Average Error: 15.3 → 14.9
Time: 13.7s
Precision: 64
\[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
↓
\[\frac{1 \cdot \frac{\frac{\left(\left(\left(1 \cdot 1 - 0.5 \cdot 0.5\right) \cdot \left(1 - 0.5\right)\right) \cdot \left(\left({1}^{3} - {0.5}^{3}\right) \cdot \left(1 - 0.5\right)\right)\right) \cdot \mathsf{hypot}\left(1, x\right) - \left(\left(1 + 0.5\right) \cdot \left(1 \cdot 1 + \left(0.5 \cdot 0.5 + 1 \cdot 0.5\right)\right)\right) \cdot \left(\left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(0.5 \cdot \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)\right)}{\left(\left(1 + 0.5\right) \cdot \left(1 \cdot 1 + \left(0.5 \cdot 0.5 + 1 \cdot 0.5\right)\right)\right) \cdot \mathsf{hypot}\left(1, x\right)}}{\left(\left(1 - 0.5\right) + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(\left(1 - 0.5\right) \cdot \left(1 - 0.5\right) + \frac{0.5}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]
1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}
↓
\frac{1 \cdot \frac{\frac{\left(\left(\left(1 \cdot 1 - 0.5 \cdot 0.5\right) \cdot \left(1 - 0.5\right)\right) \cdot \left(\left({1}^{3} - {0.5}^{3}\right) \cdot \left(1 - 0.5\right)\right)\right) \cdot \mathsf{hypot}\left(1, x\right) - \left(\left(1 + 0.5\right) \cdot \left(1 \cdot 1 + \left(0.5 \cdot 0.5 + 1 \cdot 0.5\right)\right)\right) \cdot \left(\left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(0.5 \cdot \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)\right)}{\left(\left(1 + 0.5\right) \cdot \left(1 \cdot 1 + \left(0.5 \cdot 0.5 + 1 \cdot 0.5\right)\right)\right) \cdot \mathsf{hypot}\left(1, x\right)}}{\left(\left(1 - 0.5\right) + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(\left(1 - 0.5\right) \cdot \left(1 - 0.5\right) + \frac{0.5}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}
double f(double x) {
        double r217912 = 1.0;
        double r217913 = 0.5;
        double r217914 = x;
        double r217915 = hypot(r217912, r217914);
        double r217916 = r217912 / r217915;
        double r217917 = r217912 + r217916;
        double r217918 = r217913 * r217917;
        double r217919 = sqrt(r217918);
        double r217920 = r217912 - r217919;
        return r217920;
}
↓
double f(double x) {
        double r217921 = 1.0;
        double r217922 = r217921 * r217921;
        double r217923 = 0.5;
        double r217924 = r217923 * r217923;
        double r217925 = r217922 - r217924;
        double r217926 = r217921 - r217923;
        double r217927 = r217925 * r217926;
        double r217928 = 3.0;
        double r217929 = pow(r217921, r217928);
        double r217930 = pow(r217923, r217928);
        double r217931 = r217929 - r217930;
        double r217932 = r217931 * r217926;
        double r217933 = r217927 * r217932;
        double r217934 = x;
        double r217935 = hypot(r217921, r217934);
        double r217936 = r217933 * r217935;
        double r217937 = r217921 + r217923;
        double r217938 = r217921 * r217923;
        double r217939 = r217924 + r217938;
        double r217940 = r217922 + r217939;
        double r217941 = r217937 * r217940;
        double r217942 = r217923 / r217935;
        double r217943 = r217942 * r217942;
        double r217944 = r217923 * r217942;
        double r217945 = r217943 * r217944;
        double r217946 = r217941 * r217945;
        double r217947 = r217936 - r217946;
        double r217948 = r217941 * r217935;
        double r217949 = r217947 / r217948;
        double r217950 = r217926 + r217942;
        double r217951 = r217926 * r217926;
        double r217952 = r217951 + r217943;
        double r217953 = r217950 * r217952;
        double r217954 = r217949 / r217953;
        double r217955 = r217921 * r217954;
        double r217956 = r217921 / r217935;
        double r217957 = r217921 + r217956;
        double r217958 = r217923 * r217957;
        double r217959 = sqrt(r217958);
        double r217960 = r217921 + r217959;
        double r217961 = r217955 / r217960;
        return r217961;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 15.3

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

  3. Applied flip--15.3

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

  4. Simplified14.8

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

  6. Applied flip--14.8

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

  8. Applied flip--14.8

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

  9. Applied associate-/l/14.8

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

  11. Applied associate-*l/14.8

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

  12. Applied associate-*r/14.8

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

  13. Applied flip3--14.8

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

  14. Applied associate-*l/14.8

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

  15. Applied flip--14.8

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

  16. Applied associate-*l/14.8

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

  17. Applied frac-times14.8

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

  18. Applied frac-sub14.9

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

  19. Final simplification14.9

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

Reproduce

herbie shell --seed 2020089 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1 (sqrt (* 0.5 (+ 1 (/ 1 (hypot 1 x)))))))