Average Error: 45.8 → 0.0
Time: 9.5s
Precision: 64
\[i \gt 0\]
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 13.204777387070594:\\ \;\;\;\;\frac{\left(i \cdot i\right) \cdot \frac{1}{4}}{\log \left(e^{\left(i \cdot i\right) \cdot 4}\right) - 1.0}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.015625, \frac{1}{i \cdot i}, \mathsf{fma}\left(0.00390625, \frac{\frac{1}{i \cdot i}}{i \cdot i}, \frac{1}{16}\right)\right)\\ \end{array}\]
\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) - 1.0}
↓
\begin{array}{l}
\mathbf{if}\;i \le 13.204777387070594:\\
\;\;\;\;\frac{\left(i \cdot i\right) \cdot \frac{1}{4}}{\log \left(e^{\left(i \cdot i\right) \cdot 4}\right) - 1.0}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(0.015625, \frac{1}{i \cdot i}, \mathsf{fma}\left(0.00390625, \frac{\frac{1}{i \cdot i}}{i \cdot i}, \frac{1}{16}\right)\right)\\

\end{array}
double f(double i) {
        double r1027316 = i;
        double r1027317 = r1027316 * r1027316;
        double r1027318 = r1027317 * r1027317;
        double r1027319 = 2.0;
        double r1027320 = r1027319 * r1027316;
        double r1027321 = r1027320 * r1027320;
        double r1027322 = r1027318 / r1027321;
        double r1027323 = 1.0;
        double r1027324 = r1027321 - r1027323;
        double r1027325 = r1027322 / r1027324;
        return r1027325;
}
↓
double f(double i) {
        double r1027326 = i;
        double r1027327 = 13.204777387070594;
        bool r1027328 = r1027326 <= r1027327;
        double r1027329 = r1027326 * r1027326;
        double r1027330 = 0.25;
        double r1027331 = r1027329 * r1027330;
        double r1027332 = 4.0;
        double r1027333 = r1027329 * r1027332;
        double r1027334 = exp(r1027333);
        double r1027335 = log(r1027334);
        double r1027336 = 1.0;
        double r1027337 = r1027335 - r1027336;
        double r1027338 = r1027331 / r1027337;
        double r1027339 = 0.015625;
        double r1027340 = 1.0;
        double r1027341 = r1027340 / r1027329;
        double r1027342 = 0.00390625;
        double r1027343 = r1027341 / r1027329;
        double r1027344 = 0.0625;
        double r1027345 = fma(r1027342, r1027343, r1027344);
        double r1027346 = fma(r1027339, r1027341, r1027345);
        double r1027347 = r1027328 ? r1027338 : r1027346;
        return r1027347;
}

Error

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if i < 13.204777387070594

    1. Initial program 44.7

      \[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]

    2. Simplified0.0

      \[\leadsto \color{blue}{\frac{\left(i \cdot i\right) \cdot \frac{1}{4}}{\left(i \cdot i\right) \cdot 4 - 1.0}}\]
    3. Using strategy rm

    4. Applied add-log-exp0.0

      \[\leadsto \frac{\left(i \cdot i\right) \cdot \frac{1}{4}}{\color{blue}{\log \left(e^{\left(i \cdot i\right) \cdot 4}\right)} - 1.0}\]

    if 13.204777387070594 < i

    1. Initial program 47.0

      \[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]

    2. Simplified31.5

      \[\leadsto \color{blue}{\frac{\left(i \cdot i\right) \cdot \frac{1}{4}}{\left(i \cdot i\right) \cdot 4 - 1.0}}\]
    3. Using strategy rm

    4. Applied add-log-exp62.0

      \[\leadsto \frac{\left(i \cdot i\right) \cdot \frac{1}{4}}{\color{blue}{\log \left(e^{\left(i \cdot i\right) \cdot 4}\right)} - 1.0}\]

    5. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{0.015625 \cdot \frac{1}{{i}^{2}} + \left(\frac{1}{16} + 0.00390625 \cdot \frac{1}{{i}^{4}}\right)}\]

    6. Simplified0.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.015625, \frac{1}{i \cdot i}, \mathsf{fma}\left(0.00390625, \frac{\frac{1}{i \cdot i}}{i \cdot i}, \frac{1}{16}\right)\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 13.204777387070594:\\ \;\;\;\;\frac{\left(i \cdot i\right) \cdot \frac{1}{4}}{\log \left(e^{\left(i \cdot i\right) \cdot 4}\right) - 1.0}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.015625, \frac{1}{i \cdot i}, \mathsf{fma}\left(0.00390625, \frac{\frac{1}{i \cdot i}}{i \cdot i}, \frac{1}{16}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 +o rules:numerics
(FPCore (i)
  :name "Octave 3.8, jcobi/4, as called"
  :pre (and (> i 0))
  (/ (/ (* (* i i) (* i i)) (* (* 2 i) (* 2 i))) (- (* (* 2 i) (* 2 i)) 1.0)))