Average Error: 15.4 → 0.0
Time: 12.8s
Precision: 64
\[1 - \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \le 1.0000170445415169:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right), \frac{69}{1024}, \mathsf{fma}\left(x \cdot x, \frac{1}{8}, \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{-11}{128}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \left(\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right)}{1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}\\ \end{array}\]
1 - \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\mathsf{hypot}\left(1, x\right) \le 1.0000170445415169:\\
\;\;\;\;\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right), \frac{69}{1024}, \mathsf{fma}\left(x \cdot x, \frac{1}{8}, \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{-11}{128}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{1 - \left(\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right)}{1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}\\

\end{array}
double f(double x) {
        double r2061618 = 1.0;
        double r2061619 = 0.5;
        double r2061620 = x;
        double r2061621 = hypot(r2061618, r2061620);
        double r2061622 = r2061618 / r2061621;
        double r2061623 = r2061618 + r2061622;
        double r2061624 = r2061619 * r2061623;
        double r2061625 = sqrt(r2061624);
        double r2061626 = r2061618 - r2061625;
        return r2061626;
}
↓
double f(double x) {
        double r2061627 = 1.0;
        double r2061628 = x;
        double r2061629 = hypot(r2061627, r2061628);
        double r2061630 = 1.0000170445415169;
        bool r2061631 = r2061629 <= r2061630;
        double r2061632 = r2061628 * r2061628;
        double r2061633 = r2061632 * r2061628;
        double r2061634 = r2061633 * r2061633;
        double r2061635 = 0.0673828125;
        double r2061636 = 0.125;
        double r2061637 = r2061632 * r2061632;
        double r2061638 = -0.0859375;
        double r2061639 = r2061637 * r2061638;
        double r2061640 = fma(r2061632, r2061636, r2061639);
        double r2061641 = fma(r2061634, r2061635, r2061640);
        double r2061642 = 0.5;
        double r2061643 = r2061642 / r2061629;
        double r2061644 = r2061642 + r2061643;
        double r2061645 = r2061627 - r2061644;
        double r2061646 = sqrt(r2061644);
        double r2061647 = r2061627 + r2061646;
        double r2061648 = r2061645 / r2061647;
        double r2061649 = r2061631 ? r2061641 : r2061648;
        return r2061649;
}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if (hypot 1 x) < 1.0000170445415169

    1. Initial program 29.7

      \[1 - \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]

    2. Simplified29.7

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

    3. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{\left(\frac{1}{8} \cdot {x}^{2} + \frac{69}{1024} \cdot {x}^{6}\right) - \frac{11}{128} \cdot {x}^{4}}\]

    4. Simplified0.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right), \frac{69}{1024}, \mathsf{fma}\left(x \cdot x, \frac{1}{8}, \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{-11}{128}\right)\right)}\]

    if 1.0000170445415169 < (hypot 1 x)

    1. Initial program 1.0

      \[1 - \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]

    2. Simplified1.0

      \[\leadsto \color{blue}{1 - \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}\]
    3. Using strategy rm

    4. Applied flip--1.0

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

    5. Simplified0.1

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

    6. Simplified0.1

      \[\leadsto \frac{1 - \left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}\right)}{\color{blue}{\sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}} + 1}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \le 1.0000170445415169:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right), \frac{69}{1024}, \mathsf{fma}\left(x \cdot x, \frac{1}{8}, \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{-11}{128}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \left(\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right)}{1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019154 +o rules:numerics
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  (- 1 (sqrt (* 1/2 (+ 1 (/ 1 (hypot 1 x)))))))