-\left(-1 - \alpha\right)}{\left(3 + \left(\alpha + \beta\right)\right) \cdot \beta}}\right)} - 1 \]

      4. +-commutative 50.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{-\left(-1 - \alpha\right)}{\left(3 + \color{blue}{\left(\beta + \alpha\right)}\right) \cdot \beta}\right)} - 1 \]

   8. Applied egg-rr 50.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-\left(-1 - \alpha\right)}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \beta}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 76.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-\left(-1 - \alpha\right)}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \beta}\right)\right)} \]

      2. expm1-log1p 76.8%

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

      3. associate-/r* 81.8%

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

      4. sub-neg 81.8%

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

      5. metadata-eval 81.8%

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

      6. distribute-neg-in 81.8%

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

      7. remove-double-neg 81.8%

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

      8. *-rgt-identity 81.8%

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

      9. associate-*r/ 81.8%

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

      10. remove-double-neg 81.8%

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

      11. distribute-neg-in 81.8%

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

      12. metadata-eval 81.8%

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

      13. sub-neg 81.8%

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

      14. associate-*l/ 81.8%

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

      15. associate-*r/ 81.8%

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

   10. Simplified 81.8%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.8:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \left(0.16666666666666666 + \alpha \cdot -0.1388888888888889\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{3 + \left(\beta + \alpha\right)}\\ \end{array} \]

Alternative 5: 98.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 8.2 \cdot 10^{+16}:\\ \;\;\;\;\frac{1 + \beta}{\left(\beta + 3\right) \cdot \left(4 + \beta \cdot \left(\beta + 4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{2 + \left(\beta + \alpha\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 8.2e+16)
   (/ (+ 1.0 beta) (* (+ beta 3.0) (+ 4.0 (* beta (+ beta 4.0)))))
   (/ (/ (+ 1.0 alpha) beta) (+ 2.0 (+ beta alpha)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 8.2e+16) {
		tmp = (1.0 + beta) / ((beta + 3.0) * (4.0 + (beta * (beta + 4.0))));
	} else {
		tmp = ((1.0 + alpha) / beta) / (2.0 + (beta + alpha));
	}
	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 <= 8.2d+16) then
        tmp = (1.0d0 + beta) / ((beta + 3.0d0) * (4.0d0 + (beta * (beta + 4.0d0))))
    else
        tmp = ((1.0d0 + alpha) / beta) / (2.0d0 + (beta + alpha))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 8.2e+16) {
		tmp = (1.0 + beta) / ((beta + 3.0) * (4.0 + (beta * (beta + 4.0))));
	} else {
		tmp = ((1.0 + alpha) / beta) / (2.0 + (beta + alpha));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 8.2e+16:
		tmp = (1.0 + beta) / ((beta + 3.0) * (4.0 + (beta * (beta + 4.0))))
	else:
		tmp = ((1.0 + alpha) / beta) / (2.0 + (beta + alpha))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 8.2e+16)
		tmp = Float64(Float64(1.0 + beta) / Float64(Float64(beta + 3.0) * Float64(4.0 + Float64(beta * Float64(beta + 4.0)))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(2.0 + Float64(beta + alpha)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 8.2e+16)
		tmp = (1.0 + beta) / ((beta + 3.0) * (4.0 + (beta * (beta + 4.0))));
	else
		tmp = ((1.0 + alpha) / beta) / (2.0 + (beta + alpha));
	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, 8.2e+16], N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(beta + 3.0), $MachinePrecision] * N[(4.0 + N[(beta * N[(beta + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 8.2e16

   1. Initial program 99.8%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in alpha around 0 60.9%

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

   5. Taylor expanded in beta around 0 60.9%

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

      1. +-commutative 60.9%

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

      2. unpow2 60.9%

         \[\leadsto \frac{1 + \beta}{\left(4 + \left(\color{blue}{\beta \cdot \beta} + 4 \cdot \beta\right)\right) \cdot \left(3 + \beta\right)} \]

      3. distribute-rgt-out 60.9%

         \[\leadsto \frac{1 + \beta}{\left(4 + \color{blue}{\beta \cdot \left(\beta + 4\right)}\right) \cdot \left(3 + \beta\right)} \]

   7. Simplified 60.9%

      \[\leadsto \frac{1 + \beta}{\color{blue}{\left(4 + \beta \cdot \left(\beta + 4\right)\right)} \cdot \left(3 + \beta\right)} \]

   if 8.2e16 < beta

   1. Initial program 86.4%

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

   3. Simplified 87.3%

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

   4. Taylor expanded in beta around inf 84.2%

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

      1. associate-*l/ 84.2%

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

      2. +-commutative 84.2%

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

      3. associate-+r+ 84.2%

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

      4. +-commutative 84.2%

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

      5. +-commutative 84.2%

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

   6. Applied egg-rr 84.2%

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

      1. associate-*r/ 84.2%

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

      2. *-rgt-identity 84.2%

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

      3. +-commutative 84.2%

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

   8. Simplified 84.2%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 8.2 \cdot 10^{+16}:\\ \;\;\;\;\frac{1 + \beta}{\left(\beta + 3\right) \cdot \left(4 + \beta \cdot \left(\beta + 4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{2 + \left(\beta + \alpha\right)}\\ \end{array} \]

Alternative 6: 98.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.4 \cdot 10^{+15}:\\ \;\;\;\;\frac{1 + \beta}{\left(\beta + 3\right) \cdot \left(4 + \beta \cdot \left(\beta + 4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{2 + \left(\beta + \alpha\right)}}{3 + \left(\beta + \alpha\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.4e+15)
   (/ (+ 1.0 beta) (* (+ beta 3.0) (+ 4.0 (* beta (+ beta 4.0)))))
   (/ (/ (+ 1.0 alpha) (+ 2.0 (+ beta alpha))) (+ 3.0 (+ beta alpha)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.4e+15) {
		tmp = (1.0 + beta) / ((beta + 3.0) * (4.0 + (beta * (beta + 4.0))));
	} else {
		tmp = ((1.0 + alpha) / (2.0 + (beta + alpha))) / (3.0 + (beta + alpha));
	}
	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 <= 3.4d+15) then
        tmp = (1.0d0 + beta) / ((beta + 3.0d0) * (4.0d0 + (beta * (beta + 4.0d0))))
    else
        tmp = ((1.0d0 + alpha) / (2.0d0 + (beta + alpha))) / (3.0d0 + (beta + alpha))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.4e+15) {
		tmp = (1.0 + beta) / ((beta + 3.0) * (4.0 + (beta * (beta + 4.0))));
	} else {
		tmp = ((1.0 + alpha) / (2.0 + (beta + alpha))) / (3.0 + (beta + alpha));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 3.4e+15:
		tmp = (1.0 + beta) / ((beta + 3.0) * (4.0 + (beta * (beta + 4.0))))
	else:
		tmp = ((1.0 + alpha) / (2.0 + (beta + alpha))) / (3.0 + (beta + alpha))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.4e+15)
		tmp = Float64(Float64(1.0 + beta) / Float64(Float64(beta + 3.0) * Float64(4.0 + Float64(beta * Float64(beta + 4.0)))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(2.0 + Float64(beta + alpha))) / Float64(3.0 + Float64(beta + alpha)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.4e+15)
		tmp = (1.0 + beta) / ((beta + 3.0) * (4.0 + (beta * (beta + 4.0))));
	else
		tmp = ((1.0 + alpha) / (2.0 + (beta + alpha))) / (3.0 + (beta + alpha));
	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, 3.4e+15], N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(beta + 3.0), $MachinePrecision] * N[(4.0 + N[(beta * N[(beta + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.4e15

   1. Initial program 99.8%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in alpha around 0 61.3%

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

   5. Taylor expanded in beta around 0 61.3%

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

      1. +-commutative 61.3%

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

      2. unpow2 61.3%

         \[\leadsto \frac{1 + \beta}{\left(4 + \left(\color{blue}{\beta \cdot \beta} + 4 \cdot \beta\right)\right) \cdot \left(3 + \beta\right)} \]

      3. distribute-rgt-out 61.3%

         \[\leadsto \frac{1 + \beta}{\left(4 + \color{blue}{\beta \cdot \left(\beta + 4\right)}\right) \cdot \left(3 + \beta\right)} \]

   7. Simplified 61.3%

      \[\leadsto \frac{1 + \beta}{\color{blue}{\left(4 + \beta \cdot \left(\beta + 4\right)\right)} \cdot \left(3 + \beta\right)} \]

   if 3.4e15 < beta

   1. Initial program 86.5%

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

      1. associate-/l/ 82.7%

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

      2. +-commutative 82.7%

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

      3. +-commutative 82.7%

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

      4. associate-+r+ 82.7%

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

      5. *-commutative 82.7%

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

      6. metadata-eval 82.7%

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

      7. associate-+l+ 82.7%

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

      8. metadata-eval 82.7%

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

      9. associate-+l+ 82.7%

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

      10. metadata-eval 82.7%

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

      11. metadata-eval 82.7%

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

      12. associate-+l+ 82.7%

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

   3. Simplified 82.7%

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

   4. Taylor expanded in beta around -inf 82.2%

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

      1. times-frac 83.6%

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

      2. +-commutative 83.6%

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

      3. +-commutative 83.6%

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

      4. *-commutative 83.6%

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

      5. fma-neg 83.6%

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

      6. metadata-eval 83.6%

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

      7. associate-+r+ 83.6%

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

      8. +-commutative 83.6%

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

      9. associate-+r+ 83.6%

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

   6. Applied egg-rr 83.6%

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

      1. associate-*l/ 83.7%

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

      2. mul-1-neg 83.7%

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

      3. metadata-eval 83.7%

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

      4. fma-neg 83.7%

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

      5. *-commutative 83.7%

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

      6. distribute-frac-neg 83.7%

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

      7. *-commutative 83.7%

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

      8. fma-neg 83.7%

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

      9. metadata-eval 83.7%

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

      10. fma-udef 83.7%

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

      11. *-commutative 83.7%

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

      12. mul-1-neg 83.7%

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

      13. metadata-eval 83.7%

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

      14. distribute-neg-in 83.7%

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

      15. +-commutative 83.7%

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

      16. remove-double-neg 83.7%

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

      17. associate-+l+ 83.7%

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

      18. +-commutative 83.7%

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

   8. Simplified 83.7%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.4 \cdot 10^{+15}:\\ \;\;\;\;\frac{1 + \beta}{\left(\beta + 3\right) \cdot \left(4 + \beta \cdot \left(\beta + 4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{2 + \left(\beta + \alpha\right)}}{3 + \left(\beta + \alpha\right)}\\ \end{array} \]

Alternative 7: 97.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.5:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{2 + \left(\beta + \alpha\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.5)
   (+ 0.08333333333333333 (* beta -0.027777777777777776))
   (/ (/ (+ 1.0 alpha) beta) (+ 2.0 (+ beta alpha)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.5) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = ((1.0 + alpha) / beta) / (2.0 + (beta + alpha));
	}
	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 <= 2.5d0) then
        tmp = 0.08333333333333333d0 + (beta * (-0.027777777777777776d0))
    else
        tmp = ((1.0d0 + alpha) / beta) / (2.0d0 + (beta + alpha))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.5) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = ((1.0 + alpha) / beta) / (2.0 + (beta + alpha));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.5:
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776)
	else:
		tmp = ((1.0 + alpha) / beta) / (2.0 + (beta + alpha))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.5)
		tmp = Float64(0.08333333333333333 + Float64(beta * -0.027777777777777776));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(2.0 + Float64(beta + alpha)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.5)
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	else
		tmp = ((1.0 + alpha) / beta) / (2.0 + (beta + alpha));
	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, 2.5], N[(0.08333333333333333 + N[(beta * -0.027777777777777776), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.5

   1. Initial program 99.8%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in alpha around 0 61.2%

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

   5. Taylor expanded in beta around 0 60.4%

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

      1. *-commutative 60.4%

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

   7. Simplified 60.4%

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

   if 2.5 < beta

   1. Initial program 86.9%

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

   3. Simplified 87.9%

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

   4. Taylor expanded in beta around inf 81.8%

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

      1. associate-*l/ 81.8%

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

      2. +-commutative 81.8%

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

      3. associate-+r+ 81.8%

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

      4. +-commutative 81.8%

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

      5. +-commutative 81.8%

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

   6. Applied egg-rr 81.8%

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

      1. associate-*r/ 81.8%

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

      2. *-rgt-identity 81.8%

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

      3. +-commutative 81.8%

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

   8. Simplified 81.8%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.5:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{2 + \left(\beta + \alpha\right)}\\ \end{array} \]

Alternative 8: 97.2% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if beta < 2.4500000000000002

   1. Initial program 99.8%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in alpha around 0 61.2%

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

   5. Taylor expanded in beta around 0 60.4%

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

      1. *-commutative 60.4%

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

   7. Simplified 60.4%

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

   if 2.4500000000000002 < beta

   1. Initial program 86.9%

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

   2. Taylor expanded in beta around -inf 81.8%

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

      1. expm1-log1p-u 81.8%

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

      2. expm1-udef 50.6%

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

      3. mul-1-neg 50.6%

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

      4. *-commutative 50.6%

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

      5. fma-neg 50.6%

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

      6. metadata-eval 50.6%

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

      7. metadata-eval 50.6%

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

      8. associate-+l+ 50.6%

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

      9. metadata-eval 50.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{-\frac{\mathsf{fma}\left(\alpha, -1, -1\right)}{\beta}}{\left(\alpha + \beta\right) + \color{blue}{3}}\right)} - 1 \]

      10. +-commutative 50.6%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{-\frac{\mathsf{fma}\left(\alpha, -1, -1\right)}{\beta}}{\color{blue}{\left(\beta + \alpha\right)} + 3}\right)} - 1 \]

   4. Applied egg-rr 50.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-\frac{\mathsf{fma}\left(\alpha, -1, -1\right)}{\beta}}{\left(\beta + \alpha\right) + 3}\right)} - 1} \]
   5. Step-by-step derivation

      1. expm1-def 81.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-\frac{\mathsf{fma}\left(\alpha, -1, -1\right)}{\beta}}{\left(\beta + \alpha\right) + 3}\right)\right)} \]

      2. expm1-log1p 81.8%

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

      3. distribute-neg-frac 81.8%

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

      4. fma-udef 81.8%

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

      5. distribute-lft1-in 81.8%

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

      6. +-commutative 81.8%

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

      7. *-commutative 81.8%

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

      8. distribute-lft-in 81.8%

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

      9. metadata-eval 81.8%

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

      10. neg-mul-1 81.8%

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

      11. unsub-neg 81.8%

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

      12. +-commutative 81.8%

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

      13. +-commutative 81.8%

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

   6. Simplified 81.8%

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

      1. expm1-log1p-u 81.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{-\left(-1 - \alpha\right)}{\beta}}{3 + \left(\alpha + \beta\right)}\right)\right)} \]

      2. expm1-udef 50.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{-\left(-1 - \alpha\right)}{\beta}}{3 + \left(\alpha + \beta\right)}\right)} - 1} \]

      3. associate-/l/ 50.6%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{-\left(-1 - \alpha\right)}{\left(3 + \left(\alpha + \beta\right)\right) \cdot \beta}}\right)} - 1 \]

      4. +-commutative 50.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{-\left(-1 - \alpha\right)}{\left(3 + \color{blue}{\left(\beta + \alpha\right)}\right) \cdot \beta}\right)} - 1 \]

   8. Applied egg-rr 50.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-\left(-1 - \alpha\right)}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \beta}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 76.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-\left(-1 - \alpha\right)}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \beta}\right)\right)} \]

      2. expm1-log1p 76.8%

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

      3. associate-/r* 81.8%

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

      4. sub-neg 81.8%

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

      5. metadata-eval 81.8%

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

      6. distribute-neg-in 81.8%

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

      7. remove-double-neg 81.8%

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

      8. *-rgt-identity 81.8%

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

      9. associate-*r/ 81.8%

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

      10. remove-double-neg 81.8%

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

      11. distribute-neg-in 81.8%

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

      12. metadata-eval 81.8%

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

      13. sub-neg 81.8%

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

      14. associate-*l/ 81.8%

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

      15. associate-*r/ 81.8%

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

   10. Simplified 81.8%

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

4. Final simplification 68.1%

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

Alternative 9: 97.1% accurate, 3.2× speedup

Math

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

FPCore

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

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.8) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = ((1.0 + alpha) / beta) * (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 <= 2.8d0) then
        tmp = 0.08333333333333333d0 + (beta * (-0.027777777777777776d0))
    else
        tmp = ((1.0d0 + alpha) / beta) * (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 <= 2.8) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = ((1.0 + alpha) / beta) * (1.0 / beta);
	}
	return tmp;
}

Python

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if beta < 2.7999999999999998

   1. Initial program 99.8%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in alpha around 0 61.2%

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

   5. Taylor expanded in beta around 0 60.4%

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

      1. *-commutative 60.4%

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

   7. Simplified 60.4%

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

   if 2.7999999999999998 < beta

   1. Initial program 86.9%

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

   3. Simplified 87.9%

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

   4. Taylor expanded in beta around inf 81.8%

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

   5. Taylor expanded in beta around inf 81.6%

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

4. Final simplification 68.0%

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

Alternative 10: 91.6% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if beta < 2.4500000000000002

   1. Initial program 99.8%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in alpha around 0 61.2%

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

   5. Taylor expanded in beta around 0 60.4%

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

      1. *-commutative 60.4%

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

   7. Simplified 60.4%

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

   if 2.4500000000000002 < beta

   1. Initial program 86.9%

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

   2. Taylor expanded in beta around -inf 81.8%

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

   3. Taylor expanded in alpha around 0 69.7%

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

4. Final simplification 63.7%

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

Alternative 11: 92.0% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if beta < 2.4500000000000002

   1. Initial program 99.8%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in alpha around 0 61.2%

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

   5. Taylor expanded in beta around 0 60.4%

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

      1. *-commutative 60.4%

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

   7. Simplified 60.4%

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

   if 2.4500000000000002 < beta

   1. Initial program 86.9%

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

   2. Taylor expanded in beta around -inf 81.8%

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

      1. expm1-log1p-u 81.8%

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

      2. expm1-udef 50.6%

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

      3. mul-1-neg 50.6%

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

      4. *-commutative 50.6%

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

      5. fma-neg 50.6%

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

      6. metadata-eval 50.6%

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

      7. metadata-eval 50.6%

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

      8. associate-+l+ 50.6%

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

      9. metadata-eval 50.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{-\frac{\mathsf{fma}\left(\alpha, -1, -1\right)}{\beta}}{\left(\alpha + \beta\right) + \color{blue}{3}}\right)} - 1 \]

      10. +-commutative 50.6%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{-\frac{\mathsf{fma}\left(\alpha, -1, -1\right)}{\beta}}{\color{blue}{\left(\beta + \alpha\right)} + 3}\right)} - 1 \]

   4. Applied egg-rr 50.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-\frac{\mathsf{fma}\left(\alpha, -1, -1\right)}{\beta}}{\left(\beta + \alpha\right) + 3}\right)} - 1} \]
   5. Step-by-step derivation

      1. expm1-def 81.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-\frac{\mathsf{fma}\left(\alpha, -1, -1\right)}{\beta}}{\left(\beta + \alpha\right) + 3}\right)\right)} \]

      2. expm1-log1p 81.8%

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

      3. distribute-neg-frac 81.8%

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

      4. fma-udef 81.8%

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

      5. distribute-lft1-in 81.8%

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

      6. +-commutative 81.8%

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

      7. *-commutative 81.8%

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

      8. distribute-lft-in 81.8%

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

      9. metadata-eval 81.8%

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

      10. neg-mul-1 81.8%

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

      11. unsub-neg 81.8%

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

      12. +-commutative 81.8%

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

      13. +-commutative 81.8%

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

   6. Simplified 81.8%

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

   7. Taylor expanded in alpha around 0 69.7%

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

      1. associate-/r* 70.9%

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

   9. Simplified 70.9%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.45:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta + 3}\\ \end{array} \]

Alternative 12: 47.3% accurate, 5.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.95:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.95)
   (+ 0.08333333333333333 (* beta -0.027777777777777776))
   (/ 1.0 beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.95) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} 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 <= 2.95d0) then
        tmp = 0.08333333333333333d0 + (beta * (-0.027777777777777776d0))
    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 <= 2.95) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = 1.0 / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.95:
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776)
	else:
		tmp = 1.0 / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.95)
		tmp = Float64(0.08333333333333333 + Float64(beta * -0.027777777777777776));
	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 <= 2.95)
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	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, 2.95], N[(0.08333333333333333 + N[(beta * -0.027777777777777776), $MachinePrecision]), $MachinePrecision], N[(1.0 / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.9500000000000002

   1. Initial program 99.8%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in alpha around 0 61.2%

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

   5. Taylor expanded in beta around 0 60.4%

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

      1. *-commutative 60.4%

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

   7. Simplified 60.4%

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

   if 2.9500000000000002 < beta

   1. Initial program 86.9%

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

   2. Taylor expanded in beta around -inf 81.8%

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

      1. expm1-log1p-u 81.8%

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

      2. expm1-udef 50.6%

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

      3. mul-1-neg 50.6%

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

      4. *-commutative 50.6%

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

      5. fma-neg 50.6%

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

      6. metadata-eval 50.6%

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

      7. metadata-eval 50.6%

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

      8. associate-+l+ 50.6%

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

      9. metadata-eval 50.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{-\frac{\mathsf{fma}\left(\alpha, -1, -1\right)}{\beta}}{\left(\alpha + \beta\right) + \color{blue}{3}}\right)} - 1 \]

      10. +-commutative 50.6%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{-\frac{\mathsf{fma}\left(\alpha, -1, -1\right)}{\beta}}{\color{blue}{\left(\beta + \alpha\right)} + 3}\right)} - 1 \]

   4. Applied egg-rr 50.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-\frac{\mathsf{fma}\left(\alpha, -1, -1\right)}{\beta}}{\left(\beta + \alpha\right) + 3}\right)} - 1} \]
   5. Step-by-step derivation

      1. expm1-def 81.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-\frac{\mathsf{fma}\left(\alpha, -1, -1\right)}{\beta}}{\left(\beta + \alpha\right) + 3}\right)\right)} \]

      2. expm1-log1p 81.8%

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

      3. distribute-neg-frac 81.8%

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

      4. fma-udef 81.8%

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

      5. distribute-lft1-in 81.8%

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

      6. +-commutative 81.8%

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

      7. *-commutative 81.8%

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

      8. distribute-lft-in 81.8%

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

      9. metadata-eval 81.8%

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

      10. neg-mul-1 81.8%

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

      11. unsub-neg 81.8%

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

      12. +-commutative 81.8%

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

      13. +-commutative 81.8%

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

   6. Simplified 81.8%

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

      1. expm1-log1p-u 81.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{-\left(-1 - \alpha\right)}{\beta}}{3 + \left(\alpha + \beta\right)}\right)\right)} \]

      2. expm1-udef 50.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{-\left(-1 - \alpha\right)}{\beta}}{3 + \left(\alpha + \beta\right)}\right)} - 1} \]

      3. associate-/l/ 50.6%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{-\left(-1 - \alpha\right)}{\left(3 + \left(\alpha + \beta\right)\right) \cdot \beta}}\right)} - 1 \]

      4. +-commutative 50.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{-\left(-1 - \alpha\right)}{\left(3 + \color{blue}{\left(\beta + \alpha\right)}\right) \cdot \beta}\right)} - 1 \]

   8. Applied egg-rr 50.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-\left(-1 - \alpha\right)}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \beta}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 76.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-\left(-1 - \alpha\right)}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \beta}\right)\right)} \]

      2. expm1-log1p 76.8%

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

      3. associate-/r* 81.8%

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

      4. sub-neg 81.8%

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

      5. metadata-eval 81.8%

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

      6. distribute-neg-in 81.8%

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

      7. remove-double-neg 81.8%

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

      8. *-rgt-identity 81.8%

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

      9. associate-*r/ 81.8%

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

      10. remove-double-neg 81.8%

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

      11. distribute-neg-in 81.8%

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

      12. metadata-eval 81.8%

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

      13. sub-neg 81.8%

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

      14. associate-*l/ 81.8%

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

      15. associate-*r/ 81.8%

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

   10. Simplified 81.8%

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

   11. Taylor expanded in alpha around inf 7.0%

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

4. Final simplification 41.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.95:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta}\\ \end{array} \]

Alternative 13: 46.8% accurate, 6.9× 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 99.8%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in alpha around 0 61.2%

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

   5. Taylor expanded in beta around 0 59.9%

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

   if 12 < beta

   1. Initial program 86.9%

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

   2. Taylor expanded in beta around -inf 81.8%

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

      1. expm1-log1p-u 81.8%

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

      2. expm1-udef 50.6%

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

      3. mul-1-neg 50.6%

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

      4. *-commutative 50.6%

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

      5. fma-neg 50.6%

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

      6. metadata-eval 50.6%

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

      7. metadata-eval 50.6%

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

      8. associate-+l+ 50.6%

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

      9. metadata-eval 50.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{-\frac{\mathsf{fma}\left(\alpha, -1, -1\right)}{\beta}}{\left(\alpha + \beta\right) + \color{blue}{3}}\right)} - 1 \]

      10. +-commutative 50.6%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{-\frac{\mathsf{fma}\left(\alpha, -1, -1\right)}{\beta}}{\color{blue}{\left(\beta + \alpha\right)} + 3}\right)} - 1 \]

   4. Applied egg-rr 50.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-\frac{\mathsf{fma}\left(\alpha, -1, -1\right)}{\beta}}{\left(\beta + \alpha\right) + 3}\right)} - 1} \]
   5. Step-by-step derivation

      1. expm1-def 81.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-\frac{\mathsf{fma}\left(\alpha, -1, -1\right)}{\beta}}{\left(\beta + \alpha\right) + 3}\right)\right)} \]

      2. expm1-log1p 81.8%

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

      3. distribute-neg-frac 81.8%

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

      4. fma-udef 81.8%

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

      5. distribute-lft1-in 81.8%

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

      6. +-commutative 81.8%

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

      7. *-commutative 81.8%

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

      8. distribute-lft-in 81.8%

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

      9. metadata-eval 81.8%

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

      10. neg-mul-1 81.8%

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

      11. unsub-neg 81.8%

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

      12. +-commutative 81.8%

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

      13. +-commutative 81.8%

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

   6. Simplified 81.8%

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

      1. expm1-log1p-u 81.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{-\left(-1 - \alpha\right)}{\beta}}{3 + \left(\alpha + \beta\right)}\right)\right)} \]

      2. expm1-udef 50.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{-\left(-1 - \alpha\right)}{\beta}}{3 + \left(\alpha + \beta\right)}\right)} - 1} \]

      3. associate-/l/ 50.6%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{-\left(-1 - \alpha\right)}{\left(3 + \left(\alpha + \beta\right)\right) \cdot \beta}}\right)} - 1 \]

      4. +-commutative 50.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{-\left(-1 - \alpha\right)}{\left(3 + \color{blue}{\left(\beta + \alpha\right)}\right) \cdot \beta}\right)} - 1 \]

   8. Applied egg-rr 50.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-\left(-1 - \alpha\right)}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \beta}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 76.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-\left(-1 - \alpha\right)}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \beta}\right)\right)} \]

      2. expm1-log1p 76.8%

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

      3. associate-/r* 81.8%

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

      4. sub-neg 81.8%

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

      5. metadata-eval 81.8%

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

      6. distribute-neg-in 81.8%

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

      7. remove-double-neg 81.8%

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

      8. *-rgt-identity 81.8%

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

      9. associate-*r/ 81.8%

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

      10. remove-double-neg 81.8%

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

      11. distribute-neg-in 81.8%

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

      12. metadata-eval 81.8%

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

      13. sub-neg 81.8%

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

      14. associate-*l/ 81.8%

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

      15. associate-*r/ 81.8%

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

   10. Simplified 81.8%

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

   11. Taylor expanded in alpha around inf 7.0%

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

4. Final simplification 40.9%

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

Alternative 14: 46.9% accurate, 7.0× speedup

Math

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

FPCore

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

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return 0.16666666666666666 / (beta + 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 / (beta + 2.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 95.2%

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

3. Simplified 95.1%

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

4. Taylor expanded in beta around 0 67.8%

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

   1. +-commutative 67.8%

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

6. Simplified 67.8%

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

7. Taylor expanded in alpha around 0 40.9%

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

8. Final simplification 40.9%

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

Alternative 15: 45.1% 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 95.2%

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

3. Simplified 95.1%

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

4. Taylor expanded in alpha around 0 61.3%

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

5. Taylor expanded in beta around 0 39.8%

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

6. Final simplification 39.8%

   \[\leadsto 0.08333333333333333 \]

Reproduce

herbie shell --seed 2023336 
(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)))