Average Error: 15.2 → 14.8
Time: 18.6s
Precision: 64
\[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
↓
\[\frac{\log \left(e^{{\left(1 \cdot \left(1 - 0.5\right)\right)}^{3} - {\left(0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}\right)}{\left(\left(0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(1 \cdot \left(1 - 0.5\right) + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) + \left(1 \cdot \left(1 - 0.5\right)\right) \cdot \left(1 \cdot \left(1 - 0.5\right)\right)\right) \cdot \left(1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right)}\]
1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}
↓
\frac{\log \left(e^{{\left(1 \cdot \left(1 - 0.5\right)\right)}^{3} - {\left(0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}\right)}{\left(\left(0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(1 \cdot \left(1 - 0.5\right) + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) + \left(1 \cdot \left(1 - 0.5\right)\right) \cdot \left(1 \cdot \left(1 - 0.5\right)\right)\right) \cdot \left(1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right)}
double f(double x) {
        double r107713 = 1.0;
        double r107714 = 0.5;
        double r107715 = x;
        double r107716 = hypot(r107713, r107715);
        double r107717 = r107713 / r107716;
        double r107718 = r107713 + r107717;
        double r107719 = r107714 * r107718;
        double r107720 = sqrt(r107719);
        double r107721 = r107713 - r107720;
        return r107721;
}
↓
double f(double x) {
        double r107722 = 1.0;
        double r107723 = 0.5;
        double r107724 = r107722 - r107723;
        double r107725 = r107722 * r107724;
        double r107726 = 3.0;
        double r107727 = pow(r107725, r107726);
        double r107728 = x;
        double r107729 = hypot(r107722, r107728);
        double r107730 = r107722 / r107729;
        double r107731 = r107723 * r107730;
        double r107732 = pow(r107731, r107726);
        double r107733 = r107727 - r107732;
        double r107734 = exp(r107733);
        double r107735 = log(r107734);
        double r107736 = r107725 + r107731;
        double r107737 = r107731 * r107736;
        double r107738 = r107725 * r107725;
        double r107739 = r107737 + r107738;
        double r107740 = r107722 + r107730;
        double r107741 = r107723 * r107740;
        double r107742 = sqrt(r107741);
        double r107743 = r107722 + r107742;
        double r107744 = r107739 * r107743;
        double r107745 = r107735 / r107744;
        return r107745;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 15.2

    \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
  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 distribute-lft-in14.8

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

  7. Applied associate--r+14.8

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

  8. Simplified14.8

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

  10. Applied flip3--14.8

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

  11. Applied associate-/l/14.8

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

  12. Simplified14.8

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

  14. Applied add-log-exp14.8

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

  15. Applied add-log-exp14.8

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

  16. Applied diff-log14.8

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

  17. Simplified14.8

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

  18. Final simplification14.8

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

Reproduce

herbie shell --seed 2019326 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1 (sqrt (* 0.5 (+ 1 (/ 1 (hypot 1 x)))))))