Octave 3.8, jcobi/4, as called

Percentage Accurate: 27.6% → 99.6%

Time: 3.3s

Alternatives: 3

Speedup: 25.0×

Specification

\[i > 0\]

Math

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

FPCore

(FPCore (i)
 :precision binary64
 (let* ((t_0 (* (* 2.0 i) (* 2.0 i))))
   (/ (/ (* (* i i) (* i i)) t_0) (- t_0 1.0))))

C

double code(double i) {
	double t_0 = (2.0 * i) * (2.0 * i);
	return (((i * i) * (i * i)) / t_0) / (t_0 - 1.0);
}

Fortran

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

Java

public static double code(double i) {
	double t_0 = (2.0 * i) * (2.0 * i);
	return (((i * i) * (i * i)) / t_0) / (t_0 - 1.0);
}

Python

def code(i):
	t_0 = (2.0 * i) * (2.0 * i)
	return (((i * i) * (i * i)) / t_0) / (t_0 - 1.0)

Julia

function code(i)
	t_0 = Float64(Float64(2.0 * i) * Float64(2.0 * i))
	return Float64(Float64(Float64(Float64(i * i) * Float64(i * i)) / t_0) / Float64(t_0 - 1.0))
end

MATLAB

function tmp = code(i)
	t_0 = (2.0 * i) * (2.0 * i);
	tmp = (((i * i) * (i * i)) / t_0) / (t_0 - 1.0);
end

Wolfram

code[i_] := Block[{t$95$0 = N[(N[(2.0 * i), $MachinePrecision] * N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(i * i), $MachinePrecision] * N[(i * i), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 - 1.0), $MachinePrecision]), $MachinePrecision]]

Initial Program: 27.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (i)
 :precision binary64
 (let* ((t_0 (* (* 2.0 i) (* 2.0 i))))
   (/ (/ (* (* i i) (* i i)) t_0) (- t_0 1.0))))

C

double code(double i) {
	double t_0 = (2.0 * i) * (2.0 * i);
	return (((i * i) * (i * i)) / t_0) / (t_0 - 1.0);
}

Fortran

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

Java

public static double code(double i) {
	double t_0 = (2.0 * i) * (2.0 * i);
	return (((i * i) * (i * i)) / t_0) / (t_0 - 1.0);
}

Python

def code(i):
	t_0 = (2.0 * i) * (2.0 * i)
	return (((i * i) * (i * i)) / t_0) / (t_0 - 1.0)

Julia

function code(i)
	t_0 = Float64(Float64(2.0 * i) * Float64(2.0 * i))
	return Float64(Float64(Float64(Float64(i * i) * Float64(i * i)) / t_0) / Float64(t_0 - 1.0))
end

MATLAB

function tmp = code(i)
	t_0 = (2.0 * i) * (2.0 * i);
	tmp = (((i * i) * (i * i)) / t_0) / (t_0 - 1.0);
end

Wolfram

code[i_] := Block[{t$95$0 = N[(N[(2.0 * i), $MachinePrecision] * N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(i * i), $MachinePrecision] * N[(i * i), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 - 1.0), $MachinePrecision]), $MachinePrecision]]

Alternative 1: 99.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \frac{i}{i \cdot 16 + 4 \cdot \frac{-1}{i}} \end{array} \]

FPCore

(FPCore (i) :precision binary64 (/ i (+ (* i 16.0) (* 4.0 (/ -1.0 i)))))

C

double code(double i) {
	return i / ((i * 16.0) + (4.0 * (-1.0 / i)));
}

Fortran

real(8) function code(i)
    real(8), intent (in) :: i
    code = i / ((i * 16.0d0) + (4.0d0 * ((-1.0d0) / i)))
end function

Java

public static double code(double i) {
	return i / ((i * 16.0) + (4.0 * (-1.0 / i)));
}

Python

def code(i):
	return i / ((i * 16.0) + (4.0 * (-1.0 / i)))

Julia

function code(i)
	return Float64(i / Float64(Float64(i * 16.0) + Float64(4.0 * Float64(-1.0 / i))))
end

MATLAB

function tmp = code(i)
	tmp = i / ((i * 16.0) + (4.0 * (-1.0 / i)));
end

Wolfram

code[i_] := N[(i / N[(N[(i * 16.0), $MachinePrecision] + N[(4.0 * N[(-1.0 / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 26.3%

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

   1. sqr-neg 26.3%

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

   2. associate-*l* 26.2%

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

   3. swap-sqr 26.2%

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

   4. sqr-neg 26.2%

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

   5. times-frac 34.1%

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

   6. associate-/l* 50.8%

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

   7. *-inverses 76.2%

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

   8. /-rgt-identity 76.2%

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

   9. associate-/r/ 76.0%

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

   10. swap-sqr 76.0%

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

   11. sqr-neg 76.0%

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

   12. swap-sqr 76.0%

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

   13. associate-/l/ 76.7%

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

3. Simplified 76.7%

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

4. Taylor expanded in i around 0 99.5%

   \[\leadsto \frac{i}{\color{blue}{16 \cdot i - 4 \cdot \frac{1}{i}}} \]

5. Final simplification 99.5%

   \[\leadsto \frac{i}{i \cdot 16 + 4 \cdot \frac{-1}{i}} \]

Alternative 2: 99.0% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 0.5:\\ \;\;\;\;i \cdot \left(i \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \end{array} \]

FPCore

(FPCore (i) :precision binary64 (if (<= i 0.5) (* i (* i -0.25)) 0.0625))

C

double code(double i) {
	double tmp;
	if (i <= 0.5) {
		tmp = i * (i * -0.25);
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Fortran

real(8) function code(i)
    real(8), intent (in) :: i
    real(8) :: tmp
    if (i <= 0.5d0) then
        tmp = i * (i * (-0.25d0))
    else
        tmp = 0.0625d0
    end if
    code = tmp
end function

Java

public static double code(double i) {
	double tmp;
	if (i <= 0.5) {
		tmp = i * (i * -0.25);
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Python

def code(i):
	tmp = 0
	if i <= 0.5:
		tmp = i * (i * -0.25)
	else:
		tmp = 0.0625
	return tmp

Julia

function code(i)
	tmp = 0.0
	if (i <= 0.5)
		tmp = Float64(i * Float64(i * -0.25));
	else
		tmp = 0.0625;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i)
	tmp = 0.0;
	if (i <= 0.5)
		tmp = i * (i * -0.25);
	else
		tmp = 0.0625;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_] := If[LessEqual[i, 0.5], N[(i * N[(i * -0.25), $MachinePrecision]), $MachinePrecision], 0.0625]

Derivation

1. Split input into 2 regimes

2. if i < 0.5

   1. Initial program 28.9%

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

      1. sqr-neg 28.9%

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

      2. associate-*l* 28.9%

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

      3. swap-sqr 28.9%

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

      4. sqr-neg 28.9%

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

      5. times-frac 35.9%

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

      6. associate-/l* 52.2%

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

      7. *-inverses 100.0%

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

      8. /-rgt-identity 100.0%

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

      9. associate-/r/ 99.8%

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

      10. swap-sqr 99.8%

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

      11. sqr-neg 99.8%

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

      12. swap-sqr 99.8%

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

      13. associate-/l/ 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in i around 0 97.8%

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

      1. associate-/r/ 98.0%

         \[\leadsto \color{blue}{\frac{i}{-4} \cdot i} \]

      2. div-inv 98.0%

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

      3. metadata-eval 98.0%

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

   6. Applied egg-rr 98.0%

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

   if 0.5 < i

   1. Initial program 23.3%

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

      1. sqr-neg 23.3%

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

      2. associate-*l* 23.1%

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

      3. swap-sqr 23.1%

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

      4. sqr-neg 23.1%

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

      5. times-frac 32.1%

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

      6. associate-/l* 49.2%

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

      7. *-inverses 49.2%

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

      8. /-rgt-identity 49.2%

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

      9. associate-/r/ 49.0%

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

      10. swap-sqr 49.0%

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

      11. sqr-neg 49.0%

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

      12. swap-sqr 49.0%

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

      13. associate-/l/ 50.6%

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

   3. Simplified 50.6%

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

   4. Taylor expanded in i around inf 99.0%

      \[\leadsto \color{blue}{0.0625} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 0.5:\\ \;\;\;\;i \cdot \left(i \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

Alternative 3: 51.2% accurate, 25.0× speedup

Math

\[\begin{array}{l} \\ 0.0625 \end{array} \]

FPCore

(FPCore (i) :precision binary64 0.0625)

C

double code(double i) {
	return 0.0625;
}

Fortran

real(8) function code(i)
    real(8), intent (in) :: i
    code = 0.0625d0
end function

Java

public static double code(double i) {
	return 0.0625;
}

Python

def code(i):
	return 0.0625

Julia

function code(i)
	return 0.0625
end

MATLAB

function tmp = code(i)
	tmp = 0.0625;
end

Wolfram

code[i_] := 0.0625

Derivation

1. Initial program 26.3%

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

   1. sqr-neg 26.3%

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

   2. associate-*l* 26.2%

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

   3. swap-sqr 26.2%

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

   4. sqr-neg 26.2%

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

   5. times-frac 34.1%

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

   6. associate-/l* 50.8%

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

   7. *-inverses 76.2%

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

   8. /-rgt-identity 76.2%

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

   9. associate-/r/ 76.0%

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

   10. swap-sqr 76.0%

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

   11. sqr-neg 76.0%

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

   12. swap-sqr 76.0%

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

   13. associate-/l/ 76.7%

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

3. Simplified 76.7%

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

4. Taylor expanded in i around inf 47.8%

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

5. Final simplification 47.8%

   \[\leadsto 0.0625 \]

Reproduce

herbie shell --seed 2023334 
(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)))