Average Error: 16.4 → 15.9
Time: 1.9m
Precision: 64
\[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
↓
\[\frac{\frac{\left(1 \cdot \left({1}^{3} - {0.5}^{3}\right)\right) \cdot \sqrt{\mathsf{hypot}\left(1, x\right)} - \left(1 \cdot 1 + \left(0.5 \cdot 0.5 + 1 \cdot 0.5\right)\right) \cdot \left(0.5 \cdot \frac{1}{\sqrt{\mathsf{hypot}\left(1, x\right)}}\right)}{\sqrt{\mathsf{hypot}\left(1, x\right)} \cdot \left(0.5 \cdot \left(0.5 + 1\right) + 1 \cdot 1\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]
1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}
↓
\frac{\frac{\left(1 \cdot \left({1}^{3} - {0.5}^{3}\right)\right) \cdot \sqrt{\mathsf{hypot}\left(1, x\right)} - \left(1 \cdot 1 + \left(0.5 \cdot 0.5 + 1 \cdot 0.5\right)\right) \cdot \left(0.5 \cdot \frac{1}{\sqrt{\mathsf{hypot}\left(1, x\right)}}\right)}{\sqrt{\mathsf{hypot}\left(1, x\right)} \cdot \left(0.5 \cdot \left(0.5 + 1\right) + 1 \cdot 1\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) (1.0 * ((double) (((double) pow(1.0, 3.0)) - ((double) pow(0.5, 3.0)))))) * ((double) sqrt(((double) hypot(1.0, x)))))) - ((double) (((double) (((double) (1.0 * 1.0)) + ((double) (((double) (0.5 * 0.5)) + ((double) (1.0 * 0.5)))))) * ((double) (0.5 * ((double) (1.0 / ((double) sqrt(((double) hypot(1.0, x)))))))))))) / ((double) (((double) sqrt(((double) hypot(1.0, x)))) * ((double) (((double) (0.5 * ((double) (0.5 + 1.0)))) + ((double) (1.0 * 1.0)))))))) / ((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 16.4

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

  3. Applied flip--16.4

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

  4. Simplified15.9

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

  6. Applied add-sqr-sqrt15.9

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

  7. Applied associate-/r*15.9

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

  8. Applied associate-*r/15.9

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

  9. Applied flip3--15.9

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

  10. Applied associate-*r/15.9

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

  11. Applied frac-sub15.9

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

  12. Simplified15.9

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

  13. Final simplification15.9

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

Reproduce

herbie shell --seed 2020113 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1 (sqrt (* 0.5 (+ 1 (/ 1 (hypot 1 x)))))))