Average Error: 46.8 → 0.0
Time: 2.3s
Precision: 64
\[i \gt 0.0\]
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 215.890116719305212:\\ \;\;\;\;\frac{i \cdot i}{\left(\mathsf{fma}\left(2, i, \sqrt{1}\right) \cdot \left(2 \cdot i - \sqrt{1}\right)\right) \cdot \left(2 \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{{\left(\sqrt{1}\right)}^{4}}{{i}^{4}}, 0.00390625, 0.0625 + \frac{0.015625 \cdot 1}{{i}^{2}}\right)\\ \end{array}\]
\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}
\mathbf{if}\;i \le 215.890116719305212:\\
\;\;\;\;\frac{i \cdot i}{\left(\mathsf{fma}\left(2, i, \sqrt{1}\right) \cdot \left(2 \cdot i - \sqrt{1}\right)\right) \cdot \left(2 \cdot 2\right)}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{{\left(\sqrt{1}\right)}^{4}}{{i}^{4}}, 0.00390625, 0.0625 + \frac{0.015625 \cdot 1}{{i}^{2}}\right)\\

\end{array}
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 temp;
	if ((i <= 215.8901167193052)) {
		temp = ((i * i) / ((fma(2.0, i, sqrt(1.0)) * ((2.0 * i) - sqrt(1.0))) * (2.0 * 2.0)));
	} else {
		temp = fma((pow(sqrt(1.0), 4.0) / pow(i, 4.0)), 0.00390625, (0.0625 + ((0.015625 * 1.0) / pow(i, 2.0))));
	}
	return temp;
}

Error

Bits error versus i

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if i < 215.8901167193052

    1. Initial program 45.2

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

    2. Simplified0.0

      \[\leadsto \color{blue}{\frac{i \cdot i}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)}}\]
    3. Using strategy rm

    4. Applied add-sqr-sqrt0.0

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

    5. Applied difference-of-squares0.0

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

    6. Simplified0.0

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

    if 215.8901167193052 < i

    1. Initial program 48.5

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

    2. Simplified32.0

      \[\leadsto \color{blue}{\frac{i \cdot i}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)}}\]
    3. Using strategy rm

    4. Applied add-sqr-sqrt32.0

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

    5. Applied difference-of-squares32.0

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

    6. Simplified32.0

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

    7. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{0.00390625 \cdot \frac{{\left(\sqrt{1}\right)}^{4}}{{i}^{4}} + \left(0.015625 \cdot \frac{{\left(\sqrt{1}\right)}^{2}}{{i}^{2}} + 0.0625\right)}\]

    8. Simplified0.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(\sqrt{1}\right)}^{4}}{{i}^{4}}, 0.00390625, 0.0625 + \frac{0.015625 \cdot 1}{{i}^{2}}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 215.890116719305212:\\ \;\;\;\;\frac{i \cdot i}{\left(\mathsf{fma}\left(2, i, \sqrt{1}\right) \cdot \left(2 \cdot i - \sqrt{1}\right)\right) \cdot \left(2 \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{{\left(\sqrt{1}\right)}^{4}}{{i}^{4}}, 0.00390625, 0.0625 + \frac{0.015625 \cdot 1}{{i}^{2}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 +o rules:numerics
(FPCore (i)
  :name "Octave 3.8, jcobi/4, as called"
  :precision binary64
  :pre (and (> i 0.0))
  (/ (/ (* (* i i) (* i i)) (* (* 2 i) (* 2 i))) (- (* (* 2 i) (* 2 i)) 1)))