Average Error: 15.0 → 14.6
Time: 9.2s
Precision: binary64
\[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
↓
\[e^{\log \left(\frac{0.125 - \frac{0.125}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}}{\left(1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right) \cdot \left(0.25 + \frac{0.25 + \frac{0.25}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)}\right)}\right)}\]
1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}
↓
e^{\log \left(\frac{0.125 - \frac{0.125}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}}{\left(1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right) \cdot \left(0.25 + \frac{0.25 + \frac{0.25}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)}\right)}\right)}
(FPCore (x)
 :precision binary64
 (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))
↓
(FPCore (x)
 :precision binary64
 (exp
  (log
   (/
    (- 0.125 (/ 0.125 (pow (hypot 1.0 x) 3.0)))
    (*
     (+ 1.0 (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 x)))))
     (+ 0.25 (/ (+ 0.25 (/ 0.25 (hypot 1.0 x))) (hypot 1.0 x))))))))
double code(double x) {
	return 1.0 - sqrt(0.5 * (1.0 + (1.0 / hypot(1.0, x))));
}
↓
double code(double x) {
	return exp(log((0.125 - (0.125 / pow(hypot(1.0, x), 3.0))) / ((1.0 + sqrt(0.5 + (0.5 / hypot(1.0, x)))) * (0.25 + ((0.25 + (0.25 / hypot(1.0, x))) / 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 15.0

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

  2. Simplified15.0

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

  4. Applied flip--_binary64_141715.1

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

  5. Simplified14.6

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

  7. Applied flip3--_binary64_144614.6

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

  8. Applied associate-/l/_binary64_138914.6

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

  9. Simplified14.6

    \[\leadsto \frac{{0.5}^{3} - {\left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{\color{blue}{\left(1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right) \cdot \left(0.25 + \frac{0.25 + \frac{0.25}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)}\right)}}\]
  10. Using strategy rm

  11. Applied add-exp-log_binary64_148014.6

    \[\leadsto \frac{{0.5}^{3} - {\left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{\left(1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right) \cdot \color{blue}{e^{\log \left(0.25 + \frac{0.25 + \frac{0.25}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)}\right)}}}\]

  12. Applied add-exp-log_binary64_148014.6

    \[\leadsto \frac{{0.5}^{3} - {\left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{\color{blue}{e^{\log \left(1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right)}} \cdot e^{\log \left(0.25 + \frac{0.25 + \frac{0.25}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)}\right)}}\]

  13. Applied prod-exp_binary64_149114.6

    \[\leadsto \frac{{0.5}^{3} - {\left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{\color{blue}{e^{\log \left(1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right) + \log \left(0.25 + \frac{0.25 + \frac{0.25}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)}\right)}}}\]

  14. Applied add-exp-log_binary64_148015.0

    \[\leadsto \frac{\color{blue}{e^{\log \left({0.5}^{3} - {\left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}}}{e^{\log \left(1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right) + \log \left(0.25 + \frac{0.25 + \frac{0.25}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)}\right)}}\]

  15. Applied div-exp_binary64_149315.0

    \[\leadsto \color{blue}{e^{\log \left({0.5}^{3} - {\left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right) - \left(\log \left(1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right) + \log \left(0.25 + \frac{0.25 + \frac{0.25}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)}\right)\right)}}\]

  16. Simplified14.6

    \[\leadsto e^{\color{blue}{\log \left(\frac{0.125 - \frac{0.125}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}}{\left(1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right) \cdot \left(0.25 + \frac{0.25 + \frac{0.25}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)}\right)}\right)}}\]

  17. Final simplification14.6

    \[\leadsto e^{\log \left(\frac{0.125 - \frac{0.125}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}}{\left(1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right) \cdot \left(0.25 + \frac{0.25 + \frac{0.25}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)}\right)}\right)}\]

Reproduce

herbie shell --seed 2021028 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))