Average Error: 15.8 → 15.3
Time: 3.7s
Precision: 64
\[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
↓
\[\frac{\frac{1 \cdot \left(\left({1}^{3} - {0.5}^{3}\right) \cdot \mathsf{hypot}\left(1, x\right) - 0.5 \cdot \left(0.5 \cdot \left(0.5 + 1\right) + 1 \cdot 1\right)\right)}{\mathsf{hypot}\left(1, x\right) \cdot \left(0.5 \cdot \left(0.5 + 1\right) + 1 \cdot 1\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]
1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}
↓
\frac{\frac{1 \cdot \left(\left({1}^{3} - {0.5}^{3}\right) \cdot \mathsf{hypot}\left(1, x\right) - 0.5 \cdot \left(0.5 \cdot \left(0.5 + 1\right) + 1 \cdot 1\right)\right)}{\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 (1.0 - sqrt((0.5 * (1.0 + (1.0 / hypot(1.0, x))))));
}
↓
double code(double x) {
	return (((1.0 * (((pow(1.0, 3.0) - pow(0.5, 3.0)) * hypot(1.0, x)) - (0.5 * ((0.5 * (0.5 + 1.0)) + (1.0 * 1.0))))) / (hypot(1.0, x) * ((0.5 * (0.5 + 1.0)) + (1.0 * 1.0)))) / (1.0 + sqrt((0.5 * (1.0 + (1.0 / 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 15.8

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

  3. Applied flip--15.8

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

  4. Simplified15.3

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

  6. Applied associate-*r/15.3

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

  7. Applied flip3--15.3

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

  8. Applied associate-*r/15.3

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

  9. Applied frac-sub15.3

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

  10. Simplified15.3

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

  11. Simplified15.3

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

  12. Final simplification15.3

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

Reproduce

herbie shell --seed 2020106 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1 (sqrt (* 0.5 (+ 1 (/ 1 (hypot 1 x)))))))