Average Error: 15.3 → 0.0
Time: 24.2s
Precision: 64
\[1 - \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \le 1.0000091892545029:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right), \frac{69}{1024}, \left(\frac{1}{8} - \left(x \cdot x\right) \cdot \frac{11}{128}\right) \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2} - \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}{1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}\\ \end{array}\]
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 1.0000091892545029:\\
\;\;\;\;\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}, \left(\frac{1}{8} - \left(x \cdot x\right) \cdot \frac{11}{128}\right) \cdot \left(x \cdot x\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{\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)}}}\\

\end{array}
double f(double x) {
        double r4328368 = 1.0;
        double r4328369 = 0.5;
        double r4328370 = x;
        double r4328371 = hypot(r4328368, r4328370);
        double r4328372 = r4328368 / r4328371;
        double r4328373 = r4328368 + r4328372;
        double r4328374 = r4328369 * r4328373;
        double r4328375 = sqrt(r4328374);
        double r4328376 = r4328368 - r4328375;
        return r4328376;
}
↓
double f(double x) {
        double r4328377 = 1.0;
        double r4328378 = x;
        double r4328379 = hypot(r4328377, r4328378);
        double r4328380 = 1.0000091892545029;
        bool r4328381 = r4328379 <= r4328380;
        double r4328382 = r4328378 * r4328378;
        double r4328383 = r4328382 * r4328378;
        double r4328384 = r4328383 * r4328383;
        double r4328385 = 0.0673828125;
        double r4328386 = 0.125;
        double r4328387 = 0.0859375;
        double r4328388 = r4328382 * r4328387;
        double r4328389 = r4328386 - r4328388;
        double r4328390 = r4328389 * r4328382;
        double r4328391 = fma(r4328384, r4328385, r4328390);
        double r4328392 = 0.5;
        double r4328393 = r4328392 / r4328379;
        double r4328394 = r4328392 - r4328393;
        double r4328395 = r4328392 + r4328393;
        double r4328396 = sqrt(r4328395);
        double r4328397 = r4328377 + r4328396;
        double r4328398 = r4328394 / r4328397;
        double r4328399 = r4328381 ? r4328391 : r4328398;
        return r4328399;
}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if (hypot 1 x) < 1.0000091892545029

    1. Initial program 29.9

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

    2. Simplified29.9

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

    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

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

    if 1.0000091892545029 < (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

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

    7. Applied *-un-lft-identity0.1

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

    8. Applied associate-/r*0.1

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

    9. Simplified0.1

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \le 1.0000091892545029:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right), \frac{69}{1024}, \left(\frac{1}{8} - \left(x \cdot x\right) \cdot \frac{11}{128}\right) \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2} - \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}{1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019151 +o rules:numerics
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  (- 1 (sqrt (* 1/2 (+ 1 (/ 1 (hypot 1 x)))))))