Average Error: 15.1 → 14.6
Time: 3.3s
Precision: 64
\[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
↓
\[\frac{0.5 - 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}}{\left(\sqrt{0.5} \cdot \sqrt{1 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)} + 1}\right) \cdot \left(\sqrt{0.5} \cdot \sqrt{1 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)} + 1}\right) - 1 \cdot 1} \cdot \left(\sqrt{0.5} \cdot \sqrt{1 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)} + 1} - 1\right)\]
1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}
↓
\frac{0.5 - 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}}{\left(\sqrt{0.5} \cdot \sqrt{1 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)} + 1}\right) \cdot \left(\sqrt{0.5} \cdot \sqrt{1 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)} + 1}\right) - 1 \cdot 1} \cdot \left(\sqrt{0.5} \cdot \sqrt{1 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)} + 1} - 1\right)
double code(double x) {
	return (1.0 - sqrt((0.5 * (1.0 + (1.0 / hypot(1.0, x))))));
}
↓
double code(double x) {
	return (((0.5 - (0.5 * (1.0 / hypot(1.0, x)))) / (((sqrt(0.5) * sqrt(((1.0 * (1.0 / hypot(1.0, x))) + 1.0))) * (sqrt(0.5) * sqrt(((1.0 * (1.0 / hypot(1.0, x))) + 1.0)))) - (1.0 * 1.0))) * ((sqrt(0.5) * sqrt(((1.0 * (1.0 / hypot(1.0, x))) + 1.0))) - 1.0));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 15.1

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

  3. Applied flip--15.1

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

  4. Simplified14.6

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

  5. Taylor expanded around 0 14.6

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

  7. Applied flip-+14.6

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

  8. Applied associate-/r/14.6

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

  9. Final simplification14.6

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

Reproduce

herbie shell --seed 2020078 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1 (sqrt (* 0.5 (+ 1 (/ 1 (hypot 1 x)))))))