Average Error: 15.3 → 14.8
Time: 4.4s
Precision: 64
\[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
↓
\[\frac{1 \cdot \left(1 - 0.5\right) - 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}}{1 + \frac{\sqrt{0.5 \cdot \left({1}^{3} + {\left(\frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}}{\sqrt{1 \cdot 1 + \left(\frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)} - 1 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}}\]
1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}
↓
\frac{1 \cdot \left(1 - 0.5\right) - 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}}{1 + \frac{\sqrt{0.5 \cdot \left({1}^{3} + {\left(\frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}}{\sqrt{1 \cdot 1 + \left(\frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)} - 1 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}}
double f(double x) {
        double r152978 = 1.0;
        double r152979 = 0.5;
        double r152980 = x;
        double r152981 = hypot(r152978, r152980);
        double r152982 = r152978 / r152981;
        double r152983 = r152978 + r152982;
        double r152984 = r152979 * r152983;
        double r152985 = sqrt(r152984);
        double r152986 = r152978 - r152985;
        return r152986;
}
↓
double f(double x) {
        double r152987 = 1.0;
        double r152988 = 0.5;
        double r152989 = r152987 - r152988;
        double r152990 = r152987 * r152989;
        double r152991 = x;
        double r152992 = hypot(r152987, r152991);
        double r152993 = r152987 / r152992;
        double r152994 = r152988 * r152993;
        double r152995 = r152990 - r152994;
        double r152996 = 3.0;
        double r152997 = pow(r152987, r152996);
        double r152998 = pow(r152993, r152996);
        double r152999 = r152997 + r152998;
        double r153000 = r152988 * r152999;
        double r153001 = sqrt(r153000);
        double r153002 = r152987 * r152987;
        double r153003 = r152993 * r152993;
        double r153004 = r152987 * r152993;
        double r153005 = r153003 - r153004;
        double r153006 = r153002 + r153005;
        double r153007 = sqrt(r153006);
        double r153008 = r153001 / r153007;
        double r153009 = r152987 + r153008;
        double r153010 = r152995 / r153009;
        return r153010;
}

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

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

  6. Applied flip3-+14.8

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

  7. Applied associate-*r/14.8

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

  8. Applied sqrt-div14.8

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

  9. Final simplification14.8

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

Reproduce

herbie shell --seed 2020036 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1 (sqrt (* 0.5 (+ 1 (/ 1 (hypot 1 x)))))))