Average Error: 15.1 → 14.6
Time: 3.6s
Precision: binary64
\[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
↓
\[\frac{\frac{\frac{3.814697265625 \cdot 10^{-06} - {\left({\left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}^{6}}{0.000244140625 + \left({\left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{12} + 0.015625 \cdot \frac{0.015625}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{6}}\right)}}{\left({\left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3} + 0.125\right) \cdot \left(0.25 + \frac{0.25 + \frac{0.25}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)}\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\]
1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}
↓
\frac{\frac{\frac{3.814697265625 \cdot 10^{-06} - {\left({\left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}^{6}}{0.000244140625 + \left({\left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{12} + 0.015625 \cdot \frac{0.015625}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{6}}\right)}}{\left({\left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3} + 0.125\right) \cdot \left(0.25 + \frac{0.25 + \frac{0.25}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)}\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}
(FPCore (x)
 :precision binary64
 (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))
↓
(FPCore (x)
 :precision binary64
 (/
  (/
   (/
    (- 3.814697265625e-06 (pow (pow (/ 0.5 (hypot 1.0 x)) 3.0) 6.0))
    (+
     0.000244140625
     (+
      (pow (/ 0.5 (hypot 1.0 x)) 12.0)
      (* 0.015625 (/ 0.015625 (pow (hypot 1.0 x) 6.0))))))
   (*
    (+ (pow (/ 0.5 (hypot 1.0 x)) 3.0) 0.125)
    (+ 0.25 (/ (+ 0.25 (/ 0.25 (hypot 1.0 x))) (hypot 1.0 x)))))
  (+ 1.0 (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 x)))))))
double code(double x) {
	return ((double) (1.0 - ((double) sqrt(((double) (0.5 * ((double) (1.0 + (1.0 / ((double) hypot(1.0, x)))))))))));
}
↓
double code(double x) {
	return (((((double) (3.814697265625e-06 - ((double) pow(((double) pow((0.5 / ((double) hypot(1.0, x))), 3.0)), 6.0)))) / ((double) (0.000244140625 + ((double) (((double) pow((0.5 / ((double) hypot(1.0, x))), 12.0)) + ((double) (0.015625 * (0.015625 / ((double) pow(((double) hypot(1.0, x)), 6.0)))))))))) / ((double) (((double) (((double) pow((0.5 / ((double) hypot(1.0, x))), 3.0)) + 0.125)) * ((double) (0.25 + (((double) (0.25 + (0.25 / ((double) hypot(1.0, x))))) / ((double) hypot(1.0, x)))))))) / ((double) (1.0 + ((double) sqrt(((double) (0.5 + (0.5 / ((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 15.1

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

  2. Simplified15.1

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

  4. Applied flip--_binary6415.1

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

  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--_binary6414.6

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

  8. Simplified14.6

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

  9. Simplified14.6

    \[\leadsto \frac{\frac{0.125 - {\left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{\color{blue}{0.25 + \frac{0.25 + \frac{0.25}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)}}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\]
  10. Using strategy rm

  11. Applied flip--_binary6414.6

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

  12. Applied associate-/l/_binary6414.6

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

  13. Simplified14.6

    \[\leadsto \frac{\frac{0.125 \cdot 0.125 - {\left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3} \cdot {\left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{\color{blue}{\left(0.125 + {\left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right) \cdot \left(0.25 + \frac{0.25 + \frac{0.25}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)}\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\]
  14. Using strategy rm

  15. Applied flip3--_binary6414.6

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

  16. Simplified14.6

    \[\leadsto \frac{\frac{\frac{\color{blue}{3.814697265625 \cdot 10^{-06} - {\left({\left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}^{6}}}{\left(0.125 \cdot 0.125\right) \cdot \left(0.125 \cdot 0.125\right) + \left(\left({\left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3} \cdot {\left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right) \cdot \left({\left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3} \cdot {\left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right) + \left(0.125 \cdot 0.125\right) \cdot \left({\left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3} \cdot {\left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)\right)}}{\left(0.125 + {\left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right) \cdot \left(0.25 + \frac{0.25 + \frac{0.25}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)}\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\]

  17. Simplified14.6

    \[\leadsto \frac{\frac{\frac{3.814697265625 \cdot 10^{-06} - {\left({\left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}^{6}}{\color{blue}{0.000244140625 + \left({\left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{12} + 0.015625 \cdot \frac{0.015625}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{6}}\right)}}}{\left(0.125 + {\left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right) \cdot \left(0.25 + \frac{0.25 + \frac{0.25}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)}\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\]

  18. Final simplification14.6

    \[\leadsto \frac{\frac{\frac{3.814697265625 \cdot 10^{-06} - {\left({\left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}^{6}}{0.000244140625 + \left({\left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{12} + 0.015625 \cdot \frac{0.015625}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{6}}\right)}}{\left({\left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3} + 0.125\right) \cdot \left(0.25 + \frac{0.25 + \frac{0.25}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)}\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\]

Reproduce

herbie shell --seed 2020219 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))