Average Error: 16.0 → 15.5
Time: 7.1s
Precision: binary64
\[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
↓
\[\frac{1 \cdot \frac{\frac{{\left(1 - 0.5\right)}^{6} - {\left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{6}}{{\left(1 - 0.5\right)}^{3} + {\left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}}{\left(1 - 0.5\right) \cdot \left(1 - 0.5\right) + \frac{0.5}{\mathsf{hypot}\left(1, x\right)} \cdot \left(\left(1 - 0.5\right) + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]
1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}
↓
\frac{1 \cdot \frac{\frac{{\left(1 - 0.5\right)}^{6} - {\left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{6}}{{\left(1 - 0.5\right)}^{3} + {\left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}}{\left(1 - 0.5\right) \cdot \left(1 - 0.5\right) + \frac{0.5}{\mathsf{hypot}\left(1, x\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)}}
double code(double x) {
	return ((double) (1.0 - ((double) sqrt(((double) (0.5 * ((double) (1.0 + ((double) (1.0 / ((double) hypot(1.0, x))))))))))));
}
↓
double code(double x) {
	return ((double) (((double) (1.0 * ((double) (((double) (((double) (((double) pow(((double) (1.0 - 0.5)), 6.0)) - ((double) pow(((double) (0.5 / ((double) hypot(1.0, x)))), 6.0)))) / ((double) (((double) pow(((double) (1.0 - 0.5)), 3.0)) + ((double) pow(((double) (0.5 / ((double) hypot(1.0, x)))), 3.0)))))) / ((double) (((double) (((double) (1.0 - 0.5)) * ((double) (1.0 - 0.5)))) + ((double) (((double) (0.5 / ((double) hypot(1.0, x)))) * ((double) (((double) (1.0 - 0.5)) + ((double) (0.5 / ((double) hypot(1.0, x)))))))))))))) / ((double) (1.0 + ((double) sqrt(((double) (0.5 * ((double) (1.0 + ((double) (1.0 / ((double) hypot(1.0, x))))))))))))));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 16.0

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

  3. Applied flip--16.0

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

  4. Simplified15.5

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

  6. Applied flip3--15.5

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

  7. Simplified15.5

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

  9. Applied flip--15.5

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

  10. Simplified15.5

    \[\leadsto \frac{1 \cdot \frac{\frac{\color{blue}{{\left(1 - 0.5\right)}^{6} - {\left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{6}}}{{\left(1 - 0.5\right)}^{3} + {\left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}}{\left(1 - 0.5\right) \cdot \left(1 - 0.5\right) + \frac{0.5}{\mathsf{hypot}\left(1, x\right)} \cdot \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + \left(1 - 0.5\right)\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]

  11. Final simplification15.5

    \[\leadsto \frac{1 \cdot \frac{\frac{{\left(1 - 0.5\right)}^{6} - {\left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{6}}{{\left(1 - 0.5\right)}^{3} + {\left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}}{\left(1 - 0.5\right) \cdot \left(1 - 0.5\right) + \frac{0.5}{\mathsf{hypot}\left(1, x\right)} \cdot \left(\left(1 - 0.5\right) + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]

Reproduce

herbie shell --seed 2020184 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))