Average Error: 15.6 → 15.1
Time: 10.8s
Precision: 64
\[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
↓
\[\frac{1}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}} \cdot \left(\left(1 - 0.5\right) - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)\]
1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}
↓
\frac{1}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}} \cdot \left(\left(1 - 0.5\right) - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)
double f(double x) {
        double r116862 = 1.0;
        double r116863 = 0.5;
        double r116864 = x;
        double r116865 = hypot(r116862, r116864);
        double r116866 = r116862 / r116865;
        double r116867 = r116862 + r116866;
        double r116868 = r116863 * r116867;
        double r116869 = sqrt(r116868);
        double r116870 = r116862 - r116869;
        return r116870;
}
↓
double f(double x) {
        double r116871 = 1.0;
        double r116872 = 0.5;
        double r116873 = x;
        double r116874 = hypot(r116871, r116873);
        double r116875 = r116871 / r116874;
        double r116876 = r116871 + r116875;
        double r116877 = r116872 * r116876;
        double r116878 = sqrt(r116877);
        double r116879 = r116871 + r116878;
        double r116880 = r116871 / r116879;
        double r116881 = r116871 - r116872;
        double r116882 = r116872 / r116874;
        double r116883 = r116881 - r116882;
        double r116884 = r116880 * r116883;
        return r116884;
}

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 add-cube-cbrt15.2

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

  7. Final simplification15.1

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

Reproduce

herbie shell --seed 2019291 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1 (sqrt (* 0.5 (+ 1 (/ 1 (hypot 1 x)))))))