Average Error: 15.7 → 15.2
Time: 27.0s
Precision: 64
\[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
↓
\[\frac{\frac{{1}^{6} \cdot {1}^{6} - \mathsf{log1p}\left(\mathsf{expm1}\left({\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right)}^{3}\right)\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left({\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right)}^{3}\right)\right)}{\mathsf{fma}\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5, \mathsf{fma}\left(1, 1, 0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right), 1 \cdot {1}^{3}\right) \cdot \left({1}^{6} + {\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right)}^{3}\right)}}{\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} + 1}\]
1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}
↓
\frac{\frac{{1}^{6} \cdot {1}^{6} - \mathsf{log1p}\left(\mathsf{expm1}\left({\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right)}^{3}\right)\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left({\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right)}^{3}\right)\right)}{\mathsf{fma}\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5, \mathsf{fma}\left(1, 1, 0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right), 1 \cdot {1}^{3}\right) \cdot \left({1}^{6} + {\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right)}^{3}\right)}}{\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} + 1}
double f(double x) {
        double r215893 = 1.0;
        double r215894 = 0.5;
        double r215895 = x;
        double r215896 = hypot(r215893, r215895);
        double r215897 = r215893 / r215896;
        double r215898 = r215893 + r215897;
        double r215899 = r215894 * r215898;
        double r215900 = sqrt(r215899);
        double r215901 = r215893 - r215900;
        return r215901;
}
↓
double f(double x) {
        double r215902 = 1.0;
        double r215903 = 6.0;
        double r215904 = pow(r215902, r215903);
        double r215905 = r215904 * r215904;
        double r215906 = x;
        double r215907 = hypot(r215902, r215906);
        double r215908 = r215902 / r215907;
        double r215909 = r215902 + r215908;
        double r215910 = 0.5;
        double r215911 = r215909 * r215910;
        double r215912 = 3.0;
        double r215913 = pow(r215911, r215912);
        double r215914 = expm1(r215913);
        double r215915 = log1p(r215914);
        double r215916 = r215915 * r215915;
        double r215917 = r215905 - r215916;
        double r215918 = r215910 * r215909;
        double r215919 = fma(r215902, r215902, r215918);
        double r215920 = pow(r215902, r215912);
        double r215921 = r215902 * r215920;
        double r215922 = fma(r215911, r215919, r215921);
        double r215923 = r215904 + r215913;
        double r215924 = r215922 * r215923;
        double r215925 = r215917 / r215924;
        double r215926 = sqrt(r215918);
        double r215927 = r215926 + r215902;
        double r215928 = r215925 / r215927;
        return r215928;
}

Error

Bits error versus x

Derivation

  1. Initial program 15.7

    \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
  2. Using strategy rm

  3. Applied flip--15.7

    \[\leadsto \color{blue}{\frac{1 \cdot 1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \cdot \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}}\]

  4. Simplified15.2

    \[\leadsto \frac{\color{blue}{1 \cdot 1 - \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]

  5. Simplified15.2

    \[\leadsto \frac{1 \cdot 1 - \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5}{\color{blue}{\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} + 1}}\]
  6. Using strategy rm

  7. Applied flip3--15.2

    \[\leadsto \frac{\color{blue}{\frac{{\left(1 \cdot 1\right)}^{3} - {\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right)}^{3}}{\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right) + \left(\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right) \cdot \left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right) + \left(1 \cdot 1\right) \cdot \left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right)\right)}}}{\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} + 1}\]

  8. Simplified15.2

    \[\leadsto \frac{\frac{\color{blue}{{1}^{6} - {\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right)}^{3}}}{\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right) + \left(\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right) \cdot \left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right) + \left(1 \cdot 1\right) \cdot \left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right)\right)}}{\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} + 1}\]

  9. Simplified15.2

    \[\leadsto \frac{\frac{{1}^{6} - {\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right)}^{3}}{\color{blue}{\mathsf{fma}\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5, \mathsf{fma}\left(1, 1, 0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right), 1 \cdot {1}^{3}\right)}}}{\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} + 1}\]
  10. Using strategy rm

  11. Applied log1p-expm1-u15.2

    \[\leadsto \frac{\frac{{1}^{6} - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left({\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right)}^{3}\right)\right)}}{\mathsf{fma}\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5, \mathsf{fma}\left(1, 1, 0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right), 1 \cdot {1}^{3}\right)}}{\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} + 1}\]
  12. Using strategy rm

  13. Applied flip--15.2

    \[\leadsto \frac{\frac{\color{blue}{\frac{{1}^{6} \cdot {1}^{6} - \mathsf{log1p}\left(\mathsf{expm1}\left({\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right)}^{3}\right)\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left({\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right)}^{3}\right)\right)}{{1}^{6} + \mathsf{log1p}\left(\mathsf{expm1}\left({\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right)}^{3}\right)\right)}}}{\mathsf{fma}\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5, \mathsf{fma}\left(1, 1, 0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right), 1 \cdot {1}^{3}\right)}}{\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} + 1}\]

  14. Applied associate-/l/15.2

    \[\leadsto \frac{\color{blue}{\frac{{1}^{6} \cdot {1}^{6} - \mathsf{log1p}\left(\mathsf{expm1}\left({\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right)}^{3}\right)\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left({\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right)}^{3}\right)\right)}{\mathsf{fma}\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5, \mathsf{fma}\left(1, 1, 0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right), 1 \cdot {1}^{3}\right) \cdot \left({1}^{6} + \mathsf{log1p}\left(\mathsf{expm1}\left({\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right)}^{3}\right)\right)\right)}}}{\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} + 1}\]

  15. Simplified15.2

    \[\leadsto \frac{\frac{{1}^{6} \cdot {1}^{6} - \mathsf{log1p}\left(\mathsf{expm1}\left({\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right)}^{3}\right)\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left({\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right)}^{3}\right)\right)}{\color{blue}{\mathsf{fma}\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5, \mathsf{fma}\left(1, 1, 0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right), 1 \cdot {1}^{3}\right) \cdot \left({1}^{6} + {\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right)}^{3}\right)}}}{\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} + 1}\]

  16. Final simplification15.2

    \[\leadsto \frac{\frac{{1}^{6} \cdot {1}^{6} - \mathsf{log1p}\left(\mathsf{expm1}\left({\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right)}^{3}\right)\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left({\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right)}^{3}\right)\right)}{\mathsf{fma}\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5, \mathsf{fma}\left(1, 1, 0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right), 1 \cdot {1}^{3}\right) \cdot \left({1}^{6} + {\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right)}^{3}\right)}}{\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} + 1}\]

Reproduce

herbie shell --seed 2019209 +o rules:numerics
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1 (sqrt (* 0.5 (+ 1 (/ 1 (hypot 1 x)))))))