Average Error: 15.7 → 15.2
Time: 4.1s
Precision: 64
\[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
↓
\[\frac{\frac{\frac{\frac{{\left({\left({\left(1 \cdot \left(1 - 0.5\right)\right)}^{3}\right)}^{3}\right)}^{3} - {\left({\left({\left(0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}^{3}\right)}^{3}}{\left({\left({\left(1 \cdot \left(1 - 0.5\right)\right)}^{3}\right)}^{6} + {\left({\left(0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}^{6}\right) + {\left({\left(1 \cdot \left(1 - 0.5\right)\right)}^{3}\right)}^{3} \cdot {\left({\left(0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}^{3}}}{\left({\left(1 \cdot \left(1 - 0.5\right)\right)}^{6} + {\left(0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{6}\right) + {\left(1 \cdot \left(1 - 0.5\right)\right)}^{3} \cdot {\left(0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}}{\left(0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)} + 1 \cdot \left(1 - 0.5\right)\right) + \left(1 \cdot \left(1 - 0.5\right)\right) \cdot \left(1 \cdot \left(1 - 0.5\right)\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]
1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}
↓
\frac{\frac{\frac{\frac{{\left({\left({\left(1 \cdot \left(1 - 0.5\right)\right)}^{3}\right)}^{3}\right)}^{3} - {\left({\left({\left(0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}^{3}\right)}^{3}}{\left({\left({\left(1 \cdot \left(1 - 0.5\right)\right)}^{3}\right)}^{6} + {\left({\left(0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}^{6}\right) + {\left({\left(1 \cdot \left(1 - 0.5\right)\right)}^{3}\right)}^{3} \cdot {\left({\left(0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}^{3}}}{\left({\left(1 \cdot \left(1 - 0.5\right)\right)}^{6} + {\left(0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{6}\right) + {\left(1 \cdot \left(1 - 0.5\right)\right)}^{3} \cdot {\left(0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}}{\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)} + 1 \cdot \left(1 - 0.5\right)\right) + \left(1 \cdot \left(1 - 0.5\right)\right) \cdot \left(1 \cdot \left(1 - 0.5\right)\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 (((((pow(pow(pow((1.0 * (1.0 - 0.5)), 3.0), 3.0), 3.0) - pow(pow(pow((0.5 * (1.0 / hypot(1.0, x))), 3.0), 3.0), 3.0)) / ((pow(pow((1.0 * (1.0 - 0.5)), 3.0), 6.0) + pow(pow((0.5 * (1.0 / hypot(1.0, x))), 3.0), 6.0)) + (pow(pow((1.0 * (1.0 - 0.5)), 3.0), 3.0) * pow(pow((0.5 * (1.0 / hypot(1.0, x))), 3.0), 3.0)))) / ((pow((1.0 * (1.0 - 0.5)), 6.0) + pow((0.5 * (1.0 / hypot(1.0, x))), 6.0)) + (pow((1.0 * (1.0 - 0.5)), 3.0) * pow((0.5 * (1.0 / hypot(1.0, x))), 3.0)))) / (((0.5 * (1.0 / hypot(1.0, x))) * ((0.5 * (1.0 / hypot(1.0, x))) + (1.0 * (1.0 - 0.5)))) + ((1.0 * (1.0 - 0.5)) * (1.0 * (1.0 - 0.5))))) / (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 15.7

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

  3. Applied flip--15.7

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

  4. Simplified15.2

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

  6. Applied flip3--15.2

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

  7. Simplified15.2

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

  9. Applied flip3--15.2

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

  10. Simplified15.2

    \[\leadsto \frac{\frac{\frac{{\left({\left(1 \cdot \left(1 - 0.5\right)\right)}^{3}\right)}^{3} - {\left({\left(0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}^{3}}{\color{blue}{\left({\left(1 \cdot \left(1 - 0.5\right)\right)}^{6} + {\left(0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{6}\right) + {\left(1 \cdot \left(1 - 0.5\right)\right)}^{3} \cdot {\left(0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}}}{\left(0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)} + 1 \cdot \left(1 - 0.5\right)\right) + \left(1 \cdot \left(1 - 0.5\right)\right) \cdot \left(1 \cdot \left(1 - 0.5\right)\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]
  11. Using strategy rm

  12. Applied flip3--15.2

    \[\leadsto \frac{\frac{\frac{\color{blue}{\frac{{\left({\left({\left(1 \cdot \left(1 - 0.5\right)\right)}^{3}\right)}^{3}\right)}^{3} - {\left({\left({\left(0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}^{3}\right)}^{3}}{{\left({\left(1 \cdot \left(1 - 0.5\right)\right)}^{3}\right)}^{3} \cdot {\left({\left(1 \cdot \left(1 - 0.5\right)\right)}^{3}\right)}^{3} + \left({\left({\left(0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}^{3} \cdot {\left({\left(0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}^{3} + {\left({\left(1 \cdot \left(1 - 0.5\right)\right)}^{3}\right)}^{3} \cdot {\left({\left(0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}^{3}\right)}}}{\left({\left(1 \cdot \left(1 - 0.5\right)\right)}^{6} + {\left(0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{6}\right) + {\left(1 \cdot \left(1 - 0.5\right)\right)}^{3} \cdot {\left(0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}}{\left(0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)} + 1 \cdot \left(1 - 0.5\right)\right) + \left(1 \cdot \left(1 - 0.5\right)\right) \cdot \left(1 \cdot \left(1 - 0.5\right)\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]

  13. Simplified15.2

    \[\leadsto \frac{\frac{\frac{\frac{{\left({\left({\left(1 \cdot \left(1 - 0.5\right)\right)}^{3}\right)}^{3}\right)}^{3} - {\left({\left({\left(0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}^{3}\right)}^{3}}{\color{blue}{\left({\left({\left(1 \cdot \left(1 - 0.5\right)\right)}^{3}\right)}^{6} + {\left({\left(0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}^{6}\right) + {\left({\left(1 \cdot \left(1 - 0.5\right)\right)}^{3}\right)}^{3} \cdot {\left({\left(0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}^{3}}}}{\left({\left(1 \cdot \left(1 - 0.5\right)\right)}^{6} + {\left(0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{6}\right) + {\left(1 \cdot \left(1 - 0.5\right)\right)}^{3} \cdot {\left(0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}}{\left(0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)} + 1 \cdot \left(1 - 0.5\right)\right) + \left(1 \cdot \left(1 - 0.5\right)\right) \cdot \left(1 \cdot \left(1 - 0.5\right)\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]

  14. Final simplification15.2

    \[\leadsto \frac{\frac{\frac{\frac{{\left({\left({\left(1 \cdot \left(1 - 0.5\right)\right)}^{3}\right)}^{3}\right)}^{3} - {\left({\left({\left(0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}^{3}\right)}^{3}}{\left({\left({\left(1 \cdot \left(1 - 0.5\right)\right)}^{3}\right)}^{6} + {\left({\left(0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}^{6}\right) + {\left({\left(1 \cdot \left(1 - 0.5\right)\right)}^{3}\right)}^{3} \cdot {\left({\left(0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}^{3}}}{\left({\left(1 \cdot \left(1 - 0.5\right)\right)}^{6} + {\left(0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{6}\right) + {\left(1 \cdot \left(1 - 0.5\right)\right)}^{3} \cdot {\left(0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}}{\left(0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)} + 1 \cdot \left(1 - 0.5\right)\right) + \left(1 \cdot \left(1 - 0.5\right)\right) \cdot \left(1 \cdot \left(1 - 0.5\right)\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]

Reproduce

herbie shell --seed 2020105 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1 (sqrt (* 0.5 (+ 1 (/ 1 (hypot 1 x)))))))