Average Error: 47.1 → 0.0
Time: 1.9s
Precision: 64
\[i \gt 0.0\]
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 200.9309000121655515158636262640357017517:\\ \;\;\;\;\frac{i}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \cdot \frac{i}{2 \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.00390625, \frac{1}{{i}^{4}}, \mathsf{fma}\left(0.015625, \frac{1}{{i}^{2}}, 0.0625\right)\right)\\ \end{array}\]
\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 200.9309000121655515158636262640357017517:\\
\;\;\;\;\frac{i}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \cdot \frac{i}{2 \cdot 2}\\

\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 r48559 = i;
        double r48560 = r48559 * r48559;
        double r48561 = r48560 * r48560;
        double r48562 = 2.0;
        double r48563 = r48562 * r48559;
        double r48564 = r48563 * r48563;
        double r48565 = r48561 / r48564;
        double r48566 = 1.0;
        double r48567 = r48564 - r48566;
        double r48568 = r48565 / r48567;
        return r48568;
}
↓
double f(double i) {
        double r48569 = i;
        double r48570 = 200.93090001216555;
        bool r48571 = r48569 <= r48570;
        double r48572 = 2.0;
        double r48573 = r48572 * r48569;
        double r48574 = r48573 * r48573;
        double r48575 = 1.0;
        double r48576 = r48574 - r48575;
        double r48577 = r48569 / r48576;
        double r48578 = r48572 * r48572;
        double r48579 = r48569 / r48578;
        double r48580 = r48577 * r48579;
        double r48581 = 0.00390625;
        double r48582 = 1.0;
        double r48583 = 4.0;
        double r48584 = pow(r48569, r48583);
        double r48585 = r48582 / r48584;
        double r48586 = 0.015625;
        double r48587 = 2.0;
        double r48588 = pow(r48569, r48587);
        double r48589 = r48582 / r48588;
        double r48590 = 0.0625;
        double r48591 = fma(r48586, r48589, r48590);
        double r48592 = fma(r48581, r48585, r48591);
        double r48593 = r48571 ? r48580 : r48592;
        return r48593;
}

Error

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if i < 200.93090001216555

    1. Initial program 45.3

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

    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 times-frac0.0

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

    if 200.93090001216555 < i

    1. Initial program 48.8

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

    2. Simplified33.4

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

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 200.9309000121655515158636262640357017517:\\ \;\;\;\;\frac{i}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \cdot \frac{i}{2 \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.00390625, \frac{1}{{i}^{4}}, \mathsf{fma}\left(0.015625, \frac{1}{{i}^{2}}, 0.0625\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 +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)))