Average Error: 46.3 → 0.0
Time: 21.1s
Precision: 64
\[i \gt 0\]
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 227.07557223901145:\\ \;\;\;\;\log_* (1 + (e^{\frac{i \cdot i}{\left(\left(i \cdot i\right) \cdot 4 - 1.0\right) \cdot 4}} - 1)^*)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{0.00390625}{i}}{i} + 0.015625}{i \cdot i} + \frac{1}{16}\\ \end{array}\]
\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 227.07557223901145:\\
\;\;\;\;\log_* (1 + (e^{\frac{i \cdot i}{\left(\left(i \cdot i\right) \cdot 4 - 1.0\right) \cdot 4}} - 1)^*)\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\frac{0.00390625}{i}}{i} + 0.015625}{i \cdot i} + \frac{1}{16}\\

\end{array}
double f(double i) {
        double r2489298 = i;
        double r2489299 = r2489298 * r2489298;
        double r2489300 = r2489299 * r2489299;
        double r2489301 = 2.0;
        double r2489302 = r2489301 * r2489298;
        double r2489303 = r2489302 * r2489302;
        double r2489304 = r2489300 / r2489303;
        double r2489305 = 1.0;
        double r2489306 = r2489303 - r2489305;
        double r2489307 = r2489304 / r2489306;
        return r2489307;
}
↓
double f(double i) {
        double r2489308 = i;
        double r2489309 = 227.07557223901145;
        bool r2489310 = r2489308 <= r2489309;
        double r2489311 = r2489308 * r2489308;
        double r2489312 = 4.0;
        double r2489313 = r2489311 * r2489312;
        double r2489314 = 1.0;
        double r2489315 = r2489313 - r2489314;
        double r2489316 = r2489315 * r2489312;
        double r2489317 = r2489311 / r2489316;
        double r2489318 = expm1(r2489317);
        double r2489319 = log1p(r2489318);
        double r2489320 = 0.00390625;
        double r2489321 = r2489320 / r2489308;
        double r2489322 = r2489321 / r2489308;
        double r2489323 = 0.015625;
        double r2489324 = r2489322 + r2489323;
        double r2489325 = r2489324 / r2489311;
        double r2489326 = 0.0625;
        double r2489327 = r2489325 + r2489326;
        double r2489328 = r2489310 ? r2489319 : r2489327;
        return r2489328;
}

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 < 227.07557223901145

    1. Initial program 45.6

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

    2. Simplified0.0

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

    4. Applied log1p-expm1-u0.0

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

    if 227.07557223901145 < 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. Simplified31.3

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

    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

      \[\leadsto \color{blue}{\frac{0.015625 + \frac{\frac{0.00390625}{i}}{i}}{i \cdot i} + \frac{1}{16}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 227.07557223901145:\\ \;\;\;\;\log_* (1 + (e^{\frac{i \cdot i}{\left(\left(i \cdot i\right) \cdot 4 - 1.0\right) \cdot 4}} - 1)^*)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{0.00390625}{i}}{i} + 0.015625}{i \cdot i} + \frac{1}{16}\\ \end{array}\]

Reproduce

herbie shell --seed 2019104 +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)))