Average Error: 15.3 → 0.3
Time: 16.3s
Precision: 64
\[1 - \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \le 1.000000000000252:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right), \frac{69}{1024}, \left(\frac{1}{8} - \left(x \cdot x\right) \cdot \frac{11}{128}\right) \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2} - \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}{1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}\\ \end{array}\]
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 1.000000000000252:\\
\;\;\;\;\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 r3101884 = 1.0;
        double r3101885 = 0.5;
        double r3101886 = x;
        double r3101887 = hypot(r3101884, r3101886);
        double r3101888 = r3101884 / r3101887;
        double r3101889 = r3101884 + r3101888;
        double r3101890 = r3101885 * r3101889;
        double r3101891 = sqrt(r3101890);
        double r3101892 = r3101884 - r3101891;
        return r3101892;
}
↓
double f(double x) {
        double r3101893 = 1.0;
        double r3101894 = x;
        double r3101895 = hypot(r3101893, r3101894);
        double r3101896 = 1.000000000000252;
        bool r3101897 = r3101895 <= r3101896;
        double r3101898 = r3101894 * r3101894;
        double r3101899 = r3101898 * r3101894;
        double r3101900 = r3101899 * r3101899;
        double r3101901 = 0.0673828125;
        double r3101902 = 0.125;
        double r3101903 = 0.0859375;
        double r3101904 = r3101898 * r3101903;
        double r3101905 = r3101902 - r3101904;
        double r3101906 = r3101905 * r3101898;
        double r3101907 = fma(r3101900, r3101901, r3101906);
        double r3101908 = 0.5;
        double r3101909 = r3101908 / r3101895;
        double r3101910 = r3101908 - r3101909;
        double r3101911 = r3101908 + r3101909;
        double r3101912 = sqrt(r3101911);
        double r3101913 = r3101893 + r3101912;
        double r3101914 = r3101910 / r3101913;
        double r3101915 = r3101897 ? r3101907 : r3101914;
        return r3101915;
}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if (hypot 1 x) < 1.000000000000252

    1. Initial program 30.6

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

    2. Simplified30.6

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

    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

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

    if 1.000000000000252 < (hypot 1 x)

    1. Initial program 1.4

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

    2. Simplified1.4

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

    4. Applied flip--1.4

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

    5. Simplified0.5

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \le 1.000000000000252:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right), \frac{69}{1024}, \left(\frac{1}{8} - \left(x \cdot x\right) \cdot \frac{11}{128}\right) \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2} - \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}{1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019143 +o rules:numerics
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  (- 1 (sqrt (* 1/2 (+ 1 (/ 1 (hypot 1 x)))))))