Octave 3.8, jcobi/4

Percentage Accurate: 16.5% → 85.8%

Time: 13.5s

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: 16.5% 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, 1.6× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \beta + i \cdot 2\\ \mathbf{if}\;\beta \leq 4.3 \cdot 10^{+148}:\\ \;\;\;\;\frac{i + \left(\beta + \alpha\right)}{\alpha + \left(t\_0 + 1\right)} \cdot \frac{i \cdot 0.25}{\alpha + \left(t\_0 + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i + \alpha}{\beta}}{\frac{\beta}{i}}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ beta (* i 2.0))))
   (if (<= beta 4.3e+148)
     (*
      (/ (+ i (+ beta alpha)) (+ alpha (+ t_0 1.0)))
      (/ (* i 0.25) (+ alpha (+ t_0 -1.0))))
     (/ (/ (+ i alpha) beta) (/ beta i)))))

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 <= 4.3e+148) {
		tmp = ((i + (beta + alpha)) / (alpha + (t_0 + 1.0))) * ((i * 0.25) / (alpha + (t_0 + -1.0)));
	} 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) :: t_0
    real(8) :: tmp
    t_0 = beta + (i * 2.0d0)
    if (beta <= 4.3d+148) then
        tmp = ((i + (beta + alpha)) / (alpha + (t_0 + 1.0d0))) * ((i * 0.25d0) / (alpha + (t_0 + (-1.0d0))))
    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 t_0 = beta + (i * 2.0);
	double tmp;
	if (beta <= 4.3e+148) {
		tmp = ((i + (beta + alpha)) / (alpha + (t_0 + 1.0))) * ((i * 0.25) / (alpha + (t_0 + -1.0)));
	} else {
		tmp = ((i + alpha) / beta) / (beta / i);
	}
	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 <= 4.3e+148:
		tmp = ((i + (beta + alpha)) / (alpha + (t_0 + 1.0))) * ((i * 0.25) / (alpha + (t_0 + -1.0)))
	else:
		tmp = ((i + alpha) / beta) / (beta / i)
	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 <= 4.3e+148)
		tmp = Float64(Float64(Float64(i + Float64(beta + alpha)) / Float64(alpha + Float64(t_0 + 1.0))) * Float64(Float64(i * 0.25) / Float64(alpha + Float64(t_0 + -1.0))));
	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)
	t_0 = beta + (i * 2.0);
	tmp = 0.0;
	if (beta <= 4.3e+148)
		tmp = ((i + (beta + alpha)) / (alpha + (t_0 + 1.0))) * ((i * 0.25) / (alpha + (t_0 + -1.0)));
	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_] := Block[{t$95$0 = N[(beta + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 4.3e+148], N[(N[(N[(i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(i * 0.25), $MachinePrecision] / N[(alpha + N[(t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(beta / i), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.3000000000000002e148

   1. Initial program 22.4%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. associate-*l* N/A

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

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

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

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

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

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

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

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

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

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

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

   4. Applied egg-rr 45.7%

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

   5. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 39.8%

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

   7. Simplified 39.8%

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

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

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

      2. times-frac N/A

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

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

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

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

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

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

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

      6. +-commutative N/A

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

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

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

      8. associate-+l+ N/A

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

      9. associate-+l+ N/A

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

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

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

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

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

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

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

      13. *-commutative N/A

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

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

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

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

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

   9. Applied egg-rr 83.1%

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

   if 4.3000000000000002e148 < beta

   1. Initial program 0.1%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   2. 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 19.2%

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

   7. Simplified 19.2%

      \[\leadsto \color{blue}{\frac{i \cdot \left(i + \alpha\right)}{\beta \cdot \beta}} \]
   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 66.4%

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

   9. Applied egg-rr 66.4%

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

       1. associate-/l* N/A

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

       2. clear-num N/A

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

       3. associate-*l/ N/A

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

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

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

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

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

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

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

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

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

       8. /-lowering-/.f64 66.7%

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

   11. Applied egg-rr 66.7%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.3 \cdot 10^{+148}:\\ \;\;\;\;\frac{i + \left(\beta + \alpha\right)}{\alpha + \left(\left(\beta + i \cdot 2\right) + 1\right)} \cdot \frac{i \cdot 0.25}{\alpha + \left(\left(\beta + i \cdot 2\right) + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i + \alpha}{\beta}}{\frac{\beta}{i}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.5% accurate, 3.8× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.8 \cdot 10^{+148}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i + \alpha}{\beta}}{\frac{\beta}{i}}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 3.8e+148)
   (+ 0.0625 (/ 0.015625 (* i i)))
   (/ (/ (+ i alpha) beta) (/ beta i))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 3.8e+148) {
		tmp = 0.0625 + (0.015625 / (i * i));
	} 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 <= 3.8d+148) then
        tmp = 0.0625d0 + (0.015625d0 / (i * i))
    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 <= 3.8e+148) {
		tmp = 0.0625 + (0.015625 / (i * i));
	} 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 <= 3.8e+148:
		tmp = 0.0625 + (0.015625 / (i * i))
	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 <= 3.8e+148)
		tmp = Float64(0.0625 + Float64(0.015625 / Float64(i * i)));
	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 <= 3.8e+148)
		tmp = 0.0625 + (0.015625 / (i * i));
	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, 3.8e+148], N[(0.0625 + N[(0.015625 / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(beta / i), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.7999999999999998e148

   1. Initial program 22.4%

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

   3. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 40.6%

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

   5. Simplified 40.6%

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

   6. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), \frac{1}{4}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left({i}^{2}\right), 4\right), 1\right)\right) \]

      3. unpow2 N/A

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

      4. *-lowering-*.f64 38.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), \frac{1}{4}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), 4\right), 1\right)\right) \]

   8. Simplified 38.7%

      \[\leadsto \frac{\left(i \cdot i\right) \cdot 0.25}{\color{blue}{\left(i \cdot i\right) \cdot 4} - 1} \]

   9. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{1}{16} + \frac{1}{64} \cdot \frac{1}{{i}^{2}}} \]
   10. Step-by-step derivation

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

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{16}, \color{blue}{\left(\frac{1}{64} \cdot \frac{1}{{i}^{2}}\right)}\right) \]

       2. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{16}, \left(\frac{\frac{1}{64} \cdot 1}{\color{blue}{{i}^{2}}}\right)\right) \]

       3. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{16}, \left(\frac{\frac{1}{64}}{{\color{blue}{i}}^{2}}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{16}, \mathsf{/.f64}\left(\frac{1}{64}, \color{blue}{\left({i}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{16}, \mathsf{/.f64}\left(\frac{1}{64}, \left(i \cdot \color{blue}{i}\right)\right)\right) \]

       6. *-lowering-*.f64 82.6%

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{16}, \mathsf{/.f64}\left(\frac{1}{64}, \mathsf{*.f64}\left(i, \color{blue}{i}\right)\right)\right) \]

   11. Simplified 82.6%

       \[\leadsto \color{blue}{0.0625 + \frac{0.015625}{i \cdot i}} \]

   if 3.7999999999999998e148 < beta

   1. Initial program 0.1%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   2. 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 19.2%

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

   7. Simplified 19.2%

      \[\leadsto \color{blue}{\frac{i \cdot \left(i + \alpha\right)}{\beta \cdot \beta}} \]
   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 66.4%

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

   9. Applied egg-rr 66.4%

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

       1. associate-/l* N/A

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

       2. clear-num N/A

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

       3. associate-*l/ N/A

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

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

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

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

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

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

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

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

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

       8. /-lowering-/.f64 66.7%

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

   11. Applied egg-rr 66.7%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.8 \cdot 10^{+148}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i + \alpha}{\beta}}{\frac{\beta}{i}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.5% accurate, 3.8× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.1 \cdot 10^{+148}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 4.1e+148)
   (+ 0.0625 (/ 0.015625 (* i i)))
   (* (/ (+ i alpha) beta) (/ i beta))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 4.1e+148) {
		tmp = 0.0625 + (0.015625 / (i * i));
	} 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 <= 4.1d+148) then
        tmp = 0.0625d0 + (0.015625d0 / (i * i))
    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 <= 4.1e+148) {
		tmp = 0.0625 + (0.015625 / (i * i));
	} 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 <= 4.1e+148:
		tmp = 0.0625 + (0.015625 / (i * i))
	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 <= 4.1e+148)
		tmp = Float64(0.0625 + Float64(0.015625 / Float64(i * i)));
	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 <= 4.1e+148)
		tmp = 0.0625 + (0.015625 / (i * i));
	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, 4.1e+148], N[(0.0625 + N[(0.015625 / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.0999999999999998e148

   1. Initial program 22.4%

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

   3. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 40.6%

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

   5. Simplified 40.6%

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

   6. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), \frac{1}{4}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left({i}^{2}\right), 4\right), 1\right)\right) \]

      3. unpow2 N/A

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

      4. *-lowering-*.f64 38.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), \frac{1}{4}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), 4\right), 1\right)\right) \]

   8. Simplified 38.7%

      \[\leadsto \frac{\left(i \cdot i\right) \cdot 0.25}{\color{blue}{\left(i \cdot i\right) \cdot 4} - 1} \]

   9. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{1}{16} + \frac{1}{64} \cdot \frac{1}{{i}^{2}}} \]
   10. Step-by-step derivation

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

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{16}, \color{blue}{\left(\frac{1}{64} \cdot \frac{1}{{i}^{2}}\right)}\right) \]

       2. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{16}, \left(\frac{\frac{1}{64} \cdot 1}{\color{blue}{{i}^{2}}}\right)\right) \]

       3. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{16}, \left(\frac{\frac{1}{64}}{{\color{blue}{i}}^{2}}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{16}, \mathsf{/.f64}\left(\frac{1}{64}, \color{blue}{\left({i}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{16}, \mathsf{/.f64}\left(\frac{1}{64}, \left(i \cdot \color{blue}{i}\right)\right)\right) \]

       6. *-lowering-*.f64 82.6%

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{16}, \mathsf{/.f64}\left(\frac{1}{64}, \mathsf{*.f64}\left(i, \color{blue}{i}\right)\right)\right) \]

   11. Simplified 82.6%

       \[\leadsto \color{blue}{0.0625 + \frac{0.015625}{i \cdot i}} \]

   if 4.0999999999999998e148 < beta

   1. Initial program 0.1%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   2. 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 19.2%

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

   7. Simplified 19.2%

      \[\leadsto \color{blue}{\frac{i \cdot \left(i + \alpha\right)}{\beta \cdot \beta}} \]
   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 66.6%

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

   9. Applied egg-rr 66.6%

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

Alternative 4: 83.4% accurate, 4.4× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.1 \cdot 10^{+148}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\beta}}{\frac{\beta}{i}}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 4.1e+148)
   (+ 0.0625 (/ 0.015625 (* i i)))
   (/ (/ i beta) (/ beta i))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 4.1e+148) {
		tmp = 0.0625 + (0.015625 / (i * i));
	} else {
		tmp = (i / 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 <= 4.1d+148) then
        tmp = 0.0625d0 + (0.015625d0 / (i * i))
    else
        tmp = (i / 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 <= 4.1e+148) {
		tmp = 0.0625 + (0.015625 / (i * i));
	} else {
		tmp = (i / beta) / (beta / i);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 4.1e+148:
		tmp = 0.0625 + (0.015625 / (i * i))
	else:
		tmp = (i / beta) / (beta / i)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 4.1e+148)
		tmp = Float64(0.0625 + Float64(0.015625 / Float64(i * i)));
	else
		tmp = Float64(Float64(i / 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 <= 4.1e+148)
		tmp = 0.0625 + (0.015625 / (i * i));
	else
		tmp = (i / 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, 4.1e+148], N[(0.0625 + N[(0.015625 / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(i / beta), $MachinePrecision] / N[(beta / i), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.0999999999999998e148

   1. Initial program 22.4%

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

   3. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 40.6%

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

   5. Simplified 40.6%

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

   6. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), \frac{1}{4}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left({i}^{2}\right), 4\right), 1\right)\right) \]

      3. unpow2 N/A

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

      4. *-lowering-*.f64 38.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), \frac{1}{4}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), 4\right), 1\right)\right) \]

   8. Simplified 38.7%

      \[\leadsto \frac{\left(i \cdot i\right) \cdot 0.25}{\color{blue}{\left(i \cdot i\right) \cdot 4} - 1} \]

   9. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{1}{16} + \frac{1}{64} \cdot \frac{1}{{i}^{2}}} \]
   10. Step-by-step derivation

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

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{16}, \color{blue}{\left(\frac{1}{64} \cdot \frac{1}{{i}^{2}}\right)}\right) \]

       2. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{16}, \left(\frac{\frac{1}{64} \cdot 1}{\color{blue}{{i}^{2}}}\right)\right) \]

       3. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{16}, \left(\frac{\frac{1}{64}}{{\color{blue}{i}}^{2}}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{16}, \mathsf{/.f64}\left(\frac{1}{64}, \color{blue}{\left({i}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{16}, \mathsf{/.f64}\left(\frac{1}{64}, \left(i \cdot \color{blue}{i}\right)\right)\right) \]

       6. *-lowering-*.f64 82.6%

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{16}, \mathsf{/.f64}\left(\frac{1}{64}, \mathsf{*.f64}\left(i, \color{blue}{i}\right)\right)\right) \]

   11. Simplified 82.6%

       \[\leadsto \color{blue}{0.0625 + \frac{0.015625}{i \cdot i}} \]

   if 4.0999999999999998e148 < beta

   1. Initial program 0.1%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   2. 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 19.2%

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

   7. Simplified 19.2%

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

   8. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{{i}^{2}}{{\beta}^{2}}} \]
   9. Step-by-step derivation

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

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

      2. unpow2 N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 19.4%

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

   10. Simplified 19.4%

       \[\leadsto \color{blue}{\frac{i \cdot i}{\beta \cdot \beta}} \]
   11. Step-by-step derivation

       1. times-frac N/A

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

       2. clear-num N/A

          \[\leadsto \frac{i}{\beta} \cdot \frac{1}{\color{blue}{\frac{\beta}{i}}} \]

       3. un-div-inv N/A

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

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

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

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

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

       6. /-lowering-/.f64 60.8%

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

   12. Applied egg-rr 60.8%

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

Alternative 5: 83.4% accurate, 4.4× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.5 \cdot 10^{+148}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 4.5e+148)
   (+ 0.0625 (/ 0.015625 (* i i)))
   (* (/ i beta) (/ i beta))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 4.5e+148) {
		tmp = 0.0625 + (0.015625 / (i * i));
	} 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 <= 4.5d+148) then
        tmp = 0.0625d0 + (0.015625d0 / (i * i))
    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 <= 4.5e+148) {
		tmp = 0.0625 + (0.015625 / (i * i));
	} 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 <= 4.5e+148:
		tmp = 0.0625 + (0.015625 / (i * i))
	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 <= 4.5e+148)
		tmp = Float64(0.0625 + Float64(0.015625 / Float64(i * i)));
	else
		tmp = Float64(Float64(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 <= 4.5e+148)
		tmp = 0.0625 + (0.015625 / (i * i));
	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, 4.5e+148], N[(0.0625 + N[(0.015625 / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(i / beta), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.49999999999999994e148

   1. Initial program 22.4%

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

   3. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 40.6%

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

   5. Simplified 40.6%

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

   6. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), \frac{1}{4}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left({i}^{2}\right), 4\right), 1\right)\right) \]

      3. unpow2 N/A

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

      4. *-lowering-*.f64 38.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), \frac{1}{4}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), 4\right), 1\right)\right) \]

   8. Simplified 38.7%

      \[\leadsto \frac{\left(i \cdot i\right) \cdot 0.25}{\color{blue}{\left(i \cdot i\right) \cdot 4} - 1} \]

   9. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{1}{16} + \frac{1}{64} \cdot \frac{1}{{i}^{2}}} \]
   10. Step-by-step derivation

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

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{16}, \color{blue}{\left(\frac{1}{64} \cdot \frac{1}{{i}^{2}}\right)}\right) \]

       2. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{16}, \left(\frac{\frac{1}{64} \cdot 1}{\color{blue}{{i}^{2}}}\right)\right) \]

       3. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{16}, \left(\frac{\frac{1}{64}}{{\color{blue}{i}}^{2}}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{16}, \mathsf{/.f64}\left(\frac{1}{64}, \color{blue}{\left({i}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{16}, \mathsf{/.f64}\left(\frac{1}{64}, \left(i \cdot \color{blue}{i}\right)\right)\right) \]

       6. *-lowering-*.f64 82.6%

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{16}, \mathsf{/.f64}\left(\frac{1}{64}, \mathsf{*.f64}\left(i, \color{blue}{i}\right)\right)\right) \]

   11. Simplified 82.6%

       \[\leadsto \color{blue}{0.0625 + \frac{0.015625}{i \cdot i}} \]

   if 4.49999999999999994e148 < beta

   1. Initial program 0.1%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   2. 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 19.2%

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

   7. Simplified 19.2%

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

   8. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{{i}^{2}}{{\beta}^{2}}} \]
   9. Step-by-step derivation

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

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

      2. unpow2 N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 19.4%

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

   10. Simplified 19.4%

       \[\leadsto \color{blue}{\frac{i \cdot i}{\beta \cdot \beta}} \]
   11. Step-by-step derivation

       1. times-frac N/A

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

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

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

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

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

       4. /-lowering-/.f64 60.7%

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

   12. Applied egg-rr 60.7%

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

Alternative 6: 83.1% accurate, 4.4× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2 \cdot 10^{+148}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 2e+148) 0.0625 (* (/ i beta) (/ i beta))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2e+148) {
		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 <= 2d+148) 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 <= 2e+148) {
		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 <= 2e+148:
		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 <= 2e+148)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(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 <= 2e+148)
		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, 2e+148], 0.0625, N[(N[(i / beta), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.0000000000000001e148

   1. Initial program 22.4%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   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 19.0%

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

   5. Taylor expanded in i around inf

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

   7. Simplified 82.3%

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

   if 2.0000000000000001e148 < beta

   1. Initial program 0.1%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   2. 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 19.2%

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

   7. Simplified 19.2%

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

   8. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{{i}^{2}}{{\beta}^{2}}} \]
   9. Step-by-step derivation

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

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

      2. unpow2 N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 19.4%

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

   10. Simplified 19.4%

       \[\leadsto \color{blue}{\frac{i \cdot i}{\beta \cdot \beta}} \]
   11. Step-by-step derivation

       1. times-frac N/A

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

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

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

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

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

       4. /-lowering-/.f64 60.7%

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

   12. Applied egg-rr 60.7%

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

Alternative 7: 71.7% 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 18.1%

   \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
2. 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 15.3%

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

5. Taylor expanded in i around inf

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

7. Simplified 71.8%

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

Reproduce

herbie shell --seed 2024145 
(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)))