Average Error: 15.4 → 15.4
Time: 3.6s
Precision: binary64
\[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
↓
\[\frac{\left(\sqrt[3]{\left(\sqrt{0.5} + \sqrt{\frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right) \cdot \left(\sqrt{0.5} - \sqrt{\frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right)} \cdot \sqrt[3]{\left(\sqrt{0.5} + \sqrt{\frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right) \cdot \left(\sqrt{0.5} - \sqrt{\frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right)}\right) \cdot \sqrt[3]{\left(\sqrt{0.5} + \sqrt{\frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right) \cdot \left(\sqrt{0.5} - \sqrt{\frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]
1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}
↓
\frac{\left(\sqrt[3]{\left(\sqrt{0.5} + \sqrt{\frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right) \cdot \left(\sqrt{0.5} - \sqrt{\frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right)} \cdot \sqrt[3]{\left(\sqrt{0.5} + \sqrt{\frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right) \cdot \left(\sqrt{0.5} - \sqrt{\frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right)}\right) \cdot \sqrt[3]{\left(\sqrt{0.5} + \sqrt{\frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right) \cdot \left(\sqrt{0.5} - \sqrt{\frac{0.5}{\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) cbrt(((double) (((double) (((double) sqrt(0.5)) + ((double) sqrt(((double) (0.5 / ((double) hypot(1.0, x)))))))) * ((double) (((double) sqrt(0.5)) - ((double) sqrt(((double) (0.5 / ((double) hypot(1.0, x)))))))))))) * ((double) cbrt(((double) (((double) (((double) sqrt(0.5)) + ((double) sqrt(((double) (0.5 / ((double) hypot(1.0, x)))))))) * ((double) (((double) sqrt(0.5)) - ((double) sqrt(((double) (0.5 / ((double) hypot(1.0, x)))))))))))))) * ((double) cbrt(((double) (((double) (((double) sqrt(0.5)) + ((double) sqrt(((double) (0.5 / ((double) hypot(1.0, x)))))))) * ((double) (((double) sqrt(0.5)) - ((double) sqrt(((double) (0.5 / ((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 15.4

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

  3. Applied flip--15.4

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

  4. Simplified14.9

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

  5. Taylor expanded around 0 14.9

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

  6. Simplified14.9

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

  8. Applied add-sqr-sqrt30.5

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

  9. Applied add-sqr-sqrt15.4

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

  10. Applied difference-of-squares15.4

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

  12. Applied add-cube-cbrt15.4

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

  13. Final simplification15.4

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

Reproduce

herbie shell --seed 2020156 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))