Octave 3.8, jcobi/4

Percentage Accurate: 16.3% → 85.4%

Time: 16.1s

Alternatives: 7

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: 16.3% 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: 85.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \beta + i \cdot 2\\ \mathbf{if}\;\beta \leq 4.8 \cdot 10^{+148}:\\ \;\;\;\;{\left(\frac{\alpha + \left(t\_0 + 1\right)}{i}\right)}^{-1} \cdot {\left(\frac{\alpha + \left(t\_0 + -1\right)}{i \cdot \left(0.25 + \left(2 \cdot \left(\beta + \alpha\right) - \left(\beta + \alpha\right)\right) \cdot \frac{0.25}{i}\right)}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ beta (* i 2.0))))
   (if (<= beta 4.8e+148)
     (*
      (pow (/ (+ alpha (+ t_0 1.0)) i) -1.0)
      (pow
       (/
        (+ alpha (+ t_0 -1.0))
        (*
         i
         (+ 0.25 (* (- (* 2.0 (+ beta alpha)) (+ beta alpha)) (/ 0.25 i)))))
       -1.0))
     (* (/ (+ alpha i) beta) (/ i beta)))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = beta + (i * 2.0);
	double tmp;
	if (beta <= 4.8e+148) {
		tmp = pow(((alpha + (t_0 + 1.0)) / i), -1.0) * pow(((alpha + (t_0 + -1.0)) / (i * (0.25 + (((2.0 * (beta + alpha)) - (beta + alpha)) * (0.25 / i))))), -1.0);
	} else {
		tmp = ((alpha + 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) :: t_0
    real(8) :: tmp
    t_0 = beta + (i * 2.0d0)
    if (beta <= 4.8d+148) then
        tmp = (((alpha + (t_0 + 1.0d0)) / i) ** (-1.0d0)) * (((alpha + (t_0 + (-1.0d0))) / (i * (0.25d0 + (((2.0d0 * (beta + alpha)) - (beta + alpha)) * (0.25d0 / i))))) ** (-1.0d0))
    else
        tmp = ((alpha + 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 t_0 = beta + (i * 2.0);
	double tmp;
	if (beta <= 4.8e+148) {
		tmp = Math.pow(((alpha + (t_0 + 1.0)) / i), -1.0) * Math.pow(((alpha + (t_0 + -1.0)) / (i * (0.25 + (((2.0 * (beta + alpha)) - (beta + alpha)) * (0.25 / i))))), -1.0);
	} else {
		tmp = ((alpha + i) / beta) * (i / beta);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	t_0 = beta + (i * 2.0)
	tmp = 0
	if beta <= 4.8e+148:
		tmp = math.pow(((alpha + (t_0 + 1.0)) / i), -1.0) * math.pow(((alpha + (t_0 + -1.0)) / (i * (0.25 + (((2.0 * (beta + alpha)) - (beta + alpha)) * (0.25 / i))))), -1.0)
	else:
		tmp = ((alpha + i) / beta) * (i / beta)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = Float64(beta + Float64(i * 2.0))
	tmp = 0.0
	if (beta <= 4.8e+148)
		tmp = Float64((Float64(Float64(alpha + Float64(t_0 + 1.0)) / i) ^ -1.0) * (Float64(Float64(alpha + Float64(t_0 + -1.0)) / Float64(i * Float64(0.25 + Float64(Float64(Float64(2.0 * Float64(beta + alpha)) - Float64(beta + alpha)) * Float64(0.25 / i))))) ^ -1.0));
	else
		tmp = Float64(Float64(Float64(alpha + i) / beta) * Float64(i / beta));
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	t_0 = beta + (i * 2.0);
	tmp = 0.0;
	if (beta <= 4.8e+148)
		tmp = (((alpha + (t_0 + 1.0)) / i) ^ -1.0) * (((alpha + (t_0 + -1.0)) / (i * (0.25 + (((2.0 * (beta + alpha)) - (beta + alpha)) * (0.25 / i))))) ^ -1.0);
	else
		tmp = ((alpha + 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_] := Block[{t$95$0 = N[(beta + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 4.8e+148], N[(N[Power[N[(N[(alpha + N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision], -1.0], $MachinePrecision] * N[Power[N[(N[(alpha + N[(t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision] / N[(i * N[(0.25 + N[(N[(N[(2.0 * N[(beta + alpha), $MachinePrecision]), $MachinePrecision] - N[(beta + alpha), $MachinePrecision]), $MachinePrecision] * N[(0.25 / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + i), $MachinePrecision] / beta), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.79999999999999989e148

   1. Initial program 22.9%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   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 35.2%

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

   7. Applied egg-rr 79.6%

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

   if 4.79999999999999989e148 < 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. Add Preprocessing

   3. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{\frac{i \cdot \left(\alpha + i\right)}{{\beta}^{2}}} \]
   4. 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. +-lowering-+.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 15.3%

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

   5. Simplified 15.3%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

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

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

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

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

      5. +-commutative N/A

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

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

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

      7. /-lowering-/.f64 64.6%

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

   7. Applied egg-rr 64.6%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.8 \cdot 10^{+148}:\\ \;\;\;\;{\left(\frac{\alpha + \left(\left(\beta + i \cdot 2\right) + 1\right)}{i}\right)}^{-1} \cdot {\left(\frac{\alpha + \left(\left(\beta + i \cdot 2\right) + -1\right)}{i \cdot \left(0.25 + \left(2 \cdot \left(\beta + \alpha\right) - \left(\beta + \alpha\right)\right) \cdot \frac{0.25}{i}\right)}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.05 \cdot 10^{+149}:\\ \;\;\;\;\frac{i \cdot 0.25}{\beta + \left(i \cdot 2 + 1\right)} \cdot \frac{\beta + i}{\beta + \left(i \cdot 2 + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 1.05e+149)
   (*
    (/ (* i 0.25) (+ beta (+ (* i 2.0) 1.0)))
    (/ (+ beta i) (+ beta (+ (* i 2.0) -1.0))))
   (* (/ (+ alpha i) beta) (/ i beta))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.05e+149) {
		tmp = ((i * 0.25) / (beta + ((i * 2.0) + 1.0))) * ((beta + i) / (beta + ((i * 2.0) + -1.0)));
	} else {
		tmp = ((alpha + 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 <= 1.05d+149) then
        tmp = ((i * 0.25d0) / (beta + ((i * 2.0d0) + 1.0d0))) * ((beta + i) / (beta + ((i * 2.0d0) + (-1.0d0))))
    else
        tmp = ((alpha + 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 <= 1.05e+149) {
		tmp = ((i * 0.25) / (beta + ((i * 2.0) + 1.0))) * ((beta + i) / (beta + ((i * 2.0) + -1.0)));
	} else {
		tmp = ((alpha + i) / beta) * (i / beta);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 1.05e+149:
		tmp = ((i * 0.25) / (beta + ((i * 2.0) + 1.0))) * ((beta + i) / (beta + ((i * 2.0) + -1.0)))
	else:
		tmp = ((alpha + i) / beta) * (i / beta)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1.05e+149)
		tmp = Float64(Float64(Float64(i * 0.25) / Float64(beta + Float64(Float64(i * 2.0) + 1.0))) * Float64(Float64(beta + i) / Float64(beta + Float64(Float64(i * 2.0) + -1.0))));
	else
		tmp = Float64(Float64(Float64(alpha + i) / 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 <= 1.05e+149)
		tmp = ((i * 0.25) / (beta + ((i * 2.0) + 1.0))) * ((beta + i) / (beta + ((i * 2.0) + -1.0)));
	else
		tmp = ((alpha + 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, 1.05e+149], N[(N[(N[(i * 0.25), $MachinePrecision] / N[(beta + N[(N[(i * 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(beta + i), $MachinePrecision] / N[(beta + N[(N[(i * 2.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + i), $MachinePrecision] / beta), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.0500000000000001e149

   1. Initial program 22.9%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   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 35.2%

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

   6. Taylor expanded in alpha around 0

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

      1. div-sub N/A

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

      2. associate-*r/ N/A

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

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

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

   8. Simplified 34.0%

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

   9. Taylor expanded in beta around 0

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

       1. distribute-lft-out N/A

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

       2. unpow2 N/A

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

       3. distribute-rgt-in N/A

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

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

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

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

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

       6. +-commutative N/A

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

       7. +-lowering-+.f64 34.0%

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

   11. Simplified 34.0%

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

       1. associate-*r* N/A

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

       2. +-commutative N/A

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

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

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. associate-+r+ N/A

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

       7. times-frac N/A

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

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

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

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

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

       10. *-commutative N/A

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

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

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

       12. associate-+r+ N/A

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

       13. +-commutative N/A

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

       14. associate-+l+ N/A

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

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

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

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

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

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

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

   13. Applied egg-rr 79.3%

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

   if 1.0500000000000001e149 < 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. Add Preprocessing

   3. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{\frac{i \cdot \left(\alpha + i\right)}{{\beta}^{2}}} \]
   4. 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. +-lowering-+.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 15.3%

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

   5. Simplified 15.3%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

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

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

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

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

      5. +-commutative N/A

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

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

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

      7. /-lowering-/.f64 64.6%

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

   7. Applied egg-rr 64.6%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.05 \cdot 10^{+149}:\\ \;\;\;\;\frac{i \cdot 0.25}{\beta + \left(i \cdot 2 + 1\right)} \cdot \frac{\beta + i}{\beta + \left(i \cdot 2 + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.9% accurate, 3.8× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 9.5 \cdot 10^{+139}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 9.5e+139) 0.0625 (* (/ (+ alpha i) beta) (/ i beta))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 9.5e+139) {
		tmp = 0.0625;
	} else {
		tmp = ((alpha + 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 <= 9.5d+139) then
        tmp = 0.0625d0
    else
        tmp = ((alpha + 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 <= 9.5e+139) {
		tmp = 0.0625;
	} else {
		tmp = ((alpha + i) / beta) * (i / beta);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 9.5e+139:
		tmp = 0.0625
	else:
		tmp = ((alpha + i) / beta) * (i / beta)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 9.5e+139)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(alpha + i) / 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 <= 9.5e+139)
		tmp = 0.0625;
	else
		tmp = ((alpha + 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, 9.5e+139], 0.0625, N[(N[(N[(alpha + i), $MachinePrecision] / beta), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 9.5000000000000002e139

   1. Initial program 22.9%

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

   3. Taylor expanded in i around inf

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

   5. Simplified 78.9%

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

   if 9.5000000000000002e139 < 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. Add Preprocessing

   3. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{\frac{i \cdot \left(\alpha + i\right)}{{\beta}^{2}}} \]
   4. 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. +-lowering-+.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 15.3%

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

   5. Simplified 15.3%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

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

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

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

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

      5. +-commutative N/A

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

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

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

      7. /-lowering-/.f64 64.6%

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

   7. Applied egg-rr 64.6%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 9.5 \cdot 10^{+139}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.0% accurate, 4.4× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.5 \cdot 10^{+148}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \frac{i}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 2.5e+148) 0.0625 (/ (* i (/ i beta)) beta)))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2.5e+148) {
		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 <= 2.5d+148) 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 <= 2.5e+148) {
		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 <= 2.5e+148:
		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 <= 2.5e+148)
		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 <= 2.5e+148)
		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, 2.5e+148], 0.0625, N[(N[(i * N[(i / beta), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.50000000000000012e148

   1. Initial program 22.9%

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

   3. Taylor expanded in i around inf

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

   5. Simplified 78.9%

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

   if 2.50000000000000012e148 < 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. Add Preprocessing

   3. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{\frac{i \cdot \left(\alpha + i\right)}{{\beta}^{2}}} \]
   4. 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. +-lowering-+.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 15.3%

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

   5. Simplified 15.3%

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

      1. associate-/r* N/A

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

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

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

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

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

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

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

      5. +-commutative N/A

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

      6. +-lowering-+.f64 39.5%

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

   7. Applied egg-rr 39.5%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. +-commutative N/A

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

      6. +-lowering-+.f64 64.5%

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

   9. Applied egg-rr 64.5%

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

   10. Taylor expanded in alpha around 0

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

       1. /-lowering-/.f64 62.0%

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

   12. Simplified 62.0%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.5 \cdot 10^{+148}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \frac{i}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.4% accurate, 4.4× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2 \cdot 10^{+232}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 2e+232) 0.0625 (/ (* i (/ alpha beta)) beta)))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2e+232) {
		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 <= 2d+232) 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 <= 2e+232) {
		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 <= 2e+232:
		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 <= 2e+232)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i * Float64(alpha / 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 <= 2e+232)
		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, 2e+232], 0.0625, N[(N[(i * N[(alpha / beta), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.00000000000000011e232

   1. Initial program 21.1%

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

   3. Taylor expanded in i around inf

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

   5. Simplified 76.6%

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

   if 2.00000000000000011e232 < 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. Add Preprocessing

   3. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{\frac{i \cdot \left(\alpha + i\right)}{{\beta}^{2}}} \]
   4. 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. +-lowering-+.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 22.7%

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

   5. Simplified 22.7%

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

      1. associate-/r* N/A

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

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

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

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

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

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

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

      5. +-commutative N/A

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

      6. +-lowering-+.f64 48.6%

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

   7. Applied egg-rr 48.6%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. +-commutative N/A

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

      6. +-lowering-+.f64 85.5%

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

   9. Applied egg-rr 85.5%

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

   10. Taylor expanded in alpha around inf

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

       1. /-lowering-/.f64 25.2%

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

   12. Simplified 25.2%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2 \cdot 10^{+232}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \frac{\alpha}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.4% accurate, 4.4× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.1 \cdot 10^{+232}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha \cdot i}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 3.1e+232) 0.0625 (/ (* alpha i) (* beta beta))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 3.1e+232) {
		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 <= 3.1d+232) 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 <= 3.1e+232) {
		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 <= 3.1e+232:
		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 <= 3.1e+232)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(alpha * i) / 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 <= 3.1e+232)
		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, 3.1e+232], 0.0625, N[(N[(alpha * i), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.09999999999999983e232

   1. Initial program 21.1%

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

   3. Taylor expanded in i around inf

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

   5. Simplified 76.6%

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

   if 3.09999999999999983e232 < 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. Add Preprocessing

   3. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{\frac{i \cdot \left(\alpha + i\right)}{{\beta}^{2}}} \]
   4. 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. +-lowering-+.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 22.7%

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

   5. Simplified 22.7%

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

   6. Taylor expanded in alpha around inf

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

   8. Simplified 24.5%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.1 \cdot 10^{+232}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha \cdot i}{\beta \cdot \beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.4% 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 19.6%

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

3. Taylor expanded in i around inf

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

5. Simplified 72.0%

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

Reproduce

herbie shell --seed 2024191 
(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)))