Average Error: 15.9 → 15.4
Time: 13.8s
Precision: 64
\[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
↓
\[\frac{\frac{\frac{{\left(1 \cdot \left({1}^{3} - {\left(\frac{1}{2}\right)}^{3}\right)\right)}^{3} \cdot {2}^{3} - {\left(1 \cdot 1 + \left(\frac{1}{2} \cdot \frac{1}{2} + 1 \cdot \frac{1}{2}\right)\right)}^{3} \cdot {\left(1 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{{\left(1 \cdot 1 + \left(\frac{1}{2} \cdot \frac{1}{2} + 1 \cdot \frac{1}{2}\right)\right)}^{3} \cdot {2}^{3}}}{\left(-\frac{1}{2} \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(\left(-\frac{1}{2} \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) - 1 \cdot \left(1 - \frac{1}{2}\right)\right) + \left(1 \cdot \left(1 - \frac{1}{2}\right)\right) \cdot \left(1 \cdot \left(1 - \frac{1}{2}\right)\right)}}{1 + \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]
1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}
↓
\frac{\frac{\frac{{\left(1 \cdot \left({1}^{3} - {\left(\frac{1}{2}\right)}^{3}\right)\right)}^{3} \cdot {2}^{3} - {\left(1 \cdot 1 + \left(\frac{1}{2} \cdot \frac{1}{2} + 1 \cdot \frac{1}{2}\right)\right)}^{3} \cdot {\left(1 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{{\left(1 \cdot 1 + \left(\frac{1}{2} \cdot \frac{1}{2} + 1 \cdot \frac{1}{2}\right)\right)}^{3} \cdot {2}^{3}}}{\left(-\frac{1}{2} \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(\left(-\frac{1}{2} \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) - 1 \cdot \left(1 - \frac{1}{2}\right)\right) + \left(1 \cdot \left(1 - \frac{1}{2}\right)\right) \cdot \left(1 \cdot \left(1 - \frac{1}{2}\right)\right)}}{1 + \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}
double f(double x) {
        double r163746 = 1.0;
        double r163747 = 0.5;
        double r163748 = x;
        double r163749 = hypot(r163746, r163748);
        double r163750 = r163746 / r163749;
        double r163751 = r163746 + r163750;
        double r163752 = r163747 * r163751;
        double r163753 = sqrt(r163752);
        double r163754 = r163746 - r163753;
        return r163754;
}
↓
double f(double x) {
        double r163755 = 1.0;
        double r163756 = 3.0;
        double r163757 = pow(r163755, r163756);
        double r163758 = 2.0;
        double r163759 = r163755 / r163758;
        double r163760 = pow(r163759, r163756);
        double r163761 = r163757 - r163760;
        double r163762 = r163755 * r163761;
        double r163763 = pow(r163762, r163756);
        double r163764 = pow(r163758, r163756);
        double r163765 = r163763 * r163764;
        double r163766 = r163755 * r163755;
        double r163767 = r163759 * r163759;
        double r163768 = r163755 * r163759;
        double r163769 = r163767 + r163768;
        double r163770 = r163766 + r163769;
        double r163771 = pow(r163770, r163756);
        double r163772 = x;
        double r163773 = hypot(r163755, r163772);
        double r163774 = r163755 / r163773;
        double r163775 = r163755 * r163774;
        double r163776 = pow(r163775, r163756);
        double r163777 = r163771 * r163776;
        double r163778 = r163765 - r163777;
        double r163779 = r163771 * r163764;
        double r163780 = r163778 / r163779;
        double r163781 = r163759 * r163774;
        double r163782 = -r163781;
        double r163783 = r163755 - r163759;
        double r163784 = r163755 * r163783;
        double r163785 = r163782 - r163784;
        double r163786 = r163782 * r163785;
        double r163787 = r163784 * r163784;
        double r163788 = r163786 + r163787;
        double r163789 = r163780 / r163788;
        double r163790 = r163755 + r163774;
        double r163791 = r163759 * r163790;
        double r163792 = sqrt(r163791);
        double r163793 = r163755 + r163792;
        double r163794 = r163789 / r163793;
        return r163794;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 15.9

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

  3. Applied flip--15.9

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

  4. Simplified15.4

    \[\leadsto \frac{\color{blue}{1 \cdot 1 + \left(-\frac{1}{2} \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]

  5. Simplified15.4

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

  7. Applied distribute-lft-in15.4

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

  8. Applied distribute-neg-in15.4

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

  9. Applied associate-+r+15.4

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

  10. Simplified15.4

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

  12. Applied flip3-+15.4

    \[\leadsto \frac{\color{blue}{\frac{{\left(1 \cdot \left(1 - \frac{1}{2}\right)\right)}^{3} + {\left(-\frac{1}{2} \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{\left(1 \cdot \left(1 - \frac{1}{2}\right)\right) \cdot \left(1 \cdot \left(1 - \frac{1}{2}\right)\right) + \left(\left(-\frac{1}{2} \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(-\frac{1}{2} \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) - \left(1 \cdot \left(1 - \frac{1}{2}\right)\right) \cdot \left(-\frac{1}{2} \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}}}{1 + \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]

  13. Simplified15.4

    \[\leadsto \frac{\frac{\color{blue}{{\left(1 \cdot \left(1 - \frac{1}{2}\right)\right)}^{3} - {\left(\frac{1}{2} \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}}{\left(1 \cdot \left(1 - \frac{1}{2}\right)\right) \cdot \left(1 \cdot \left(1 - \frac{1}{2}\right)\right) + \left(\left(-\frac{1}{2} \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(-\frac{1}{2} \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) - \left(1 \cdot \left(1 - \frac{1}{2}\right)\right) \cdot \left(-\frac{1}{2} \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}}{1 + \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]

  14. Simplified15.4

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

  16. Applied associate-*l/15.4

    \[\leadsto \frac{\frac{{\left(1 \cdot \left(1 - \frac{1}{2}\right)\right)}^{3} - {\color{blue}{\left(\frac{1 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}}{2}\right)}}^{3}}{\left(-\frac{1}{2} \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(\left(-\frac{1}{2} \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) - 1 \cdot \left(1 - \frac{1}{2}\right)\right) + \left(1 \cdot \left(1 - \frac{1}{2}\right)\right) \cdot \left(1 \cdot \left(1 - \frac{1}{2}\right)\right)}}{1 + \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]

  17. Applied cube-div15.4

    \[\leadsto \frac{\frac{{\left(1 \cdot \left(1 - \frac{1}{2}\right)\right)}^{3} - \color{blue}{\frac{{\left(1 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{{2}^{3}}}}{\left(-\frac{1}{2} \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(\left(-\frac{1}{2} \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) - 1 \cdot \left(1 - \frac{1}{2}\right)\right) + \left(1 \cdot \left(1 - \frac{1}{2}\right)\right) \cdot \left(1 \cdot \left(1 - \frac{1}{2}\right)\right)}}{1 + \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]

  18. Applied flip3--15.4

    \[\leadsto \frac{\frac{{\left(1 \cdot \color{blue}{\frac{{1}^{3} - {\left(\frac{1}{2}\right)}^{3}}{1 \cdot 1 + \left(\frac{1}{2} \cdot \frac{1}{2} + 1 \cdot \frac{1}{2}\right)}}\right)}^{3} - \frac{{\left(1 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{{2}^{3}}}{\left(-\frac{1}{2} \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(\left(-\frac{1}{2} \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) - 1 \cdot \left(1 - \frac{1}{2}\right)\right) + \left(1 \cdot \left(1 - \frac{1}{2}\right)\right) \cdot \left(1 \cdot \left(1 - \frac{1}{2}\right)\right)}}{1 + \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]

  19. Applied associate-*r/15.4

    \[\leadsto \frac{\frac{{\color{blue}{\left(\frac{1 \cdot \left({1}^{3} - {\left(\frac{1}{2}\right)}^{3}\right)}{1 \cdot 1 + \left(\frac{1}{2} \cdot \frac{1}{2} + 1 \cdot \frac{1}{2}\right)}\right)}}^{3} - \frac{{\left(1 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{{2}^{3}}}{\left(-\frac{1}{2} \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(\left(-\frac{1}{2} \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) - 1 \cdot \left(1 - \frac{1}{2}\right)\right) + \left(1 \cdot \left(1 - \frac{1}{2}\right)\right) \cdot \left(1 \cdot \left(1 - \frac{1}{2}\right)\right)}}{1 + \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]

  20. Applied cube-div15.4

    \[\leadsto \frac{\frac{\color{blue}{\frac{{\left(1 \cdot \left({1}^{3} - {\left(\frac{1}{2}\right)}^{3}\right)\right)}^{3}}{{\left(1 \cdot 1 + \left(\frac{1}{2} \cdot \frac{1}{2} + 1 \cdot \frac{1}{2}\right)\right)}^{3}}} - \frac{{\left(1 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{{2}^{3}}}{\left(-\frac{1}{2} \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(\left(-\frac{1}{2} \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) - 1 \cdot \left(1 - \frac{1}{2}\right)\right) + \left(1 \cdot \left(1 - \frac{1}{2}\right)\right) \cdot \left(1 \cdot \left(1 - \frac{1}{2}\right)\right)}}{1 + \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]

  21. Applied frac-sub15.4

    \[\leadsto \frac{\frac{\color{blue}{\frac{{\left(1 \cdot \left({1}^{3} - {\left(\frac{1}{2}\right)}^{3}\right)\right)}^{3} \cdot {2}^{3} - {\left(1 \cdot 1 + \left(\frac{1}{2} \cdot \frac{1}{2} + 1 \cdot \frac{1}{2}\right)\right)}^{3} \cdot {\left(1 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{{\left(1 \cdot 1 + \left(\frac{1}{2} \cdot \frac{1}{2} + 1 \cdot \frac{1}{2}\right)\right)}^{3} \cdot {2}^{3}}}}{\left(-\frac{1}{2} \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(\left(-\frac{1}{2} \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) - 1 \cdot \left(1 - \frac{1}{2}\right)\right) + \left(1 \cdot \left(1 - \frac{1}{2}\right)\right) \cdot \left(1 \cdot \left(1 - \frac{1}{2}\right)\right)}}{1 + \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]

  22. Final simplification15.4

    \[\leadsto \frac{\frac{\frac{{\left(1 \cdot \left({1}^{3} - {\left(\frac{1}{2}\right)}^{3}\right)\right)}^{3} \cdot {2}^{3} - {\left(1 \cdot 1 + \left(\frac{1}{2} \cdot \frac{1}{2} + 1 \cdot \frac{1}{2}\right)\right)}^{3} \cdot {\left(1 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{{\left(1 \cdot 1 + \left(\frac{1}{2} \cdot \frac{1}{2} + 1 \cdot \frac{1}{2}\right)\right)}^{3} \cdot {2}^{3}}}{\left(-\frac{1}{2} \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(\left(-\frac{1}{2} \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) - 1 \cdot \left(1 - \frac{1}{2}\right)\right) + \left(1 \cdot \left(1 - \frac{1}{2}\right)\right) \cdot \left(1 \cdot \left(1 - \frac{1}{2}\right)\right)}}{1 + \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]

Reproduce

herbie shell --seed 2019303 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1 (sqrt (* 0.5 (+ 1 (/ 1 (hypot 1 x)))))))