Average Error: 45.4 → 0.0
Time: 12.0s
Precision: 64
\[i \gt 0\]
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 1645.4224050168862:\\ \;\;\;\;\frac{\left(\frac{1}{4} \cdot i\right) \cdot i}{4 \cdot \left(i \cdot i\right) - 1.0}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.015625, \frac{1}{i \cdot i}, \mathsf{fma}\left(\frac{\frac{1}{i \cdot i}}{i \cdot i}, 0.00390625, \frac{1}{16}\right)\right)\\ \end{array}\]
\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 1645.4224050168862:\\
\;\;\;\;\frac{\left(\frac{1}{4} \cdot i\right) \cdot i}{4 \cdot \left(i \cdot i\right) - 1.0}\\

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

\end{array}
double f(double i) {
        double r1119165 = i;
        double r1119166 = r1119165 * r1119165;
        double r1119167 = r1119166 * r1119166;
        double r1119168 = 2.0;
        double r1119169 = r1119168 * r1119165;
        double r1119170 = r1119169 * r1119169;
        double r1119171 = r1119167 / r1119170;
        double r1119172 = 1.0;
        double r1119173 = r1119170 - r1119172;
        double r1119174 = r1119171 / r1119173;
        return r1119174;
}
↓
double f(double i) {
        double r1119175 = i;
        double r1119176 = 1645.4224050168862;
        bool r1119177 = r1119175 <= r1119176;
        double r1119178 = 0.25;
        double r1119179 = r1119178 * r1119175;
        double r1119180 = r1119179 * r1119175;
        double r1119181 = 4.0;
        double r1119182 = r1119175 * r1119175;
        double r1119183 = r1119181 * r1119182;
        double r1119184 = 1.0;
        double r1119185 = r1119183 - r1119184;
        double r1119186 = r1119180 / r1119185;
        double r1119187 = 0.015625;
        double r1119188 = 1.0;
        double r1119189 = r1119188 / r1119182;
        double r1119190 = r1119189 / r1119182;
        double r1119191 = 0.00390625;
        double r1119192 = 0.0625;
        double r1119193 = fma(r1119190, r1119191, r1119192);
        double r1119194 = fma(r1119187, r1119189, r1119193);
        double r1119195 = r1119177 ? r1119186 : r1119194;
        return r1119195;
}

Error

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if i < 1645.4224050168862

    1. Initial program 43.9

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

    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 associate-*l*0.0

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

    if 1645.4224050168862 < i

    1. Initial program 46.9

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

    2. Simplified30.6

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

    3. Taylor expanded around inf 0.0

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

    4. Simplified0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 1645.4224050168862:\\ \;\;\;\;\frac{\left(\frac{1}{4} \cdot i\right) \cdot i}{4 \cdot \left(i \cdot i\right) - 1.0}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.015625, \frac{1}{i \cdot i}, \mathsf{fma}\left(\frac{\frac{1}{i \cdot i}}{i \cdot i}, 0.00390625, \frac{1}{16}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 +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)))