Average Error: 15.3 → 14.8
Time: 3.6s
Precision: binary64
\[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
↓
\[\frac{\frac{{\left(1 \cdot \left(1 - 0.5\right)\right)}^{3} - {\left(0.5 \cdot \frac{1}{\sqrt[3]{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}}\right)}^{3}}{\left(1 - 0.5\right) \cdot \left(\left(1 \cdot \left(1 - 0.5\right)\right) \cdot 1 + 1 \cdot \frac{0.5 \cdot 1}{\mathsf{hypot}\left(1, x\right)}\right) + \left(0.5 \cdot 0.5\right) \cdot \left(\frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]
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)}^{3} - {\left(0.5 \cdot \frac{1}{\sqrt[3]{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}}\right)}^{3}}{\left(1 - 0.5\right) \cdot \left(\left(1 \cdot \left(1 - 0.5\right)\right) \cdot 1 + 1 \cdot \frac{0.5 \cdot 1}{\mathsf{hypot}\left(1, x\right)}\right) + \left(0.5 \cdot 0.5\right) \cdot \left(\frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \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)}}
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) (((double) pow(((double) (1.0 * ((double) (1.0 - 0.5)))), 3.0)) - ((double) pow(((double) (0.5 * ((double) (1.0 / ((double) cbrt(((double) pow(((double) hypot(1.0, x)), 3.0)))))))), 3.0)))) / ((double) (((double) (((double) (1.0 - 0.5)) * ((double) (((double) (((double) (1.0 * ((double) (1.0 - 0.5)))) * 1.0)) + ((double) (1.0 * ((double) (((double) (0.5 * 1.0)) / ((double) hypot(1.0, x)))))))))) + ((double) (((double) (0.5 * 0.5)) * ((double) (((double) (1.0 / ((double) hypot(1.0, x)))) * ((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 15.3

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

  3. Applied flip--15.3

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

  4. Simplified14.8

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

  6. Applied add-cbrt-cube14.8

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

  7. Simplified14.8

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

  9. Applied flip3--14.8

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

  10. Simplified14.8

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

  11. Final simplification14.8

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

Reproduce

herbie shell --seed 2020161 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))