Average Error: 45.8 → 0.0
Time: 15.5s
Precision: 64
\[i \gt 0\]
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 3.714521008652235 \cdot 10^{-06}:\\ \;\;\;\;-\mathsf{fma}\left(i \cdot i, 0.25, \mathsf{fma}\left(4.0, \left(i \cdot i\right) \cdot \left(\left(i \cdot i\right) \cdot \left(i \cdot i\right)\right), \left(\left(i \cdot i\right) \cdot \left(i \cdot i\right)\right) \cdot 1.0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(4 + \frac{1.0}{i \cdot i}\right) \cdot \frac{\frac{1}{4}}{16 - \frac{1.0}{i \cdot i} \cdot \frac{1.0}{i \cdot i}}\\ \end{array}\]
\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 3.714521008652235 \cdot 10^{-06}:\\
\;\;\;\;-\mathsf{fma}\left(i \cdot i, 0.25, \mathsf{fma}\left(4.0, \left(i \cdot i\right) \cdot \left(\left(i \cdot i\right) \cdot \left(i \cdot i\right)\right), \left(\left(i \cdot i\right) \cdot \left(i \cdot i\right)\right) \cdot 1.0\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\left(4 + \frac{1.0}{i \cdot i}\right) \cdot \frac{\frac{1}{4}}{16 - \frac{1.0}{i \cdot i} \cdot \frac{1.0}{i \cdot i}}\\

\end{array}
double f(double i) {
        double r2265119 = i;
        double r2265120 = r2265119 * r2265119;
        double r2265121 = r2265120 * r2265120;
        double r2265122 = 2.0;
        double r2265123 = r2265122 * r2265119;
        double r2265124 = r2265123 * r2265123;
        double r2265125 = r2265121 / r2265124;
        double r2265126 = 1.0;
        double r2265127 = r2265124 - r2265126;
        double r2265128 = r2265125 / r2265127;
        return r2265128;
}
↓
double f(double i) {
        double r2265129 = i;
        double r2265130 = 3.714521008652235e-06;
        bool r2265131 = r2265129 <= r2265130;
        double r2265132 = r2265129 * r2265129;
        double r2265133 = 0.25;
        double r2265134 = 4.0;
        double r2265135 = r2265132 * r2265132;
        double r2265136 = r2265132 * r2265135;
        double r2265137 = 1.0;
        double r2265138 = r2265135 * r2265137;
        double r2265139 = fma(r2265134, r2265136, r2265138);
        double r2265140 = fma(r2265132, r2265133, r2265139);
        double r2265141 = -r2265140;
        double r2265142 = 4.0;
        double r2265143 = r2265137 / r2265132;
        double r2265144 = r2265142 + r2265143;
        double r2265145 = 0.25;
        double r2265146 = 16.0;
        double r2265147 = r2265143 * r2265143;
        double r2265148 = r2265146 - r2265147;
        double r2265149 = r2265145 / r2265148;
        double r2265150 = r2265144 * r2265149;
        double r2265151 = r2265131 ? r2265141 : r2265150;
        return r2265151;
}

Error

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if i < 3.714521008652235e-06

    1. Initial program 45.5

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

    2. Simplified0.7

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

    4. Applied flip--47.2

      \[\leadsto \frac{\frac{1}{4}}{\color{blue}{\frac{4 \cdot 4 - \frac{1.0}{i \cdot i} \cdot \frac{1.0}{i \cdot i}}{4 + \frac{1.0}{i \cdot i}}}}\]

    5. Applied associate-/r/47.2

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

    6. Simplified47.2

      \[\leadsto \color{blue}{\frac{\frac{1}{4}}{16 - \frac{1.0}{i \cdot i} \cdot \frac{1.0}{i \cdot i}}} \cdot \left(4 + \frac{1.0}{i \cdot i}\right)\]

    7. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{-\left(0.25 \cdot {i}^{2} + \left(1.0 \cdot {i}^{4} + 4.0 \cdot {i}^{6}\right)\right)}\]

    8. Simplified0.0

      \[\leadsto \color{blue}{-\mathsf{fma}\left(i \cdot i, 0.25, \mathsf{fma}\left(4.0, \left(\left(i \cdot i\right) \cdot \left(i \cdot i\right)\right) \cdot \left(i \cdot i\right), 1.0 \cdot \left(\left(i \cdot i\right) \cdot \left(i \cdot i\right)\right)\right)\right)}\]

    if 3.714521008652235e-06 < i

    1. Initial program 46.1

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

    2. Simplified0.0

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

    4. Applied flip--0.0

      \[\leadsto \frac{\frac{1}{4}}{\color{blue}{\frac{4 \cdot 4 - \frac{1.0}{i \cdot i} \cdot \frac{1.0}{i \cdot i}}{4 + \frac{1.0}{i \cdot i}}}}\]

    5. Applied associate-/r/0.0

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

    6. Simplified0.0

      \[\leadsto \color{blue}{\frac{\frac{1}{4}}{16 - \frac{1.0}{i \cdot i} \cdot \frac{1.0}{i \cdot i}}} \cdot \left(4 + \frac{1.0}{i \cdot i}\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 3.714521008652235 \cdot 10^{-06}:\\ \;\;\;\;-\mathsf{fma}\left(i \cdot i, 0.25, \mathsf{fma}\left(4.0, \left(i \cdot i\right) \cdot \left(\left(i \cdot i\right) \cdot \left(i \cdot i\right)\right), \left(\left(i \cdot i\right) \cdot \left(i \cdot i\right)\right) \cdot 1.0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(4 + \frac{1.0}{i \cdot i}\right) \cdot \frac{\frac{1}{4}}{16 - \frac{1.0}{i \cdot i} \cdot \frac{1.0}{i \cdot i}}\\ \end{array}\]

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)))