Average Error: 15.3 → 14.9
Time: 6.1s
Precision: 64
\[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
↓
\[\frac{1 \cdot \frac{\left({1}^{3} - {0.5}^{3}\right) \cdot \mathsf{hypot}\left(1, x\right) - \left(1 \cdot 1 + \left(0.5 \cdot 0.5 + 1 \cdot 0.5\right)\right) \cdot 0.5}{\mathsf{hypot}\left(1, x\right) \cdot \left(0.5 \cdot \left(0.5 + 1\right) + 1 \cdot 1\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]
1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}
↓
\frac{1 \cdot \frac{\left({1}^{3} - {0.5}^{3}\right) \cdot \mathsf{hypot}\left(1, x\right) - \left(1 \cdot 1 + \left(0.5 \cdot 0.5 + 1 \cdot 0.5\right)\right) \cdot 0.5}{\mathsf{hypot}\left(1, x\right) \cdot \left(0.5 \cdot \left(0.5 + 1\right) + 1 \cdot 1\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}
double f(double x) {
        double r269767 = 1.0;
        double r269768 = 0.5;
        double r269769 = x;
        double r269770 = hypot(r269767, r269769);
        double r269771 = r269767 / r269770;
        double r269772 = r269767 + r269771;
        double r269773 = r269768 * r269772;
        double r269774 = sqrt(r269773);
        double r269775 = r269767 - r269774;
        return r269775;
}
↓
double f(double x) {
        double r269776 = 1.0;
        double r269777 = 3.0;
        double r269778 = pow(r269776, r269777);
        double r269779 = 0.5;
        double r269780 = pow(r269779, r269777);
        double r269781 = r269778 - r269780;
        double r269782 = x;
        double r269783 = hypot(r269776, r269782);
        double r269784 = r269781 * r269783;
        double r269785 = r269776 * r269776;
        double r269786 = r269779 * r269779;
        double r269787 = r269776 * r269779;
        double r269788 = r269786 + r269787;
        double r269789 = r269785 + r269788;
        double r269790 = r269789 * r269779;
        double r269791 = r269784 - r269790;
        double r269792 = r269779 + r269776;
        double r269793 = r269779 * r269792;
        double r269794 = r269793 + r269785;
        double r269795 = r269783 * r269794;
        double r269796 = r269791 / r269795;
        double r269797 = r269776 * r269796;
        double r269798 = r269776 / r269783;
        double r269799 = r269776 + r269798;
        double r269800 = r269779 * r269799;
        double r269801 = sqrt(r269800);
        double r269802 = r269776 + r269801;
        double r269803 = r269797 / r269802;
        return r269803;
}

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 flip3--14.8

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

  7. Applied frac-sub14.9

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

  8. Simplified14.9

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

  9. Final simplification14.9

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

Reproduce

herbie shell --seed 2020033 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1 (sqrt (* 0.5 (+ 1 (/ 1 (hypot 1 x)))))))