Average Error: 15.5 → 15.5
Time: 6.8s
Precision: binary64
\[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
↓
\[\frac{\frac{\left(\sqrt{0.125} + {\left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{1.5}\right) \cdot \left(\sqrt{0.125} - {\left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{1.5}\right)}{0.25 + \frac{0.25 + \frac{0.25}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\]
1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}
↓
\frac{\frac{\left(\sqrt{0.125} + {\left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{1.5}\right) \cdot \left(\sqrt{0.125} - {\left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{1.5}\right)}{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)}}}
(FPCore (x)
 :precision binary64
 (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))
↓
(FPCore (x)
 :precision binary64
 (/
  (/
   (*
    (+ (sqrt 0.125) (pow (/ 0.5 (hypot 1.0 x)) 1.5))
    (- (sqrt 0.125) (pow (/ 0.5 (hypot 1.0 x)) 1.5)))
   (+ 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 1.0 - sqrt(0.5 * (1.0 + (1.0 / hypot(1.0, x))));
}
↓
double code(double x) {
	return (((sqrt(0.125) + pow((0.5 / hypot(1.0, x)), 1.5)) * (sqrt(0.125) - pow((0.5 / hypot(1.0, x)), 1.5))) / (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))));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 15.5

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

  2. Simplified15.5

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

  4. Applied flip--_binary64_140015.5

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

  5. Simplified15.0

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

  7. Applied flip3--_binary64_142915.0

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

  8. Simplified15.0

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

  9. Simplified15.0

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

  11. Applied add-sqr-sqrt_binary64_144631.0

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

  12. Applied unpow-prod-down_binary64_150131.0

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

  13. Applied add-sqr-sqrt_binary64_144631.5

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

  14. Applied difference-of-squares_binary64_139431.5

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

  15. Simplified31.5

    \[\leadsto \frac{\frac{\color{blue}{\left(\sqrt{0.125} + {\left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{1.5}\right)} \cdot \left(\sqrt{0.125} - {\left(\sqrt{\frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right)}^{3}\right)}{0.25 + \frac{0.25 + \frac{0.25}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\]

  16. Simplified15.5

    \[\leadsto \frac{\frac{\left(\sqrt{0.125} + {\left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{1.5}\right) \cdot \color{blue}{\left(\sqrt{0.125} - {\left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{1.5}\right)}}{0.25 + \frac{0.25 + \frac{0.25}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\]

  17. Final simplification15.5

    \[\leadsto \frac{\frac{\left(\sqrt{0.125} + {\left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{1.5}\right) \cdot \left(\sqrt{0.125} - {\left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{1.5}\right)}{0.25 + \frac{0.25 + \frac{0.25}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\]

Reproduce

herbie shell --seed 2020281 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))