Average Error: 15.3 → 14.9
Time: 3.9s
Precision: 64
\[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
↓
\[\frac{\frac{\frac{\left(-\left(0.5 \cdot \left(0.5 + 1\right) + 1 \cdot 1\right)\right) \cdot \left(\left(\left(0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(0.5 \cdot 1\right)\right) \cdot \left(0.5 \cdot \left(0.5 + 1\right) + 1 \cdot 1\right)\right) + \left(\left(1 \cdot \left({1}^{3} - {0.5}^{3}\right)\right) \cdot \left(1 \cdot \left({1}^{3} - {0.5}^{3}\right)\right)\right) \cdot \mathsf{hypot}\left(1, x\right)}{\left(\mathsf{hypot}\left(1, x\right) \cdot \left(0.5 \cdot \left(0.5 + 1\right) + 1 \cdot 1\right)\right) \cdot \left(0.5 \cdot \left(0.5 + 1\right) + 1 \cdot 1\right)}}{1 \cdot \left(1 - 0.5\right) + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]
1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}
↓
\frac{\frac{\frac{\left(-\left(0.5 \cdot \left(0.5 + 1\right) + 1 \cdot 1\right)\right) \cdot \left(\left(\left(0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(0.5 \cdot 1\right)\right) \cdot \left(0.5 \cdot \left(0.5 + 1\right) + 1 \cdot 1\right)\right) + \left(\left(1 \cdot \left({1}^{3} - {0.5}^{3}\right)\right) \cdot \left(1 \cdot \left({1}^{3} - {0.5}^{3}\right)\right)\right) \cdot \mathsf{hypot}\left(1, x\right)}{\left(\mathsf{hypot}\left(1, x\right) \cdot \left(0.5 \cdot \left(0.5 + 1\right) + 1 \cdot 1\right)\right) \cdot \left(0.5 \cdot \left(0.5 + 1\right) + 1 \cdot 1\right)}}{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)}}
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) (((double) -(((double) (((double) (0.5 * ((double) (0.5 + 1.0)))) + ((double) (1.0 * 1.0)))))) * ((double) (((double) (((double) (0.5 * ((double) (1.0 / ((double) hypot(1.0, x)))))) * ((double) (0.5 * 1.0)))) * ((double) (((double) (0.5 * ((double) (0.5 + 1.0)))) + ((double) (1.0 * 1.0)))))))) + ((double) (((double) (((double) (1.0 * ((double) (((double) pow(1.0, 3.0)) - ((double) pow(0.5, 3.0)))))) * ((double) (1.0 * ((double) (((double) pow(1.0, 3.0)) - ((double) pow(0.5, 3.0)))))))) * ((double) hypot(1.0, x)))))) / ((double) (((double) (((double) hypot(1.0, x)) * ((double) (((double) (0.5 * ((double) (0.5 + 1.0)))) + ((double) (1.0 * 1.0)))))) * ((double) (((double) (0.5 * ((double) (0.5 + 1.0)))) + ((double) (1.0 * 1.0)))))))) / ((double) (((double) (1.0 * ((double) (1.0 - 0.5)))) + ((double) (0.5 * ((double) (1.0 / ((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 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 flip--14.8

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

  8. Applied associate-*r/14.8

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

  9. Applied associate-*r/14.8

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

  10. Applied flip3--14.8

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

  11. Applied associate-*r/14.8

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

  12. Applied flip3--14.8

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

  13. Applied associate-*r/14.8

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

  14. Applied frac-times14.8

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

  15. Applied frac-sub14.8

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

  16. Simplified14.8

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

  17. Simplified14.9

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

  18. Final simplification14.9

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

Reproduce

herbie shell --seed 2020123 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1 (sqrt (* 0.5 (+ 1 (/ 1 (hypot 1 x)))))))