Average Error: 46.8 → 0.0
Time: 1.9s
Precision: 64
\[i \gt 0.0\]
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 215.890116719305212:\\ \;\;\;\;\frac{i \cdot i}{\left(\mathsf{fma}\left(2, i, \sqrt{1}\right) \cdot \left(2 \cdot i - \sqrt{1}\right)\right) \cdot \left(2 \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.00390625, \frac{1}{{i}^{4}}, \mathsf{fma}\left(0.015625, \frac{1}{{i}^{2}}, 0.0625\right)\right)\\ \end{array}\]
\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(0.00390625, \frac{1}{{i}^{4}}, \mathsf{fma}\left(0.015625, \frac{1}{{i}^{2}}, 0.0625\right)\right)\\

\end{array}
double f(double i) {
        double r52348 = i;
        double r52349 = r52348 * r52348;
        double r52350 = r52349 * r52349;
        double r52351 = 2.0;
        double r52352 = r52351 * r52348;
        double r52353 = r52352 * r52352;
        double r52354 = r52350 / r52353;
        double r52355 = 1.0;
        double r52356 = r52353 - r52355;
        double r52357 = r52354 / r52356;
        return r52357;
}
↓
double f(double i) {
        double r52358 = i;
        double r52359 = 215.8901167193052;
        bool r52360 = r52358 <= r52359;
        double r52361 = r52358 * r52358;
        double r52362 = 2.0;
        double r52363 = 1.0;
        double r52364 = sqrt(r52363);
        double r52365 = fma(r52362, r52358, r52364);
        double r52366 = r52362 * r52358;
        double r52367 = r52366 - r52364;
        double r52368 = r52365 * r52367;
        double r52369 = r52362 * r52362;
        double r52370 = r52368 * r52369;
        double r52371 = r52361 / r52370;
        double r52372 = 0.00390625;
        double r52373 = 1.0;
        double r52374 = 4.0;
        double r52375 = pow(r52358, r52374);
        double r52376 = r52373 / r52375;
        double r52377 = 0.015625;
        double r52378 = 2.0;
        double r52379 = pow(r52358, r52378);
        double r52380 = r52373 / r52379;
        double r52381 = 0.0625;
        double r52382 = fma(r52377, r52380, r52381);
        double r52383 = fma(r52372, r52376, r52382);
        double r52384 = r52360 ? r52371 : r52383;
        return r52384;
}

Error

Bits error versus i

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. Taylor expanded around inf 0.0

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

    4. Simplified0.0

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

  4. Final simplification0.0

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

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)))