Octave 3.8, jcobi/4

Percentage Accurate: 16.2% → 85.8%

Time: 13.7s

Alternatives: 9

Speedup: 53.0×

Specification

\[\left(\alpha > -1 \land \beta > -1\right) \land i > 1\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := i \cdot \left(\left(\alpha + \beta\right) + i\right)\\ t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_2 := t\_1 \cdot t\_1\\ \frac{\frac{t\_0 \cdot \left(\beta \cdot \alpha + t\_0\right)}{t\_2}}{t\_2 - 1} \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (* i (+ (+ alpha beta) i)))
        (t_1 (+ (+ alpha beta) (* 2.0 i)))
        (t_2 (* t_1 t_1)))
   (/ (/ (* t_0 (+ (* beta alpha) t_0)) t_2) (- t_2 1.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = i * ((alpha + beta) + i);
	double t_1 = (alpha + beta) + (2.0 * i);
	double t_2 = t_1 * t_1;
	return ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0);
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    t_0 = i * ((alpha + beta) + i)
    t_1 = (alpha + beta) + (2.0d0 * i)
    t_2 = t_1 * t_1
    code = ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0d0)
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = i * ((alpha + beta) + i);
	double t_1 = (alpha + beta) + (2.0 * i);
	double t_2 = t_1 * t_1;
	return ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0);
}

Python

def code(alpha, beta, i):
	t_0 = i * ((alpha + beta) + i)
	t_1 = (alpha + beta) + (2.0 * i)
	t_2 = t_1 * t_1
	return ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0)

Julia

function code(alpha, beta, i)
	t_0 = Float64(i * Float64(Float64(alpha + beta) + i))
	t_1 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_2 = Float64(t_1 * t_1)
	return Float64(Float64(Float64(t_0 * Float64(Float64(beta * alpha) + t_0)) / t_2) / Float64(t_2 - 1.0))
end

MATLAB

function tmp = code(alpha, beta, i)
	t_0 = i * ((alpha + beta) + i);
	t_1 = (alpha + beta) + (2.0 * i);
	t_2 = t_1 * t_1;
	tmp = ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0);
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(i * N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, N[(N[(N[(t$95$0 * N[(N[(beta * alpha), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] / N[(t$95$2 - 1.0), $MachinePrecision]), $MachinePrecision]]]]

Initial Program: 16.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := i \cdot \left(\left(\alpha + \beta\right) + i\right)\\ t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_2 := t\_1 \cdot t\_1\\ \frac{\frac{t\_0 \cdot \left(\beta \cdot \alpha + t\_0\right)}{t\_2}}{t\_2 - 1} \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (* i (+ (+ alpha beta) i)))
        (t_1 (+ (+ alpha beta) (* 2.0 i)))
        (t_2 (* t_1 t_1)))
   (/ (/ (* t_0 (+ (* beta alpha) t_0)) t_2) (- t_2 1.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = i * ((alpha + beta) + i);
	double t_1 = (alpha + beta) + (2.0 * i);
	double t_2 = t_1 * t_1;
	return ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0);
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    t_0 = i * ((alpha + beta) + i)
    t_1 = (alpha + beta) + (2.0d0 * i)
    t_2 = t_1 * t_1
    code = ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0d0)
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = i * ((alpha + beta) + i);
	double t_1 = (alpha + beta) + (2.0 * i);
	double t_2 = t_1 * t_1;
	return ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0);
}

Python

def code(alpha, beta, i):
	t_0 = i * ((alpha + beta) + i)
	t_1 = (alpha + beta) + (2.0 * i)
	t_2 = t_1 * t_1
	return ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0)

Julia

function code(alpha, beta, i)
	t_0 = Float64(i * Float64(Float64(alpha + beta) + i))
	t_1 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_2 = Float64(t_1 * t_1)
	return Float64(Float64(Float64(t_0 * Float64(Float64(beta * alpha) + t_0)) / t_2) / Float64(t_2 - 1.0))
end

MATLAB

function tmp = code(alpha, beta, i)
	t_0 = i * ((alpha + beta) + i);
	t_1 = (alpha + beta) + (2.0 * i);
	t_2 = t_1 * t_1;
	tmp = ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0);
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(i * N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, N[(N[(N[(t$95$0 * N[(N[(beta * alpha), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] / N[(t$95$2 - 1.0), $MachinePrecision]), $MachinePrecision]]]]

Alternative 1: 85.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := i \cdot 2 + \left(\left(\beta + \alpha\right) + 1\right)\\ \mathbf{if}\;\beta \leq 7.5 \cdot 10^{+122}:\\ \;\;\;\;\frac{\frac{i \cdot \left(0.25 + 0.25 \cdot \frac{\beta}{i}\right)}{i \cdot 2 + \left(\alpha + \left(\beta + -1\right)\right)}}{\frac{t\_0}{i}}\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{t\_0} \cdot \frac{i}{i \cdot 2 + \left(\left(\beta + \alpha\right) + -1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (* i 2.0) (+ (+ beta alpha) 1.0))))
   (if (<= beta 7.5e+122)
     (/
      (/
       (* i (+ 0.25 (* 0.25 (/ beta i))))
       (+ (* i 2.0) (+ alpha (+ beta -1.0))))
      (/ t_0 i))
     (* (/ (+ i alpha) t_0) (/ i (+ (* i 2.0) (+ (+ beta alpha) -1.0)))))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = (i * 2.0) + ((beta + alpha) + 1.0);
	double tmp;
	if (beta <= 7.5e+122) {
		tmp = ((i * (0.25 + (0.25 * (beta / i)))) / ((i * 2.0) + (alpha + (beta + -1.0)))) / (t_0 / i);
	} else {
		tmp = ((i + alpha) / t_0) * (i / ((i * 2.0) + ((beta + alpha) + -1.0)));
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (i * 2.0d0) + ((beta + alpha) + 1.0d0)
    if (beta <= 7.5d+122) then
        tmp = ((i * (0.25d0 + (0.25d0 * (beta / i)))) / ((i * 2.0d0) + (alpha + (beta + (-1.0d0))))) / (t_0 / i)
    else
        tmp = ((i + alpha) / t_0) * (i / ((i * 2.0d0) + ((beta + alpha) + (-1.0d0))))
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double t_0 = (i * 2.0) + ((beta + alpha) + 1.0);
	double tmp;
	if (beta <= 7.5e+122) {
		tmp = ((i * (0.25 + (0.25 * (beta / i)))) / ((i * 2.0) + (alpha + (beta + -1.0)))) / (t_0 / i);
	} else {
		tmp = ((i + alpha) / t_0) * (i / ((i * 2.0) + ((beta + alpha) + -1.0)));
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	t_0 = (i * 2.0) + ((beta + alpha) + 1.0)
	tmp = 0
	if beta <= 7.5e+122:
		tmp = ((i * (0.25 + (0.25 * (beta / i)))) / ((i * 2.0) + (alpha + (beta + -1.0)))) / (t_0 / i)
	else:
		tmp = ((i + alpha) / t_0) * (i / ((i * 2.0) + ((beta + alpha) + -1.0)))
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = Float64(Float64(i * 2.0) + Float64(Float64(beta + alpha) + 1.0))
	tmp = 0.0
	if (beta <= 7.5e+122)
		tmp = Float64(Float64(Float64(i * Float64(0.25 + Float64(0.25 * Float64(beta / i)))) / Float64(Float64(i * 2.0) + Float64(alpha + Float64(beta + -1.0)))) / Float64(t_0 / i));
	else
		tmp = Float64(Float64(Float64(i + alpha) / t_0) * Float64(i / Float64(Float64(i * 2.0) + Float64(Float64(beta + alpha) + -1.0))));
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	t_0 = (i * 2.0) + ((beta + alpha) + 1.0);
	tmp = 0.0;
	if (beta <= 7.5e+122)
		tmp = ((i * (0.25 + (0.25 * (beta / i)))) / ((i * 2.0) + (alpha + (beta + -1.0)))) / (t_0 / i);
	else
		tmp = ((i + alpha) / t_0) * (i / ((i * 2.0) + ((beta + alpha) + -1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(i * 2.0), $MachinePrecision] + N[(N[(beta + alpha), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 7.5e+122], N[(N[(N[(i * N[(0.25 + N[(0.25 * N[(beta / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(i * 2.0), $MachinePrecision] + N[(alpha + N[(beta + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 / i), $MachinePrecision]), $MachinePrecision], N[(N[(N[(i + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(i / N[(N[(i * 2.0), $MachinePrecision] + N[(N[(beta + alpha), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 7.5000000000000002e122

   1. Initial program 19.4%

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

   3. Taylor expanded in i around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left({i}^{2} \cdot \left(\left(\frac{1}{4} + \frac{1}{4} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{4} \cdot \frac{\alpha + \beta}{i}\right)\right)}, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right)\right), 1\right)\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({i}^{2}\right), \left(\left(\frac{1}{4} + \frac{1}{4} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{4} \cdot \frac{\alpha + \beta}{i}\right)\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right)\right)}, 1\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(i \cdot i\right), \left(\left(\frac{1}{4} + \frac{1}{4} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{4} \cdot \frac{\alpha + \beta}{i}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right)}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right)\right), 1\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), \left(\left(\frac{1}{4} + \frac{1}{4} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{4} \cdot \frac{\alpha + \beta}{i}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right)}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right)\right), 1\right)\right) \]

      4. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), \left(\frac{1}{4} + \left(\frac{1}{4} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i} - \frac{1}{4} \cdot \frac{\alpha + \beta}{i}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), \color{blue}{\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right)}\right), 1\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), \left(\frac{1}{4} + \left(\frac{\frac{1}{4} \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right)}{i} - \frac{1}{4} \cdot \frac{\alpha + \beta}{i}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(\color{blue}{2}, i\right)\right)\right), 1\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), \left(\frac{1}{4} + \left(\frac{\frac{1}{4} \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right)}{i} - \frac{\frac{1}{4} \cdot \left(\alpha + \beta\right)}{i}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, \color{blue}{i}\right)\right)\right), 1\right)\right) \]

      7. div-sub N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), \left(\frac{1}{4} + \frac{\frac{1}{4} \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right) - \frac{1}{4} \cdot \left(\alpha + \beta\right)}{i}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \color{blue}{\mathsf{*.f64}\left(2, i\right)}\right)\right), 1\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), \mathsf{+.f64}\left(\frac{1}{4}, \left(\frac{\frac{1}{4} \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right) - \frac{1}{4} \cdot \left(\alpha + \beta\right)}{i}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), \color{blue}{\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right)}\right), 1\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\left(\frac{1}{4} \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right) - \frac{1}{4} \cdot \left(\alpha + \beta\right)\right), i\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \color{blue}{\mathsf{*.f64}\left(2, i\right)}\right)\right), 1\right)\right) \]

   5. Simplified 39.1%

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

   7. Applied egg-rr 80.4%

      \[\leadsto \color{blue}{{\left(\frac{i \cdot 2 + \left(\left(\alpha + \beta\right) + 1\right)}{i}\right)}^{-1} \cdot {\left(\frac{i \cdot 2 + \left(\left(\alpha + \beta\right) - 1\right)}{i \cdot \left(0.25 + \left(\left(2 \cdot \left(\alpha + \beta\right) - \beta\right) - \alpha\right) \cdot \frac{0.25}{i}\right)}\right)}^{-1}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

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

      2. unpow-1 N/A

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

      3. clear-num N/A

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

      4. unpow-1 N/A

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

      5. un-div-inv N/A

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

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{i \cdot \left(\frac{1}{4} + \left(\left(2 \cdot \left(\alpha + \beta\right) - \beta\right) - \alpha\right) \cdot \frac{\frac{1}{4}}{i}\right)}{i \cdot 2 + \left(\left(\alpha + \beta\right) - 1\right)}\right), \color{blue}{\left(\frac{i \cdot 2 + \left(\left(\alpha + \beta\right) + 1\right)}{i}\right)}\right) \]

   9. Applied egg-rr 80.4%

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

   10. Taylor expanded in beta around inf

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{4}, \color{blue}{\left(\frac{1}{4} \cdot \frac{\beta}{i}\right)}\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(i, 2\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, -1\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(i, 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 1\right)\right), i\right)\right) \]
   11. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\frac{1}{4}, \left(\frac{\beta}{i}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(i, 2\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, -1\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(i, 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 1\right)\right), i\right)\right) \]

       2. /-lowering-/.f64 86.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\beta, i\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(i, 2\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, -1\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(i, 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 1\right)\right), i\right)\right) \]

   12. Simplified 86.5%

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

   if 7.5000000000000002e122 < beta

   1. Initial program 2.3%

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

   3. Taylor expanded in beta around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(i \cdot \left(\alpha + i\right)\right)}, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right)\right), 1\right)\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(\alpha + i\right)\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right)\right)}, 1\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(i + \alpha\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), \color{blue}{\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right)}\right), 1\right)\right) \]

      3. +-lowering-+.f64 26.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(i, \alpha\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), \color{blue}{\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right)}\right), 1\right)\right) \]

   5. Simplified 26.6%

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

      1. *-commutative N/A

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

      2. difference-of-sqr-1 N/A

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

      3. times-frac N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{i + \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 1}\right), \color{blue}{\left(\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(i + \alpha\right), \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 1\right)\right), \left(\frac{\color{blue}{i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(i, \alpha\right), \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 1\right)\right), \left(\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(i, \alpha\right), \left(\left(2 \cdot i + \left(\alpha + \beta\right)\right) + 1\right)\right), \left(\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(i, \alpha\right), \left(\left(2 \cdot i + \left(\beta + \alpha\right)\right) + 1\right)\right), \left(\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\right)\right) \]

      9. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(i, \alpha\right), \left(2 \cdot i + \left(\left(\beta + \alpha\right) + 1\right)\right)\right), \left(\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(i, \alpha\right), \mathsf{+.f64}\left(\left(2 \cdot i\right), \left(\left(\beta + \alpha\right) + 1\right)\right)\right), \left(\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(i, \alpha\right), \mathsf{+.f64}\left(\left(i \cdot 2\right), \left(\left(\beta + \alpha\right) + 1\right)\right)\right), \left(\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(i, \alpha\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(i, 2\right), \left(\left(\beta + \alpha\right) + 1\right)\right)\right), \left(\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(i, \alpha\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(i, 2\right), \mathsf{+.f64}\left(\left(\beta + \alpha\right), 1\right)\right)\right), \left(\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\right)\right) \]

      14. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(i, \alpha\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(i, 2\right), \mathsf{+.f64}\left(\left(\alpha + \beta\right), 1\right)\right)\right), \left(\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(i, \alpha\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(i, 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 1\right)\right)\right), \left(\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(i, \alpha\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(i, 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 1\right)\right)\right), \mathsf{/.f64}\left(i, \color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)}\right)\right) \]

   7. Applied egg-rr 69.2%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 7.5 \cdot 10^{+122}:\\ \;\;\;\;\frac{\frac{i \cdot \left(0.25 + 0.25 \cdot \frac{\beta}{i}\right)}{i \cdot 2 + \left(\alpha + \left(\beta + -1\right)\right)}}{\frac{i \cdot 2 + \left(\left(\beta + \alpha\right) + 1\right)}{i}}\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{i \cdot 2 + \left(\left(\beta + \alpha\right) + 1\right)} \cdot \frac{i}{i \cdot 2 + \left(\left(\beta + \alpha\right) + -1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5 \cdot 10^{+122}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{i \cdot 2 + \left(\left(\beta + \alpha\right) + 1\right)} \cdot \frac{i}{i \cdot 2 + \left(\left(\beta + \alpha\right) + -1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 5e+122)
   0.0625
   (*
    (/ (+ i alpha) (+ (* i 2.0) (+ (+ beta alpha) 1.0)))
    (/ i (+ (* i 2.0) (+ (+ beta alpha) -1.0))))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 5e+122) {
		tmp = 0.0625;
	} else {
		tmp = ((i + alpha) / ((i * 2.0) + ((beta + alpha) + 1.0))) * (i / ((i * 2.0) + ((beta + alpha) + -1.0)));
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 5d+122) then
        tmp = 0.0625d0
    else
        tmp = ((i + alpha) / ((i * 2.0d0) + ((beta + alpha) + 1.0d0))) * (i / ((i * 2.0d0) + ((beta + alpha) + (-1.0d0))))
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 5e+122) {
		tmp = 0.0625;
	} else {
		tmp = ((i + alpha) / ((i * 2.0) + ((beta + alpha) + 1.0))) * (i / ((i * 2.0) + ((beta + alpha) + -1.0)));
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 5e+122:
		tmp = 0.0625
	else:
		tmp = ((i + alpha) / ((i * 2.0) + ((beta + alpha) + 1.0))) * (i / ((i * 2.0) + ((beta + alpha) + -1.0)))
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 5e+122)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(i + alpha) / Float64(Float64(i * 2.0) + Float64(Float64(beta + alpha) + 1.0))) * Float64(i / Float64(Float64(i * 2.0) + Float64(Float64(beta + alpha) + -1.0))));
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 5e+122)
		tmp = 0.0625;
	else
		tmp = ((i + alpha) / ((i * 2.0) + ((beta + alpha) + 1.0))) * (i / ((i * 2.0) + ((beta + alpha) + -1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 5e+122], 0.0625, N[(N[(N[(i + alpha), $MachinePrecision] / N[(N[(i * 2.0), $MachinePrecision] + N[(N[(beta + alpha), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(i / N[(N[(i * 2.0), $MachinePrecision] + N[(N[(beta + alpha), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.99999999999999989e122

   1. Initial program 19.4%

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

      1. associate-/l/ N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)\right)}\right) \]

   3. Simplified 16.8%

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

   5. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{1}{16}} \]
   6. Step-by-step derivation

   7. Simplified 79.9%

      \[\leadsto \color{blue}{0.0625} \]

   if 4.99999999999999989e122 < beta

   1. Initial program 2.3%

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

   3. Taylor expanded in beta around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(i \cdot \left(\alpha + i\right)\right)}, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right)\right), 1\right)\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(\alpha + i\right)\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right)\right)}, 1\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(i + \alpha\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), \color{blue}{\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right)}\right), 1\right)\right) \]

      3. +-lowering-+.f64 26.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(i, \alpha\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), \color{blue}{\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right)}\right), 1\right)\right) \]

   5. Simplified 26.6%

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

      1. *-commutative N/A

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

      2. difference-of-sqr-1 N/A

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

      3. times-frac N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{i + \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 1}\right), \color{blue}{\left(\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(i + \alpha\right), \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 1\right)\right), \left(\frac{\color{blue}{i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(i, \alpha\right), \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 1\right)\right), \left(\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(i, \alpha\right), \left(\left(2 \cdot i + \left(\alpha + \beta\right)\right) + 1\right)\right), \left(\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(i, \alpha\right), \left(\left(2 \cdot i + \left(\beta + \alpha\right)\right) + 1\right)\right), \left(\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\right)\right) \]

      9. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(i, \alpha\right), \left(2 \cdot i + \left(\left(\beta + \alpha\right) + 1\right)\right)\right), \left(\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(i, \alpha\right), \mathsf{+.f64}\left(\left(2 \cdot i\right), \left(\left(\beta + \alpha\right) + 1\right)\right)\right), \left(\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(i, \alpha\right), \mathsf{+.f64}\left(\left(i \cdot 2\right), \left(\left(\beta + \alpha\right) + 1\right)\right)\right), \left(\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(i, \alpha\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(i, 2\right), \left(\left(\beta + \alpha\right) + 1\right)\right)\right), \left(\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(i, \alpha\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(i, 2\right), \mathsf{+.f64}\left(\left(\beta + \alpha\right), 1\right)\right)\right), \left(\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\right)\right) \]

      14. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(i, \alpha\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(i, 2\right), \mathsf{+.f64}\left(\left(\alpha + \beta\right), 1\right)\right)\right), \left(\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(i, \alpha\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(i, 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 1\right)\right)\right), \left(\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(i, \alpha\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(i, 2\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), 1\right)\right)\right), \mathsf{/.f64}\left(i, \color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)}\right)\right) \]

   7. Applied egg-rr 69.2%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5 \cdot 10^{+122}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{i \cdot 2 + \left(\left(\beta + \alpha\right) + 1\right)} \cdot \frac{i}{i \cdot 2 + \left(\left(\beta + \alpha\right) + -1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.1% accurate, 3.8× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.4 \cdot 10^{+123}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i + \alpha}{\beta}}{\frac{\beta}{i}}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 2.4e+123) 0.0625 (/ (/ (+ i alpha) beta) (/ beta i))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2.4e+123) {
		tmp = 0.0625;
	} else {
		tmp = ((i + alpha) / beta) / (beta / i);
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 2.4d+123) then
        tmp = 0.0625d0
    else
        tmp = ((i + alpha) / beta) / (beta / i)
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2.4e+123) {
		tmp = 0.0625;
	} else {
		tmp = ((i + alpha) / beta) / (beta / i);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 2.4e+123:
		tmp = 0.0625
	else:
		tmp = ((i + alpha) / beta) / (beta / i)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 2.4e+123)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(i + alpha) / beta) / Float64(beta / i));
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 2.4e+123)
		tmp = 0.0625;
	else
		tmp = ((i + alpha) / beta) / (beta / i);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 2.4e+123], 0.0625, N[(N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(beta / i), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.39999999999999989e123

   1. Initial program 19.4%

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

      1. associate-/l/ N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)\right)}\right) \]

   3. Simplified 16.8%

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

   5. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{1}{16}} \]
   6. Step-by-step derivation

   7. Simplified 79.9%

      \[\leadsto \color{blue}{0.0625} \]

   if 2.39999999999999989e123 < beta

   1. Initial program 2.3%

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

      1. associate-/l/ N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)\right)}\right) \]

   3. Simplified 0.1%

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

   5. Taylor expanded in beta around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(i \cdot \left(\alpha + i\right)\right), \color{blue}{\left({\beta}^{2}\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(\alpha + i\right)\right), \left({\color{blue}{\beta}}^{2}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(i + \alpha\right)\right), \left({\beta}^{2}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(i, \alpha\right)\right), \left({\beta}^{2}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(i, \alpha\right)\right), \left(\beta \cdot \color{blue}{\beta}\right)\right) \]

      6. *-lowering-*.f64 26.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(i, \alpha\right)\right), \mathsf{*.f64}\left(\beta, \color{blue}{\beta}\right)\right) \]

   7. Simplified 26.7%

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

      1. times-frac N/A

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

      2. associate-*r/ N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{i}{\beta} \cdot \left(i + \alpha\right)\right), \color{blue}{\beta}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{i}{\beta}\right), \left(i + \alpha\right)\right), \beta\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(i, \beta\right), \left(i + \alpha\right)\right), \beta\right) \]

      6. +-lowering-+.f64 65.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(i, \beta\right), \mathsf{+.f64}\left(i, \alpha\right)\right), \beta\right) \]

   9. Applied egg-rr 65.5%

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

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{i}{\beta} \cdot \left(i + \alpha\right)\right), \color{blue}{\beta}\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(i + \alpha\right) \cdot \frac{i}{\beta}\right), \beta\right) \]

       3. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(i + \alpha\right) \cdot \frac{1}{\frac{\beta}{i}}\right), \beta\right) \]

       4. un-div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{i + \alpha}{\frac{\beta}{i}}\right), \beta\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(i + \alpha\right), \left(\frac{\beta}{i}\right)\right), \beta\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(i, \alpha\right), \left(\frac{\beta}{i}\right)\right), \beta\right) \]

       7. /-lowering-/.f64 65.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(i, \alpha\right), \mathsf{/.f64}\left(\beta, i\right)\right), \beta\right) \]

   11. Applied egg-rr 65.5%

       \[\leadsto \color{blue}{\frac{\frac{i + \alpha}{\frac{\beta}{i}}}{\beta}} \]
   12. Step-by-step derivation

       1. associate-/l/ N/A

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

       2. associate-/r* N/A

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

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{i + \alpha}{\beta}\right), \color{blue}{\left(\frac{\beta}{i}\right)}\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(i + \alpha\right), \beta\right), \left(\frac{\color{blue}{\beta}}{i}\right)\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(i, \alpha\right), \beta\right), \left(\frac{\beta}{i}\right)\right) \]

       6. /-lowering-/.f64 65.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(i, \alpha\right), \beta\right), \mathsf{/.f64}\left(\beta, \color{blue}{i}\right)\right) \]

   13. Applied egg-rr 65.6%

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

Alternative 4: 84.0% accurate, 3.8× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 7 \cdot 10^{+122}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 7e+122) 0.0625 (* (/ (+ i alpha) beta) (/ i beta))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 7e+122) {
		tmp = 0.0625;
	} else {
		tmp = ((i + alpha) / beta) * (i / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 7d+122) then
        tmp = 0.0625d0
    else
        tmp = ((i + alpha) / beta) * (i / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 7e+122) {
		tmp = 0.0625;
	} else {
		tmp = ((i + alpha) / beta) * (i / beta);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 7e+122:
		tmp = 0.0625
	else:
		tmp = ((i + alpha) / beta) * (i / beta)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 7e+122)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(i + alpha) / beta) * Float64(i / beta));
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 7e+122)
		tmp = 0.0625;
	else
		tmp = ((i + alpha) / beta) * (i / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 7e+122], 0.0625, N[(N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 7.00000000000000028e122

   1. Initial program 19.4%

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

      1. associate-/l/ N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)\right)}\right) \]

   3. Simplified 16.8%

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

   5. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{1}{16}} \]
   6. Step-by-step derivation

   7. Simplified 79.9%

      \[\leadsto \color{blue}{0.0625} \]

   if 7.00000000000000028e122 < beta

   1. Initial program 2.3%

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

      1. associate-/l/ N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)\right)}\right) \]

   3. Simplified 0.1%

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

   5. Taylor expanded in beta around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(i \cdot \left(\alpha + i\right)\right), \color{blue}{\left({\beta}^{2}\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(\alpha + i\right)\right), \left({\color{blue}{\beta}}^{2}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(i + \alpha\right)\right), \left({\beta}^{2}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(i, \alpha\right)\right), \left({\beta}^{2}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(i, \alpha\right)\right), \left(\beta \cdot \color{blue}{\beta}\right)\right) \]

      6. *-lowering-*.f64 26.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(i, \alpha\right)\right), \mathsf{*.f64}\left(\beta, \color{blue}{\beta}\right)\right) \]

   7. Simplified 26.7%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{i + \alpha}{\beta}\right), \color{blue}{\left(\frac{i}{\beta}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(i + \alpha\right), \beta\right), \left(\frac{\color{blue}{i}}{\beta}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(i, \alpha\right), \beta\right), \left(\frac{i}{\beta}\right)\right) \]

      6. /-lowering-/.f64 65.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(i, \alpha\right), \beta\right), \mathsf{/.f64}\left(i, \color{blue}{\beta}\right)\right) \]

   9. Applied egg-rr 65.6%

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

Alternative 5: 81.6% accurate, 4.4× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 7.5 \cdot 10^{+122}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\frac{\beta}{i}}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 7.5e+122) 0.0625 (/ (/ i (/ beta i)) beta)))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 7.5e+122) {
		tmp = 0.0625;
	} else {
		tmp = (i / (beta / i)) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 7.5d+122) then
        tmp = 0.0625d0
    else
        tmp = (i / (beta / i)) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 7.5e+122) {
		tmp = 0.0625;
	} else {
		tmp = (i / (beta / i)) / beta;
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 7.5e+122:
		tmp = 0.0625
	else:
		tmp = (i / (beta / i)) / beta
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 7.5e+122)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i / Float64(beta / i)) / beta);
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 7.5e+122)
		tmp = 0.0625;
	else
		tmp = (i / (beta / i)) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 7.5e+122], 0.0625, N[(N[(i / N[(beta / i), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 7.5000000000000002e122

   1. Initial program 19.4%

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

      1. associate-/l/ N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)\right)}\right) \]

   3. Simplified 16.8%

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

   5. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{1}{16}} \]
   6. Step-by-step derivation

   7. Simplified 79.9%

      \[\leadsto \color{blue}{0.0625} \]

   if 7.5000000000000002e122 < beta

   1. Initial program 2.3%

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

      1. associate-/l/ N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)\right)}\right) \]

   3. Simplified 0.1%

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

   5. Taylor expanded in beta around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(i \cdot \left(\alpha + i\right)\right), \color{blue}{\left({\beta}^{2}\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(\alpha + i\right)\right), \left({\color{blue}{\beta}}^{2}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(i + \alpha\right)\right), \left({\beta}^{2}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(i, \alpha\right)\right), \left({\beta}^{2}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(i, \alpha\right)\right), \left(\beta \cdot \color{blue}{\beta}\right)\right) \]

      6. *-lowering-*.f64 26.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(i, \alpha\right)\right), \mathsf{*.f64}\left(\beta, \color{blue}{\beta}\right)\right) \]

   7. Simplified 26.7%

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

      1. times-frac N/A

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

      2. associate-*r/ N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{i}{\beta} \cdot \left(i + \alpha\right)\right), \color{blue}{\beta}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{i}{\beta}\right), \left(i + \alpha\right)\right), \beta\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(i, \beta\right), \left(i + \alpha\right)\right), \beta\right) \]

      6. +-lowering-+.f64 65.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(i, \beta\right), \mathsf{+.f64}\left(i, \alpha\right)\right), \beta\right) \]

   9. Applied egg-rr 65.5%

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

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{i}{\beta} \cdot \left(i + \alpha\right)\right), \color{blue}{\beta}\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(i + \alpha\right) \cdot \frac{i}{\beta}\right), \beta\right) \]

       3. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(i + \alpha\right) \cdot \frac{1}{\frac{\beta}{i}}\right), \beta\right) \]

       4. un-div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{i + \alpha}{\frac{\beta}{i}}\right), \beta\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(i + \alpha\right), \left(\frac{\beta}{i}\right)\right), \beta\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(i, \alpha\right), \left(\frac{\beta}{i}\right)\right), \beta\right) \]

       7. /-lowering-/.f64 65.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(i, \alpha\right), \mathsf{/.f64}\left(\beta, i\right)\right), \beta\right) \]

   11. Applied egg-rr 65.5%

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

   12. Taylor expanded in i around inf

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\color{blue}{i}, \mathsf{/.f64}\left(\beta, i\right)\right), \beta\right) \]
   13. Step-by-step derivation

   14. Simplified 59.8%

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

Alternative 6: 81.7% accurate, 4.4× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.2 \cdot 10^{+123}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \frac{i}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 2.2e+123) 0.0625 (/ (* i (/ i beta)) beta)))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2.2e+123) {
		tmp = 0.0625;
	} else {
		tmp = (i * (i / beta)) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 2.2d+123) then
        tmp = 0.0625d0
    else
        tmp = (i * (i / beta)) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2.2e+123) {
		tmp = 0.0625;
	} else {
		tmp = (i * (i / beta)) / beta;
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 2.2e+123:
		tmp = 0.0625
	else:
		tmp = (i * (i / beta)) / beta
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 2.2e+123)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i * Float64(i / beta)) / beta);
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 2.2e+123)
		tmp = 0.0625;
	else
		tmp = (i * (i / beta)) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 2.2e+123], 0.0625, N[(N[(i * N[(i / beta), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.19999999999999992e123

   1. Initial program 19.4%

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

      1. associate-/l/ N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)\right)}\right) \]

   3. Simplified 16.8%

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

   5. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{1}{16}} \]
   6. Step-by-step derivation

   7. Simplified 79.9%

      \[\leadsto \color{blue}{0.0625} \]

   if 2.19999999999999992e123 < beta

   1. Initial program 2.3%

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

      1. associate-/l/ N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)\right)}\right) \]

   3. Simplified 0.1%

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

   5. Taylor expanded in beta around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(i \cdot \left(\alpha + i\right)\right), \color{blue}{\left({\beta}^{2}\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(\alpha + i\right)\right), \left({\color{blue}{\beta}}^{2}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(i + \alpha\right)\right), \left({\beta}^{2}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(i, \alpha\right)\right), \left({\beta}^{2}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(i, \alpha\right)\right), \left(\beta \cdot \color{blue}{\beta}\right)\right) \]

      6. *-lowering-*.f64 26.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(i, \alpha\right)\right), \mathsf{*.f64}\left(\beta, \color{blue}{\beta}\right)\right) \]

   7. Simplified 26.7%

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

      1. times-frac N/A

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

      2. associate-*r/ N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{i}{\beta} \cdot \left(i + \alpha\right)\right), \color{blue}{\beta}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{i}{\beta}\right), \left(i + \alpha\right)\right), \beta\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(i, \beta\right), \left(i + \alpha\right)\right), \beta\right) \]

      6. +-lowering-+.f64 65.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(i, \beta\right), \mathsf{+.f64}\left(i, \alpha\right)\right), \beta\right) \]

   9. Applied egg-rr 65.5%

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

   10. Taylor expanded in i around inf

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(i, \beta\right), \color{blue}{i}\right), \beta\right) \]
   11. Step-by-step derivation

   12. Simplified 59.8%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.2 \cdot 10^{+123}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \frac{i}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.1% accurate, 4.4× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.4 \cdot 10^{+198}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha \cdot \frac{i}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 1.4e+198) 0.0625 (/ (* alpha (/ i beta)) beta)))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.4e+198) {
		tmp = 0.0625;
	} else {
		tmp = (alpha * (i / beta)) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 1.4d+198) then
        tmp = 0.0625d0
    else
        tmp = (alpha * (i / beta)) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.4e+198) {
		tmp = 0.0625;
	} else {
		tmp = (alpha * (i / beta)) / beta;
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 1.4e+198:
		tmp = 0.0625
	else:
		tmp = (alpha * (i / beta)) / beta
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1.4e+198)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(alpha * Float64(i / beta)) / beta);
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 1.4e+198)
		tmp = 0.0625;
	else
		tmp = (alpha * (i / beta)) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 1.4e+198], 0.0625, N[(N[(alpha * N[(i / beta), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.4e198

   1. Initial program 18.0%

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

      1. associate-/l/ N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)\right)}\right) \]

   3. Simplified 15.2%

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

   5. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{1}{16}} \]
   6. Step-by-step derivation

   7. Simplified 76.2%

      \[\leadsto \color{blue}{0.0625} \]

   if 1.4e198 < beta

   1. Initial program 0.0%

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

      1. associate-/l/ N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)\right)}\right) \]

   3. Simplified 0.0%

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

   5. Taylor expanded in beta around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(i \cdot \left(\alpha + i\right)\right), \color{blue}{\left({\beta}^{2}\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(\alpha + i\right)\right), \left({\color{blue}{\beta}}^{2}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(i + \alpha\right)\right), \left({\beta}^{2}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(i, \alpha\right)\right), \left({\beta}^{2}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(i, \alpha\right)\right), \left(\beta \cdot \color{blue}{\beta}\right)\right) \]

      6. *-lowering-*.f64 21.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(i, \alpha\right)\right), \mathsf{*.f64}\left(\beta, \color{blue}{\beta}\right)\right) \]

   7. Simplified 21.5%

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

      1. times-frac N/A

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

      2. associate-*r/ N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{i}{\beta} \cdot \left(i + \alpha\right)\right), \color{blue}{\beta}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{i}{\beta}\right), \left(i + \alpha\right)\right), \beta\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(i, \beta\right), \left(i + \alpha\right)\right), \beta\right) \]

      6. +-lowering-+.f64 71.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(i, \beta\right), \mathsf{+.f64}\left(i, \alpha\right)\right), \beta\right) \]

   9. Applied egg-rr 71.9%

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

   10. Taylor expanded in i around 0

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(i, \beta\right), \color{blue}{\alpha}\right), \beta\right) \]
   11. Step-by-step derivation

   12. Simplified 31.9%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.4 \cdot 10^{+198}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha \cdot \frac{i}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.4% accurate, 4.4× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.05 \cdot 10^{+198}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \alpha}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 4.05e+198) 0.0625 (/ (* i alpha) (* beta beta))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 4.05e+198) {
		tmp = 0.0625;
	} else {
		tmp = (i * alpha) / (beta * beta);
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 4.05d+198) then
        tmp = 0.0625d0
    else
        tmp = (i * alpha) / (beta * beta)
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 4.05e+198) {
		tmp = 0.0625;
	} else {
		tmp = (i * alpha) / (beta * beta);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 4.05e+198:
		tmp = 0.0625
	else:
		tmp = (i * alpha) / (beta * beta)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 4.05e+198)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i * alpha) / Float64(beta * beta));
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 4.05e+198)
		tmp = 0.0625;
	else
		tmp = (i * alpha) / (beta * beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 4.05e+198], 0.0625, N[(N[(i * alpha), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.0499999999999999e198

   1. Initial program 18.0%

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

      1. associate-/l/ N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)\right)}\right) \]

   3. Simplified 15.2%

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

   5. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{1}{16}} \]
   6. Step-by-step derivation

   7. Simplified 76.2%

      \[\leadsto \color{blue}{0.0625} \]

   if 4.0499999999999999e198 < beta

   1. Initial program 0.0%

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

      1. associate-/l/ N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)\right)}\right) \]

   3. Simplified 0.0%

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

   5. Taylor expanded in beta around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(i \cdot \left(\alpha + i\right)\right), \color{blue}{\left({\beta}^{2}\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(\alpha + i\right)\right), \left({\color{blue}{\beta}}^{2}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(i + \alpha\right)\right), \left({\beta}^{2}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(i, \alpha\right)\right), \left({\beta}^{2}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(i, \alpha\right)\right), \left(\beta \cdot \color{blue}{\beta}\right)\right) \]

      6. *-lowering-*.f64 21.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(i, \alpha\right)\right), \mathsf{*.f64}\left(\beta, \color{blue}{\beta}\right)\right) \]

   7. Simplified 21.5%

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

   8. Taylor expanded in i around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \color{blue}{\alpha}\right), \mathsf{*.f64}\left(\beta, \beta\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 22.5%

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

Alternative 9: 70.7% accurate, 53.0× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ 0.0625 \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i) :precision binary64 0.0625)

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	return 0.0625;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    code = 0.0625d0
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	return 0.0625;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	return 0.0625

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	return 0.0625
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp = code(alpha, beta, i)
	tmp = 0.0625;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := 0.0625

Derivation

1. Initial program 16.2%

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

   1. associate-/l/ N/A

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

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)\right)}\right) \]

3. Simplified 13.7%

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

5. Taylor expanded in i around inf

   \[\leadsto \color{blue}{\frac{1}{16}} \]
6. Step-by-step derivation

7. Simplified 70.7%

   \[\leadsto \color{blue}{0.0625} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024150 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/4"
  :precision binary64
  :pre (and (and (> alpha -1.0) (> beta -1.0)) (> i 1.0))
  (/ (/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2.0 i)) (+ (+ alpha beta) (* 2.0 i)))) (- (* (+ (+ alpha beta) (* 2.0 i)) (+ (+ alpha beta) (* 2.0 i))) 1.0)))