Octave 3.8, jcobi/4, as called

Average Error: 47.0 → 0.0

Time: 1.5s

Precision: binary64

Cost: 6788

\[i > 0\]

Math

\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

↓

\[\begin{array}{l} t_0 := \left(i \cdot i\right) \cdot 4\\ \mathbf{if}\;i \leq 2 \cdot 10^{-13}:\\ \;\;\;\;-0.25 \cdot {i}^{2}\\ \mathbf{elif}\;i \leq 200000:\\ \;\;\;\;\frac{i \cdot \left(i \cdot \left(i \cdot i\right)\right)}{\left(-1 + t_0\right) \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

FPCore

(FPCore (i)
 :precision binary64
 (/
  (/ (* (* i i) (* i i)) (* (* 2.0 i) (* 2.0 i)))
  (- (* (* 2.0 i) (* 2.0 i)) 1.0)))

↓

(FPCore (i)
 :precision binary64
 (let* ((t_0 (* (* i i) 4.0)))
   (if (<= i 2e-13)
     (* -0.25 (pow i 2.0))
     (if (<= i 200000.0)
       (/ (* i (* i (* i i))) (* (+ -1.0 t_0) t_0))
       0.0625))))

C

double code(double i) {
	return (((i * i) * (i * i)) / ((2.0 * i) * (2.0 * i))) / (((2.0 * i) * (2.0 * i)) - 1.0);
}

↓

double code(double i) {
	double t_0 = (i * i) * 4.0;
	double tmp;
	if (i <= 2e-13) {
		tmp = -0.25 * pow(i, 2.0);
	} else if (i <= 200000.0) {
		tmp = (i * (i * (i * i))) / ((-1.0 + t_0) * t_0);
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Fortran

real(8) function code(i)
    real(8), intent (in) :: i
    code = (((i * i) * (i * i)) / ((2.0d0 * i) * (2.0d0 * i))) / (((2.0d0 * i) * (2.0d0 * i)) - 1.0d0)
end function

↓

real(8) function code(i)
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (i * i) * 4.0d0
    if (i <= 2d-13) then
        tmp = (-0.25d0) * (i ** 2.0d0)
    else if (i <= 200000.0d0) then
        tmp = (i * (i * (i * i))) / (((-1.0d0) + t_0) * t_0)
    else
        tmp = 0.0625d0
    end if
    code = tmp
end function

Java

public static double code(double i) {
	return (((i * i) * (i * i)) / ((2.0 * i) * (2.0 * i))) / (((2.0 * i) * (2.0 * i)) - 1.0);
}

↓

public static double code(double i) {
	double t_0 = (i * i) * 4.0;
	double tmp;
	if (i <= 2e-13) {
		tmp = -0.25 * Math.pow(i, 2.0);
	} else if (i <= 200000.0) {
		tmp = (i * (i * (i * i))) / ((-1.0 + t_0) * t_0);
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Python

def code(i):
	return (((i * i) * (i * i)) / ((2.0 * i) * (2.0 * i))) / (((2.0 * i) * (2.0 * i)) - 1.0)

↓

def code(i):
	t_0 = (i * i) * 4.0
	tmp = 0
	if i <= 2e-13:
		tmp = -0.25 * math.pow(i, 2.0)
	elif i <= 200000.0:
		tmp = (i * (i * (i * i))) / ((-1.0 + t_0) * t_0)
	else:
		tmp = 0.0625
	return tmp

Julia

function code(i)
	return Float64(Float64(Float64(Float64(i * i) * Float64(i * i)) / Float64(Float64(2.0 * i) * Float64(2.0 * i))) / Float64(Float64(Float64(2.0 * i) * Float64(2.0 * i)) - 1.0))
end

↓

function code(i)
	t_0 = Float64(Float64(i * i) * 4.0)
	tmp = 0.0
	if (i <= 2e-13)
		tmp = Float64(-0.25 * (i ^ 2.0));
	elseif (i <= 200000.0)
		tmp = Float64(Float64(i * Float64(i * Float64(i * i))) / Float64(Float64(-1.0 + t_0) * t_0));
	else
		tmp = 0.0625;
	end
	return tmp
end

MATLAB

function tmp = code(i)
	tmp = (((i * i) * (i * i)) / ((2.0 * i) * (2.0 * i))) / (((2.0 * i) * (2.0 * i)) - 1.0);
end

↓

function tmp_2 = code(i)
	t_0 = (i * i) * 4.0;
	tmp = 0.0;
	if (i <= 2e-13)
		tmp = -0.25 * (i ^ 2.0);
	elseif (i <= 200000.0)
		tmp = (i * (i * (i * i))) / ((-1.0 + t_0) * t_0);
	else
		tmp = 0.0625;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_] := N[(N[(N[(N[(i * i), $MachinePrecision] * N[(i * i), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 * i), $MachinePrecision] * N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(2.0 * i), $MachinePrecision] * N[(2.0 * i), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]

↓

code[i_] := Block[{t$95$0 = N[(N[(i * i), $MachinePrecision] * 4.0), $MachinePrecision]}, If[LessEqual[i, 2e-13], N[(-0.25 * N[Power[i, 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 200000.0], N[(N[(i * N[(i * N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(-1.0 + t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 0.0625]]]

Derivation

1. Split input into 3 regimes

2. if i < 2.0000000000000001e-13

   1. Initial program 47.2

      \[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

   2. Simplified 47.3

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

      [Start] 47.2	\[ \frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]
      rational.json-simplify-38 [=>] 47.2	\[ \color{blue}{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)\right)}} \]
      rational.json-simplify-31 [=>] 47.3	\[ \frac{\color{blue}{i \cdot \left(\left(i \cdot i\right) \cdot i\right)}}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)\right)} \]
      rational.json-simplify-2 [=>] 47.3	\[ \frac{i \cdot \color{blue}{\left(i \cdot \left(i \cdot i\right)\right)}}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)\right)} \]
      rational.json-simplify-31 [=>] 47.3	\[ \frac{i \cdot \left(i \cdot \left(i \cdot i\right)\right)}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \color{blue}{\left(2 \cdot \left(\left(2 \cdot i\right) \cdot i\right)\right)}} \]
      rational.json-simplify-41 [=>] 47.3	\[ \frac{i \cdot \left(i \cdot \left(i \cdot i\right)\right)}{\color{blue}{\left(\left(-1\right) + \left(2 \cdot i\right) \cdot \left(2 \cdot i\right)\right)} \cdot \left(2 \cdot \left(\left(2 \cdot i\right) \cdot i\right)\right)} \]
      metadata-eval [=>] 47.3	\[ \frac{i \cdot \left(i \cdot \left(i \cdot i\right)\right)}{\left(\color{blue}{-1} + \left(2 \cdot i\right) \cdot \left(2 \cdot i\right)\right) \cdot \left(2 \cdot \left(\left(2 \cdot i\right) \cdot i\right)\right)} \]
      rational.json-simplify-31 [=>] 47.3	\[ \frac{i \cdot \left(i \cdot \left(i \cdot i\right)\right)}{\left(-1 + \color{blue}{2 \cdot \left(\left(2 \cdot i\right) \cdot i\right)}\right) \cdot \left(2 \cdot \left(\left(2 \cdot i\right) \cdot i\right)\right)} \]
      rational.json-simplify-2 [=>] 47.3	\[ \frac{i \cdot \left(i \cdot \left(i \cdot i\right)\right)}{\left(-1 + 2 \cdot \color{blue}{\left(i \cdot \left(2 \cdot i\right)\right)}\right) \cdot \left(2 \cdot \left(\left(2 \cdot i\right) \cdot i\right)\right)} \]
      rational.json-simplify-31 [=>] 47.3	\[ \frac{i \cdot \left(i \cdot \left(i \cdot i\right)\right)}{\left(-1 + 2 \cdot \color{blue}{\left(2 \cdot \left(i \cdot i\right)\right)}\right) \cdot \left(2 \cdot \left(\left(2 \cdot i\right) \cdot i\right)\right)} \]
      rational.json-simplify-2 [=>] 47.3	\[ \frac{i \cdot \left(i \cdot \left(i \cdot i\right)\right)}{\left(-1 + 2 \cdot \color{blue}{\left(\left(i \cdot i\right) \cdot 2\right)}\right) \cdot \left(2 \cdot \left(\left(2 \cdot i\right) \cdot i\right)\right)} \]
      rational.json-simplify-31 [=>] 47.3	\[ \frac{i \cdot \left(i \cdot \left(i \cdot i\right)\right)}{\left(-1 + \color{blue}{\left(i \cdot i\right) \cdot \left(2 \cdot 2\right)}\right) \cdot \left(2 \cdot \left(\left(2 \cdot i\right) \cdot i\right)\right)} \]
      metadata-eval [=>] 47.3	\[ \frac{i \cdot \left(i \cdot \left(i \cdot i\right)\right)}{\left(-1 + \left(i \cdot i\right) \cdot \color{blue}{4}\right) \cdot \left(2 \cdot \left(\left(2 \cdot i\right) \cdot i\right)\right)} \]
      rational.json-simplify-2 [=>] 47.3	\[ \frac{i \cdot \left(i \cdot \left(i \cdot i\right)\right)}{\left(-1 + \left(i \cdot i\right) \cdot 4\right) \cdot \left(2 \cdot \color{blue}{\left(i \cdot \left(2 \cdot i\right)\right)}\right)} \]
      rational.json-simplify-31 [=>] 47.3	\[ \frac{i \cdot \left(i \cdot \left(i \cdot i\right)\right)}{\left(-1 + \left(i \cdot i\right) \cdot 4\right) \cdot \left(2 \cdot \color{blue}{\left(2 \cdot \left(i \cdot i\right)\right)}\right)} \]
      rational.json-simplify-2 [=>] 47.3	\[ \frac{i \cdot \left(i \cdot \left(i \cdot i\right)\right)}{\left(-1 + \left(i \cdot i\right) \cdot 4\right) \cdot \left(2 \cdot \color{blue}{\left(\left(i \cdot i\right) \cdot 2\right)}\right)} \]
      rational.json-simplify-31 [=>] 47.3	\[ \frac{i \cdot \left(i \cdot \left(i \cdot i\right)\right)}{\left(-1 + \left(i \cdot i\right) \cdot 4\right) \cdot \color{blue}{\left(\left(i \cdot i\right) \cdot \left(2 \cdot 2\right)\right)}} \]
      metadata-eval [=>] 47.3	\[ \frac{i \cdot \left(i \cdot \left(i \cdot i\right)\right)}{\left(-1 + \left(i \cdot i\right) \cdot 4\right) \cdot \left(\left(i \cdot i\right) \cdot \color{blue}{4}\right)} \]

   3. Taylor expanded in i around 0 0.0

      \[\leadsto \color{blue}{-0.25 \cdot {i}^{2}} \]

   if 2.0000000000000001e-13 < i < 2e5

   1. Initial program 0.2

      \[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

   2. Simplified 0.3

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

      [Start] 0.2	\[ \frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]
      rational.json-simplify-38 [=>] 0.3	\[ \color{blue}{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)\right)}} \]
      rational.json-simplify-31 [=>] 0.3	\[ \frac{\color{blue}{i \cdot \left(\left(i \cdot i\right) \cdot i\right)}}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)\right)} \]
      rational.json-simplify-2 [=>] 0.3	\[ \frac{i \cdot \color{blue}{\left(i \cdot \left(i \cdot i\right)\right)}}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)\right)} \]
      rational.json-simplify-31 [=>] 0.3	\[ \frac{i \cdot \left(i \cdot \left(i \cdot i\right)\right)}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \color{blue}{\left(2 \cdot \left(\left(2 \cdot i\right) \cdot i\right)\right)}} \]
      rational.json-simplify-41 [=>] 0.3	\[ \frac{i \cdot \left(i \cdot \left(i \cdot i\right)\right)}{\color{blue}{\left(\left(-1\right) + \left(2 \cdot i\right) \cdot \left(2 \cdot i\right)\right)} \cdot \left(2 \cdot \left(\left(2 \cdot i\right) \cdot i\right)\right)} \]
      metadata-eval [=>] 0.3	\[ \frac{i \cdot \left(i \cdot \left(i \cdot i\right)\right)}{\left(\color{blue}{-1} + \left(2 \cdot i\right) \cdot \left(2 \cdot i\right)\right) \cdot \left(2 \cdot \left(\left(2 \cdot i\right) \cdot i\right)\right)} \]
      rational.json-simplify-31 [=>] 0.3	\[ \frac{i \cdot \left(i \cdot \left(i \cdot i\right)\right)}{\left(-1 + \color{blue}{2 \cdot \left(\left(2 \cdot i\right) \cdot i\right)}\right) \cdot \left(2 \cdot \left(\left(2 \cdot i\right) \cdot i\right)\right)} \]
      rational.json-simplify-2 [=>] 0.3	\[ \frac{i \cdot \left(i \cdot \left(i \cdot i\right)\right)}{\left(-1 + 2 \cdot \color{blue}{\left(i \cdot \left(2 \cdot i\right)\right)}\right) \cdot \left(2 \cdot \left(\left(2 \cdot i\right) \cdot i\right)\right)} \]
      rational.json-simplify-31 [=>] 0.3	\[ \frac{i \cdot \left(i \cdot \left(i \cdot i\right)\right)}{\left(-1 + 2 \cdot \color{blue}{\left(2 \cdot \left(i \cdot i\right)\right)}\right) \cdot \left(2 \cdot \left(\left(2 \cdot i\right) \cdot i\right)\right)} \]
      rational.json-simplify-2 [=>] 0.3	\[ \frac{i \cdot \left(i \cdot \left(i \cdot i\right)\right)}{\left(-1 + 2 \cdot \color{blue}{\left(\left(i \cdot i\right) \cdot 2\right)}\right) \cdot \left(2 \cdot \left(\left(2 \cdot i\right) \cdot i\right)\right)} \]
      rational.json-simplify-31 [=>] 0.3	\[ \frac{i \cdot \left(i \cdot \left(i \cdot i\right)\right)}{\left(-1 + \color{blue}{\left(i \cdot i\right) \cdot \left(2 \cdot 2\right)}\right) \cdot \left(2 \cdot \left(\left(2 \cdot i\right) \cdot i\right)\right)} \]
      metadata-eval [=>] 0.3	\[ \frac{i \cdot \left(i \cdot \left(i \cdot i\right)\right)}{\left(-1 + \left(i \cdot i\right) \cdot \color{blue}{4}\right) \cdot \left(2 \cdot \left(\left(2 \cdot i\right) \cdot i\right)\right)} \]
      rational.json-simplify-2 [=>] 0.3	\[ \frac{i \cdot \left(i \cdot \left(i \cdot i\right)\right)}{\left(-1 + \left(i \cdot i\right) \cdot 4\right) \cdot \left(2 \cdot \color{blue}{\left(i \cdot \left(2 \cdot i\right)\right)}\right)} \]
      rational.json-simplify-31 [=>] 0.3	\[ \frac{i \cdot \left(i \cdot \left(i \cdot i\right)\right)}{\left(-1 + \left(i \cdot i\right) \cdot 4\right) \cdot \left(2 \cdot \color{blue}{\left(2 \cdot \left(i \cdot i\right)\right)}\right)} \]
      rational.json-simplify-2 [=>] 0.3	\[ \frac{i \cdot \left(i \cdot \left(i \cdot i\right)\right)}{\left(-1 + \left(i \cdot i\right) \cdot 4\right) \cdot \left(2 \cdot \color{blue}{\left(\left(i \cdot i\right) \cdot 2\right)}\right)} \]
      rational.json-simplify-31 [=>] 0.3	\[ \frac{i \cdot \left(i \cdot \left(i \cdot i\right)\right)}{\left(-1 + \left(i \cdot i\right) \cdot 4\right) \cdot \color{blue}{\left(\left(i \cdot i\right) \cdot \left(2 \cdot 2\right)\right)}} \]
      metadata-eval [=>] 0.3	\[ \frac{i \cdot \left(i \cdot \left(i \cdot i\right)\right)}{\left(-1 + \left(i \cdot i\right) \cdot 4\right) \cdot \left(\left(i \cdot i\right) \cdot \color{blue}{4}\right)} \]

   if 2e5 < i

   1. Initial program 49.6

      \[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

   2. Simplified 49.8

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

      [Start] 49.6	\[ \frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]
      rational.json-simplify-38 [=>] 49.7	\[ \color{blue}{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)\right)}} \]
      rational.json-simplify-31 [=>] 49.8	\[ \frac{\color{blue}{i \cdot \left(\left(i \cdot i\right) \cdot i\right)}}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)\right)} \]
      rational.json-simplify-2 [=>] 49.8	\[ \frac{i \cdot \color{blue}{\left(i \cdot \left(i \cdot i\right)\right)}}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)\right)} \]
      rational.json-simplify-31 [=>] 49.8	\[ \frac{i \cdot \left(i \cdot \left(i \cdot i\right)\right)}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \color{blue}{\left(2 \cdot \left(\left(2 \cdot i\right) \cdot i\right)\right)}} \]
      rational.json-simplify-41 [=>] 49.8	\[ \frac{i \cdot \left(i \cdot \left(i \cdot i\right)\right)}{\color{blue}{\left(\left(-1\right) + \left(2 \cdot i\right) \cdot \left(2 \cdot i\right)\right)} \cdot \left(2 \cdot \left(\left(2 \cdot i\right) \cdot i\right)\right)} \]
      metadata-eval [=>] 49.8	\[ \frac{i \cdot \left(i \cdot \left(i \cdot i\right)\right)}{\left(\color{blue}{-1} + \left(2 \cdot i\right) \cdot \left(2 \cdot i\right)\right) \cdot \left(2 \cdot \left(\left(2 \cdot i\right) \cdot i\right)\right)} \]
      rational.json-simplify-31 [=>] 49.8	\[ \frac{i \cdot \left(i \cdot \left(i \cdot i\right)\right)}{\left(-1 + \color{blue}{2 \cdot \left(\left(2 \cdot i\right) \cdot i\right)}\right) \cdot \left(2 \cdot \left(\left(2 \cdot i\right) \cdot i\right)\right)} \]
      rational.json-simplify-2 [=>] 49.8	\[ \frac{i \cdot \left(i \cdot \left(i \cdot i\right)\right)}{\left(-1 + 2 \cdot \color{blue}{\left(i \cdot \left(2 \cdot i\right)\right)}\right) \cdot \left(2 \cdot \left(\left(2 \cdot i\right) \cdot i\right)\right)} \]
      rational.json-simplify-31 [=>] 49.8	\[ \frac{i \cdot \left(i \cdot \left(i \cdot i\right)\right)}{\left(-1 + 2 \cdot \color{blue}{\left(2 \cdot \left(i \cdot i\right)\right)}\right) \cdot \left(2 \cdot \left(\left(2 \cdot i\right) \cdot i\right)\right)} \]
      rational.json-simplify-2 [=>] 49.8	\[ \frac{i \cdot \left(i \cdot \left(i \cdot i\right)\right)}{\left(-1 + 2 \cdot \color{blue}{\left(\left(i \cdot i\right) \cdot 2\right)}\right) \cdot \left(2 \cdot \left(\left(2 \cdot i\right) \cdot i\right)\right)} \]
      rational.json-simplify-31 [=>] 49.8	\[ \frac{i \cdot \left(i \cdot \left(i \cdot i\right)\right)}{\left(-1 + \color{blue}{\left(i \cdot i\right) \cdot \left(2 \cdot 2\right)}\right) \cdot \left(2 \cdot \left(\left(2 \cdot i\right) \cdot i\right)\right)} \]
      metadata-eval [=>] 49.8	\[ \frac{i \cdot \left(i \cdot \left(i \cdot i\right)\right)}{\left(-1 + \left(i \cdot i\right) \cdot \color{blue}{4}\right) \cdot \left(2 \cdot \left(\left(2 \cdot i\right) \cdot i\right)\right)} \]
      rational.json-simplify-2 [=>] 49.8	\[ \frac{i \cdot \left(i \cdot \left(i \cdot i\right)\right)}{\left(-1 + \left(i \cdot i\right) \cdot 4\right) \cdot \left(2 \cdot \color{blue}{\left(i \cdot \left(2 \cdot i\right)\right)}\right)} \]
      rational.json-simplify-31 [=>] 49.8	\[ \frac{i \cdot \left(i \cdot \left(i \cdot i\right)\right)}{\left(-1 + \left(i \cdot i\right) \cdot 4\right) \cdot \left(2 \cdot \color{blue}{\left(2 \cdot \left(i \cdot i\right)\right)}\right)} \]
      rational.json-simplify-2 [=>] 49.8	\[ \frac{i \cdot \left(i \cdot \left(i \cdot i\right)\right)}{\left(-1 + \left(i \cdot i\right) \cdot 4\right) \cdot \left(2 \cdot \color{blue}{\left(\left(i \cdot i\right) \cdot 2\right)}\right)} \]
      rational.json-simplify-31 [=>] 49.8	\[ \frac{i \cdot \left(i \cdot \left(i \cdot i\right)\right)}{\left(-1 + \left(i \cdot i\right) \cdot 4\right) \cdot \color{blue}{\left(\left(i \cdot i\right) \cdot \left(2 \cdot 2\right)\right)}} \]
      metadata-eval [=>] 49.8	\[ \frac{i \cdot \left(i \cdot \left(i \cdot i\right)\right)}{\left(-1 + \left(i \cdot i\right) \cdot 4\right) \cdot \left(\left(i \cdot i\right) \cdot \color{blue}{4}\right)} \]

   3. Taylor expanded in i around inf 0.0

      \[\leadsto \color{blue}{0.0625} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 2 \cdot 10^{-13}:\\ \;\;\;\;-0.25 \cdot {i}^{2}\\ \mathbf{elif}\;i \leq 200000:\\ \;\;\;\;\frac{i \cdot \left(i \cdot \left(i \cdot i\right)\right)}{\left(-1 + \left(i \cdot i\right) \cdot 4\right) \cdot \left(\left(i \cdot i\right) \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

Alternatives

Alternative 1
Error	22.3
Cost	3012

\[\begin{array}{l} t_0 := \left(i \cdot i\right) \cdot \left(i \cdot i\right)\\ t_1 := \left(2 \cdot i\right) \cdot \left(2 \cdot i\right)\\ \mathbf{if}\;\frac{\frac{t_0}{t_1}}{t_1 - 1} \leq 0.1:\\ \;\;\;\;\frac{t_0}{\left(-1 - \left(i \cdot i\right) \cdot -4\right) \cdot \left(i \cdot \left(i \cdot 4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

Alternative 2
Error	22.8
Cost	1476

\[\begin{array}{l} t_0 := \left(i \cdot i\right) \cdot 4\\ \mathbf{if}\;i \leq 200000:\\ \;\;\;\;\frac{i \cdot \left(i \cdot \left(i \cdot i\right)\right)}{\left(-1 + t_0\right) \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

Alternative 3
Error	31.5
Cost	64

\[0.0625 \]

Reproduce

herbie shell --seed 2023053 
(FPCore (i)
  :name "Octave 3.8, jcobi/4, as called"
  :precision binary64
  :pre (> i 0.0)
  (/ (/ (* (* i i) (* i i)) (* (* 2.0 i) (* 2.0 i))) (- (* (* 2.0 i) (* 2.0 i)) 1.0)))