Average Error: 16.3 → 15.8
Time: 6.3s
Precision: binary64
\[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
↓
\[\frac{\frac{\left({1}^{3} + \sqrt{{\left(1 \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{3}}\right) \cdot \left({1}^{3} - \sqrt{{\left(1 \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{3}}\right)}{{1}^{4} + 0.5 \cdot \left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(1 \cdot \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + \left(1 + 0.5\right)\right)\right)\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]
1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}
↓
\frac{\frac{\left({1}^{3} + \sqrt{{\left(1 \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{3}}\right) \cdot \left({1}^{3} - \sqrt{{\left(1 \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{3}}\right)}{{1}^{4} + 0.5 \cdot \left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(1 \cdot \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + \left(1 + 0.5\right)\right)\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) (((double) (((double) (((double) pow(1.0, 3.0)) + ((double) sqrt(((double) pow(((double) (1.0 * ((double) (0.5 + ((double) (0.5 / ((double) hypot(1.0, x)))))))), 3.0)))))) * ((double) (((double) pow(1.0, 3.0)) - ((double) sqrt(((double) pow(((double) (1.0 * ((double) (0.5 + ((double) (0.5 / ((double) hypot(1.0, x)))))))), 3.0)))))))) / ((double) (((double) pow(1.0, 4.0)) + ((double) (0.5 * ((double) (((double) (1.0 + ((double) (1.0 / ((double) hypot(1.0, x)))))) * ((double) (1.0 * ((double) (((double) (0.5 / ((double) hypot(1.0, x)))) + ((double) (1.0 + 0.5)))))))))))))) / ((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 16.3

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

  3. Applied flip--16.3

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

  4. Simplified15.8

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

  6. Applied flip3--15.8

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

  7. Simplified15.8

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

  8. Simplified15.8

    \[\leadsto \frac{\frac{{1}^{6} - {\left(1 \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{3}}{\color{blue}{{1}^{4} + 0.5 \cdot \left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(1 \cdot \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + \left(1 + 0.5\right)\right)\right)\right)}}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]
  9. Using strategy rm

  10. Applied add-sqr-sqrt15.8

    \[\leadsto \frac{\frac{{1}^{6} - \color{blue}{\sqrt{{\left(1 \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{3}} \cdot \sqrt{{\left(1 \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{3}}}}{{1}^{4} + 0.5 \cdot \left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(1 \cdot \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + \left(1 + 0.5\right)\right)\right)\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]

  11. Applied add-sqr-sqrt15.8

    \[\leadsto \frac{\frac{{\color{blue}{\left(\sqrt{1} \cdot \sqrt{1}\right)}}^{6} - \sqrt{{\left(1 \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{3}} \cdot \sqrt{{\left(1 \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{3}}}{{1}^{4} + 0.5 \cdot \left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(1 \cdot \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + \left(1 + 0.5\right)\right)\right)\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]

  12. Applied unpow-prod-down15.8

    \[\leadsto \frac{\frac{\color{blue}{{\left(\sqrt{1}\right)}^{6} \cdot {\left(\sqrt{1}\right)}^{6}} - \sqrt{{\left(1 \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{3}} \cdot \sqrt{{\left(1 \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{3}}}{{1}^{4} + 0.5 \cdot \left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(1 \cdot \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + \left(1 + 0.5\right)\right)\right)\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]

  13. Applied difference-of-squares15.8

    \[\leadsto \frac{\frac{\color{blue}{\left({\left(\sqrt{1}\right)}^{6} + \sqrt{{\left(1 \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{3}}\right) \cdot \left({\left(\sqrt{1}\right)}^{6} - \sqrt{{\left(1 \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{3}}\right)}}{{1}^{4} + 0.5 \cdot \left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(1 \cdot \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + \left(1 + 0.5\right)\right)\right)\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]

  14. Simplified15.8

    \[\leadsto \frac{\frac{\color{blue}{\left({1}^{3} + \sqrt{{\left(1 \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{3}}\right)} \cdot \left({\left(\sqrt{1}\right)}^{6} - \sqrt{{\left(1 \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{3}}\right)}{{1}^{4} + 0.5 \cdot \left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(1 \cdot \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + \left(1 + 0.5\right)\right)\right)\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]

  15. Simplified15.8

    \[\leadsto \frac{\frac{\left({1}^{3} + \sqrt{{\left(1 \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{3}}\right) \cdot \color{blue}{\left({1}^{3} - \sqrt{{\left(1 \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{3}}\right)}}{{1}^{4} + 0.5 \cdot \left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(1 \cdot \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + \left(1 + 0.5\right)\right)\right)\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]

  16. Final simplification15.8

    \[\leadsto \frac{\frac{\left({1}^{3} + \sqrt{{\left(1 \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{3}}\right) \cdot \left({1}^{3} - \sqrt{{\left(1 \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{3}}\right)}{{1}^{4} + 0.5 \cdot \left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(1 \cdot \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + \left(1 + 0.5\right)\right)\right)\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]

Reproduce

herbie shell --seed 2020185 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))