Average Error: 46.8 → 0.0
Time: 4.3s
Precision: 64
\[i \gt 0.0\]
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 1923.1709384006522:\\ \;\;\;\;\frac{i \cdot i}{\left(\mathsf{fma}\left(2, i, \sqrt{1}\right) \cdot \left(2 \cdot i - \sqrt{1}\right)\right) \cdot \left(2 \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.00390625, \frac{1}{{i}^{4}}, \mathsf{fma}\left(0.015625, \frac{1}{{i}^{2}}, 0.0625\right)\right)\\ \end{array}\]
\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 1923.1709384006522:\\
\;\;\;\;\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 r308 = i;
        double r309 = r308 * r308;
        double r310 = r309 * r309;
        double r311 = 2.0;
        double r312 = r311 * r308;
        double r313 = r312 * r312;
        double r314 = r310 / r313;
        double r315 = 1.0;
        double r316 = r313 - r315;
        double r317 = r314 / r316;
        return r317;
}
↓
double f(double i) {
        double r318 = i;
        double r319 = 1923.1709384006522;
        bool r320 = r318 <= r319;
        double r321 = r318 * r318;
        double r322 = 2.0;
        double r323 = 1.0;
        double r324 = sqrt(r323);
        double r325 = fma(r322, r318, r324);
        double r326 = r322 * r318;
        double r327 = r326 - r324;
        double r328 = r325 * r327;
        double r329 = r322 * r322;
        double r330 = r328 * r329;
        double r331 = r321 / r330;
        double r332 = 0.00390625;
        double r333 = 1.0;
        double r334 = 4.0;
        double r335 = pow(r318, r334);
        double r336 = r333 / r335;
        double r337 = 0.015625;
        double r338 = 2.0;
        double r339 = pow(r318, r338);
        double r340 = r333 / r339;
        double r341 = 0.0625;
        double r342 = fma(r337, r340, r341);
        double r343 = fma(r332, r336, r342);
        double r344 = r320 ? r331 : r343;
        return r344;
}

Error

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if i < 1923.1709384006522

    1. Initial program 44.9

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

    2. 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 1923.1709384006522 < i

    1. Initial program 48.9

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

    2. Simplified32.2

      \[\leadsto \color{blue}{\frac{i \cdot i}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)}}\]
    3. Using strategy rm

    4. Applied add-sqr-sqrt32.2

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

    5. Applied difference-of-squares32.2

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

    6. Simplified32.2

      \[\leadsto \frac{i \cdot i}{\left(\color{blue}{\mathsf{fma}\left(2, i, \sqrt{1}\right)} \cdot \left(2 \cdot i - \sqrt{1}\right)\right) \cdot \left(2 \cdot 2\right)}\]
    7. Using strategy rm

    8. Applied add-cube-cbrt32.2

      \[\leadsto \frac{i \cdot i}{\left(\mathsf{fma}\left(2, i, \sqrt{1}\right) \cdot \left(2 \cdot i - \color{blue}{\left(\sqrt[3]{\sqrt{1}} \cdot \sqrt[3]{\sqrt{1}}\right) \cdot \sqrt[3]{\sqrt{1}}}\right)\right) \cdot \left(2 \cdot 2\right)}\]

    9. Applied prod-diff32.2

      \[\leadsto \frac{i \cdot i}{\left(\mathsf{fma}\left(2, i, \sqrt{1}\right) \cdot \color{blue}{\left(\mathsf{fma}\left(2, i, -\sqrt[3]{\sqrt{1}} \cdot \left(\sqrt[3]{\sqrt{1}} \cdot \sqrt[3]{\sqrt{1}}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{\sqrt{1}}, \sqrt[3]{\sqrt{1}} \cdot \sqrt[3]{\sqrt{1}}, \sqrt[3]{\sqrt{1}} \cdot \left(\sqrt[3]{\sqrt{1}} \cdot \sqrt[3]{\sqrt{1}}\right)\right)\right)}\right) \cdot \left(2 \cdot 2\right)}\]

    10. Applied distribute-lft-in32.2

      \[\leadsto \frac{i \cdot i}{\color{blue}{\left(\mathsf{fma}\left(2, i, \sqrt{1}\right) \cdot \mathsf{fma}\left(2, i, -\sqrt[3]{\sqrt{1}} \cdot \left(\sqrt[3]{\sqrt{1}} \cdot \sqrt[3]{\sqrt{1}}\right)\right) + \mathsf{fma}\left(2, i, \sqrt{1}\right) \cdot \mathsf{fma}\left(-\sqrt[3]{\sqrt{1}}, \sqrt[3]{\sqrt{1}} \cdot \sqrt[3]{\sqrt{1}}, \sqrt[3]{\sqrt{1}} \cdot \left(\sqrt[3]{\sqrt{1}} \cdot \sqrt[3]{\sqrt{1}}\right)\right)\right)} \cdot \left(2 \cdot 2\right)}\]

    11. Simplified32.2

      \[\leadsto \frac{i \cdot i}{\left(\color{blue}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right)} + \mathsf{fma}\left(2, i, \sqrt{1}\right) \cdot \mathsf{fma}\left(-\sqrt[3]{\sqrt{1}}, \sqrt[3]{\sqrt{1}} \cdot \sqrt[3]{\sqrt{1}}, \sqrt[3]{\sqrt{1}} \cdot \left(\sqrt[3]{\sqrt{1}} \cdot \sqrt[3]{\sqrt{1}}\right)\right)\right) \cdot \left(2 \cdot 2\right)}\]

    12. Simplified32.2

      \[\leadsto \frac{i \cdot i}{\left(\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) + \color{blue}{\mathsf{fma}\left(2, i, \sqrt{1}\right) \cdot \mathsf{fma}\left(-\sqrt{1}, 1, \sqrt{1} \cdot 1\right)}\right) \cdot \left(2 \cdot 2\right)}\]

    13. Taylor expanded around inf 0.0

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

    14. 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

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 1923.1709384006522:\\ \;\;\;\;\frac{i \cdot i}{\left(\mathsf{fma}\left(2, i, \sqrt{1}\right) \cdot \left(2 \cdot i - \sqrt{1}\right)\right) \cdot \left(2 \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.00390625, \frac{1}{{i}^{4}}, \mathsf{fma}\left(0.015625, \frac{1}{{i}^{2}}, 0.0625\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 +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)))