Average Error: 15.6 → 15.1
Time: 4.7s
Precision: 64
\[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
↓
\[\frac{\frac{1 \cdot \left(\left(1 \cdot 1 - 0.5 \cdot 0.5\right) \cdot \mathsf{hypot}\left(1, x\right) - 0.5 \cdot \left(0.5 + 1\right)\right)}{\left(1 + 0.5\right) \cdot \mathsf{hypot}\left(1, x\right)}}{{1}^{3} + {\left(\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \cdot \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} - 1 \cdot \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right)\right)\]
1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}
↓
\frac{\frac{1 \cdot \left(\left(1 \cdot 1 - 0.5 \cdot 0.5\right) \cdot \mathsf{hypot}\left(1, x\right) - 0.5 \cdot \left(0.5 + 1\right)\right)}{\left(1 + 0.5\right) \cdot \mathsf{hypot}\left(1, x\right)}}{{1}^{3} + {\left(\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\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 \cdot \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right)\right)
double f(double x) {
        double r194698 = 1.0;
        double r194699 = 0.5;
        double r194700 = x;
        double r194701 = hypot(r194698, r194700);
        double r194702 = r194698 / r194701;
        double r194703 = r194698 + r194702;
        double r194704 = r194699 * r194703;
        double r194705 = sqrt(r194704);
        double r194706 = r194698 - r194705;
        return r194706;
}
↓
double f(double x) {
        double r194707 = 1.0;
        double r194708 = r194707 * r194707;
        double r194709 = 0.5;
        double r194710 = r194709 * r194709;
        double r194711 = r194708 - r194710;
        double r194712 = x;
        double r194713 = hypot(r194707, r194712);
        double r194714 = r194711 * r194713;
        double r194715 = r194709 + r194707;
        double r194716 = r194709 * r194715;
        double r194717 = r194714 - r194716;
        double r194718 = r194707 * r194717;
        double r194719 = r194707 + r194709;
        double r194720 = r194719 * r194713;
        double r194721 = r194718 / r194720;
        double r194722 = 3.0;
        double r194723 = pow(r194707, r194722);
        double r194724 = r194707 / r194713;
        double r194725 = r194707 + r194724;
        double r194726 = r194709 * r194725;
        double r194727 = sqrt(r194726);
        double r194728 = pow(r194727, r194722);
        double r194729 = r194723 + r194728;
        double r194730 = r194721 / r194729;
        double r194731 = r194727 * r194727;
        double r194732 = r194707 * r194727;
        double r194733 = r194731 - r194732;
        double r194734 = r194708 + r194733;
        double r194735 = r194730 * r194734;
        return r194735;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 15.6

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

  3. Applied flip--15.6

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

  4. Simplified15.1

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

  6. Applied associate-*r/15.1

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

  7. Applied flip--15.1

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

  8. Applied associate-*r/15.1

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

  9. Applied frac-sub15.1

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

  10. Simplified15.1

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

  12. Applied flip3-+15.6

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

  13. Applied associate-/r/15.1

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

  14. Final simplification15.1

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

Reproduce

herbie shell --seed 2020001 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1 (sqrt (* 0.5 (+ 1 (/ 1 (hypot 1 x)))))))