Average Error: 15.5 → 15.0
Time: 3.9s
Precision: 64
\[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
↓
\[\frac{\frac{e^{\log \left(\left(1 \cdot \left(1 - 0.5\right)\right) \cdot \left(1 \cdot \left(1 - 0.5\right)\right) - \left(0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}}{1 \cdot \left(1 - 0.5\right) + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]
1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}
↓
\frac{\frac{e^{\log \left(\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)\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) exp(((double) log(((double) (((double) (((double) (1.0 * ((double) (1.0 - 0.5)))) * ((double) (1.0 * ((double) (1.0 - 0.5)))))) - ((double) (((double) (0.5 * ((double) (1.0 / ((double) hypot(1.0, x)))))) * ((double) (0.5 * ((double) (1.0 / ((double) hypot(1.0, x)))))))))))))) / ((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 15.5

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

  3. Applied flip--15.5

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

  4. Simplified15.0

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

  6. Applied flip--15.0

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

  8. Applied add-exp-log15.0

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

  9. Final simplification15.0

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

Reproduce

herbie shell --seed 2020130 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))