Average Error: 15.1 → 14.6
Time: 3.7s
Precision: 64
\[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
↓
\[\frac{\frac{\left(1 \cdot \left(1 - 0.5\right)\right) \cdot \left(1 \cdot \left(1 - 0.5\right)\right) - \left(0.5 \cdot \sqrt[3]{\frac{{1}^{3}}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}}\right) \cdot \left(0.5 \cdot \sqrt[3]{\frac{{1}^{3}}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}}\right)}{1 \cdot \left(1 - 0.5\right) + 0.5 \cdot \sqrt[3]{\frac{{1}^{3}}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}}}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]
1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}
↓
\frac{\frac{\left(1 \cdot \left(1 - 0.5\right)\right) \cdot \left(1 \cdot \left(1 - 0.5\right)\right) - \left(0.5 \cdot \sqrt[3]{\frac{{1}^{3}}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}}\right) \cdot \left(0.5 \cdot \sqrt[3]{\frac{{1}^{3}}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}}\right)}{1 \cdot \left(1 - 0.5\right) + 0.5 \cdot \sqrt[3]{\frac{{1}^{3}}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}}}}{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 * (1.0 - 0.5)) * (1.0 * (1.0 - 0.5))) - ((0.5 * cbrt((pow(1.0, 3.0) / pow(hypot(1.0, x), 3.0)))) * (0.5 * cbrt((pow(1.0, 3.0) / pow(hypot(1.0, x), 3.0)))))) / ((1.0 * (1.0 - 0.5)) + (0.5 * cbrt((pow(1.0, 3.0) / pow(hypot(1.0, x), 3.0)))))) / (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 15.1

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

  6. Applied add-cbrt-cube14.6

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

  7. Applied add-cbrt-cube14.6

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

  8. Applied cbrt-undiv14.6

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

  9. Simplified14.6

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

  11. Applied cube-div14.6

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

  13. Applied flip--14.6

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

  14. Final simplification14.6

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

Reproduce

herbie shell --seed 2020075 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1 (sqrt (* 0.5 (+ 1 (/ 1 (hypot 1 x)))))))