Average Error: 46.5 → 0.0
Time: 1.9s
Precision: 64
\[i \gt 0.0\]
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 631.9542587844484842207748442888259887695:\\ \;\;\;\;\frac{i \cdot i}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.00390625, \frac{1}{{i}^{4}}, \mathsf{fma}\left(0.015625, \frac{1}{{i}^{2}}, 0.0625\right)\right)\\ \end{array}\]
\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 631.9542587844484842207748442888259887695:\\
\;\;\;\;\frac{i \cdot i}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\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 r58277 = i;
        double r58278 = r58277 * r58277;
        double r58279 = r58278 * r58278;
        double r58280 = 2.0;
        double r58281 = r58280 * r58277;
        double r58282 = r58281 * r58281;
        double r58283 = r58279 / r58282;
        double r58284 = 1.0;
        double r58285 = r58282 - r58284;
        double r58286 = r58283 / r58285;
        return r58286;
}
↓
double f(double i) {
        double r58287 = i;
        double r58288 = 631.9542587844485;
        bool r58289 = r58287 <= r58288;
        double r58290 = r58287 * r58287;
        double r58291 = 2.0;
        double r58292 = r58291 * r58287;
        double r58293 = r58292 * r58292;
        double r58294 = 1.0;
        double r58295 = r58293 - r58294;
        double r58296 = r58291 * r58291;
        double r58297 = r58295 * r58296;
        double r58298 = r58290 / r58297;
        double r58299 = 0.00390625;
        double r58300 = 1.0;
        double r58301 = 4.0;
        double r58302 = pow(r58287, r58301);
        double r58303 = r58300 / r58302;
        double r58304 = 0.015625;
        double r58305 = 2.0;
        double r58306 = pow(r58287, r58305);
        double r58307 = r58300 / r58306;
        double r58308 = 0.0625;
        double r58309 = fma(r58304, r58307, r58308);
        double r58310 = fma(r58299, r58303, r58309);
        double r58311 = r58289 ? r58298 : r58310;
        return r58311;
}

Error

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if i < 631.9542587844485

    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

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

    if 631.9542587844485 < i

    1. Initial program 47.9

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

    2. Simplified32.4

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

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 631.9542587844484842207748442888259887695:\\ \;\;\;\;\frac{i \cdot i}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.00390625, \frac{1}{{i}^{4}}, \mathsf{fma}\left(0.015625, \frac{1}{{i}^{2}}, 0.0625\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019362 +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)))