Average Error: 15.3 → 14.8
Time: 20.3s
Precision: 64
\[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
↓
\[\frac{\frac{{1}^{6} \cdot {1}^{6} - {\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{3} \cdot {\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{3}}{\frac{\left({1}^{8} - {\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{2} \cdot \left(\left(1 \cdot 1 + 0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right) \cdot \left(1 \cdot 1 + 0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)\right)\right) \cdot \left({\left({1}^{6}\right)}^{3} + {\left({\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{3}\right)}^{3}\right)}{\left({1}^{12} + {\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{3} \cdot \left({\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{3} - {1}^{6}\right)\right) \cdot \left({1}^{4} - \left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right) \cdot \left(1 \cdot 1 + 0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)\right)}}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]
1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}
↓
\frac{\frac{{1}^{6} \cdot {1}^{6} - {\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{3} \cdot {\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{3}}{\frac{\left({1}^{8} - {\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{2} \cdot \left(\left(1 \cdot 1 + 0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right) \cdot \left(1 \cdot 1 + 0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)\right)\right) \cdot \left({\left({1}^{6}\right)}^{3} + {\left({\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{3}\right)}^{3}\right)}{\left({1}^{12} + {\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{3} \cdot \left({\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{3} - {1}^{6}\right)\right) \cdot \left({1}^{4} - \left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right) \cdot \left(1 \cdot 1 + 0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)\right)}}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}
double f(double x) {
        double r101754 = 1.0;
        double r101755 = 0.5;
        double r101756 = x;
        double r101757 = hypot(r101754, r101756);
        double r101758 = r101754 / r101757;
        double r101759 = r101754 + r101758;
        double r101760 = r101755 * r101759;
        double r101761 = sqrt(r101760);
        double r101762 = r101754 - r101761;
        return r101762;
}
↓
double f(double x) {
        double r101763 = 1.0;
        double r101764 = 6.0;
        double r101765 = pow(r101763, r101764);
        double r101766 = r101765 * r101765;
        double r101767 = 0.5;
        double r101768 = x;
        double r101769 = hypot(r101763, r101768);
        double r101770 = r101763 / r101769;
        double r101771 = r101763 + r101770;
        double r101772 = r101767 * r101771;
        double r101773 = 3.0;
        double r101774 = pow(r101772, r101773);
        double r101775 = r101774 * r101774;
        double r101776 = r101766 - r101775;
        double r101777 = 8.0;
        double r101778 = pow(r101763, r101777);
        double r101779 = 2.0;
        double r101780 = pow(r101772, r101779);
        double r101781 = r101763 * r101763;
        double r101782 = r101781 + r101772;
        double r101783 = r101782 * r101782;
        double r101784 = r101780 * r101783;
        double r101785 = r101778 - r101784;
        double r101786 = pow(r101765, r101773);
        double r101787 = pow(r101774, r101773);
        double r101788 = r101786 + r101787;
        double r101789 = r101785 * r101788;
        double r101790 = 12.0;
        double r101791 = pow(r101763, r101790);
        double r101792 = r101774 - r101765;
        double r101793 = r101774 * r101792;
        double r101794 = r101791 + r101793;
        double r101795 = 4.0;
        double r101796 = pow(r101763, r101795);
        double r101797 = r101771 * r101767;
        double r101798 = r101797 * r101782;
        double r101799 = r101796 - r101798;
        double r101800 = r101794 * r101799;
        double r101801 = r101789 / r101800;
        double r101802 = r101776 / r101801;
        double r101803 = sqrt(r101772);
        double r101804 = r101763 + r101803;
        double r101805 = r101802 / r101804;
        return r101805;
}

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

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

  6. Applied flip3--14.8

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

  7. Simplified14.8

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

  8. Simplified14.8

    \[\leadsto \frac{\frac{{1}^{6} - {\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{3}}{\color{blue}{{1}^{4} + \left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right) \cdot \left(1 \cdot 1 + 0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]
  9. Using strategy rm

  10. Applied flip--14.8

    \[\leadsto \frac{\frac{\color{blue}{\frac{{1}^{6} \cdot {1}^{6} - {\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{3} \cdot {\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{3}}{{1}^{6} + {\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{3}}}}{{1}^{4} + \left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right) \cdot \left(1 \cdot 1 + 0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]

  11. Applied associate-/l/14.8

    \[\leadsto \frac{\color{blue}{\frac{{1}^{6} \cdot {1}^{6} - {\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{3} \cdot {\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{3}}{\left({1}^{4} + \left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right) \cdot \left(1 \cdot 1 + 0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)\right) \cdot \left({1}^{6} + {\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{3}\right)}}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]
  12. Using strategy rm

  13. Applied flip3-+14.8

    \[\leadsto \frac{\frac{{1}^{6} \cdot {1}^{6} - {\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{3} \cdot {\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{3}}{\left({1}^{4} + \left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right) \cdot \left(1 \cdot 1 + 0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)\right) \cdot \color{blue}{\frac{{\left({1}^{6}\right)}^{3} + {\left({\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{3}\right)}^{3}}{{1}^{6} \cdot {1}^{6} + \left({\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{3} \cdot {\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{3} - {1}^{6} \cdot {\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{3}\right)}}}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]

  14. Applied flip-+14.8

    \[\leadsto \frac{\frac{{1}^{6} \cdot {1}^{6} - {\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{3} \cdot {\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{3}}{\color{blue}{\frac{{1}^{4} \cdot {1}^{4} - \left(\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right) \cdot \left(1 \cdot 1 + 0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)\right) \cdot \left(\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right) \cdot \left(1 \cdot 1 + 0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)\right)}{{1}^{4} - \left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right) \cdot \left(1 \cdot 1 + 0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}} \cdot \frac{{\left({1}^{6}\right)}^{3} + {\left({\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{3}\right)}^{3}}{{1}^{6} \cdot {1}^{6} + \left({\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{3} \cdot {\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{3} - {1}^{6} \cdot {\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{3}\right)}}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]

  15. Applied frac-times14.8

    \[\leadsto \frac{\frac{{1}^{6} \cdot {1}^{6} - {\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{3} \cdot {\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{3}}{\color{blue}{\frac{\left({1}^{4} \cdot {1}^{4} - \left(\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right) \cdot \left(1 \cdot 1 + 0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)\right) \cdot \left(\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right) \cdot \left(1 \cdot 1 + 0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)\right)\right) \cdot \left({\left({1}^{6}\right)}^{3} + {\left({\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{3}\right)}^{3}\right)}{\left({1}^{4} - \left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right) \cdot \left(1 \cdot 1 + 0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)\right) \cdot \left({1}^{6} \cdot {1}^{6} + \left({\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{3} \cdot {\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{3} - {1}^{6} \cdot {\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{3}\right)\right)}}}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]

  16. Simplified14.8

    \[\leadsto \frac{\frac{{1}^{6} \cdot {1}^{6} - {\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{3} \cdot {\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{3}}{\frac{\color{blue}{\left({1}^{8} - {\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{2} \cdot \left(\left(1 \cdot 1 + 0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right) \cdot \left(1 \cdot 1 + 0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)\right)\right) \cdot \left({\left({1}^{6}\right)}^{3} + {\left({\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{3}\right)}^{3}\right)}}{\left({1}^{4} - \left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right) \cdot \left(1 \cdot 1 + 0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)\right) \cdot \left({1}^{6} \cdot {1}^{6} + \left({\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{3} \cdot {\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{3} - {1}^{6} \cdot {\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{3}\right)\right)}}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]

  17. Simplified14.8

    \[\leadsto \frac{\frac{{1}^{6} \cdot {1}^{6} - {\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{3} \cdot {\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{3}}{\frac{\left({1}^{8} - {\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{2} \cdot \left(\left(1 \cdot 1 + 0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right) \cdot \left(1 \cdot 1 + 0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)\right)\right) \cdot \left({\left({1}^{6}\right)}^{3} + {\left({\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{3}\right)}^{3}\right)}{\color{blue}{\left({1}^{12} + {\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{3} \cdot \left({\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{3} - {1}^{6}\right)\right) \cdot \left({1}^{4} - \left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right) \cdot \left(1 \cdot 1 + 0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)\right)}}}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]

  18. Final simplification14.8

    \[\leadsto \frac{\frac{{1}^{6} \cdot {1}^{6} - {\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{3} \cdot {\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{3}}{\frac{\left({1}^{8} - {\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{2} \cdot \left(\left(1 \cdot 1 + 0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right) \cdot \left(1 \cdot 1 + 0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)\right)\right) \cdot \left({\left({1}^{6}\right)}^{3} + {\left({\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{3}\right)}^{3}\right)}{\left({1}^{12} + {\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{3} \cdot \left({\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{3} - {1}^{6}\right)\right) \cdot \left({1}^{4} - \left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right) \cdot \left(1 \cdot 1 + 0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)\right)}}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]

Reproduce

herbie shell --seed 2019323 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1 (sqrt (* 0.5 (+ 1 (/ 1 (hypot 1 x)))))))