Average Error: 31.0 → 0.3
Time: 1.9s
Precision: binary64
\[\sqrt{2 \cdot \left(x \cdot x\right)}\]
↓
\[\sqrt{\sqrt{\sqrt{2}}} \cdot \left(\left|x\right| \cdot {\left(2 \cdot \sqrt{2}\right)}^{0.25}\right)\]
\sqrt{2 \cdot \left(x \cdot x\right)}
↓
\sqrt{\sqrt{\sqrt{2}}} \cdot \left(\left|x\right| \cdot {\left(2 \cdot \sqrt{2}\right)}^{0.25}\right)
(FPCore (x) :precision binary64 (sqrt (* 2.0 (* x x))))
↓
(FPCore (x)
 :precision binary64
 (* (sqrt (sqrt (sqrt 2.0))) (* (fabs x) (pow (* 2.0 (sqrt 2.0)) 0.25))))
double code(double x) {
	return ((double) sqrt(((double) (2.0 * ((double) (x * x))))));
}
↓
double code(double x) {
	return ((double) (((double) sqrt(((double) sqrt(((double) sqrt(2.0)))))) * ((double) (((double) fabs(x)) * ((double) pow(((double) (2.0 * ((double) sqrt(2.0)))), 0.25))))));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program Error: 31.0 bits

    \[\sqrt{2 \cdot \left(x \cdot x\right)}\]
  2. Using strategy rm

  3. Applied sqrt-prodError: 31.1 bits

    \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{x \cdot x}}\]

  4. SimplifiedError: 0.4 bits

    \[\leadsto \sqrt{2} \cdot \color{blue}{\left|x\right|}\]
  5. Using strategy rm

  6. Applied add-sqr-sqrtError: 0.4 bits

    \[\leadsto \sqrt{\color{blue}{\sqrt{2} \cdot \sqrt{2}}} \cdot \left|x\right|\]

  7. Applied sqrt-prodError: 0.6 bits

    \[\leadsto \color{blue}{\left(\sqrt{\sqrt{2}} \cdot \sqrt{\sqrt{2}}\right)} \cdot \left|x\right|\]

  8. Applied associate-*l*Error: 0.4 bits

    \[\leadsto \color{blue}{\sqrt{\sqrt{2}} \cdot \left(\sqrt{\sqrt{2}} \cdot \left|x\right|\right)}\]

  9. SimplifiedError: 0.4 bits

    \[\leadsto \sqrt{\sqrt{2}} \cdot \color{blue}{\left(\left|x\right| \cdot \sqrt{\sqrt{2}}\right)}\]
  10. Using strategy rm

  11. Applied add-sqr-sqrtError: 0.4 bits

    \[\leadsto \sqrt{\sqrt{\color{blue}{\sqrt{2} \cdot \sqrt{2}}}} \cdot \left(\left|x\right| \cdot \sqrt{\sqrt{2}}\right)\]

  12. Applied sqrt-prodError: 0.4 bits

    \[\leadsto \sqrt{\color{blue}{\sqrt{\sqrt{2}} \cdot \sqrt{\sqrt{2}}}} \cdot \left(\left|x\right| \cdot \sqrt{\sqrt{2}}\right)\]

  13. Applied sqrt-prodError: 0.4 bits

    \[\leadsto \color{blue}{\left(\sqrt{\sqrt{\sqrt{2}}} \cdot \sqrt{\sqrt{\sqrt{2}}}\right)} \cdot \left(\left|x\right| \cdot \sqrt{\sqrt{2}}\right)\]

  14. Applied associate-*l*Error: 0.4 bits

    \[\leadsto \color{blue}{\sqrt{\sqrt{\sqrt{2}}} \cdot \left(\sqrt{\sqrt{\sqrt{2}}} \cdot \left(\left|x\right| \cdot \sqrt{\sqrt{2}}\right)\right)}\]

  15. SimplifiedError: 0.4 bits

    \[\leadsto \sqrt{\sqrt{\sqrt{2}}} \cdot \color{blue}{\left(\sqrt{\sqrt{2}} \cdot \left(\left|x\right| \cdot \sqrt{\sqrt{\sqrt{2}}}\right)\right)}\]

  16. Taylor expanded around 0 Error: 0.3 bits

    \[\leadsto \sqrt{\sqrt{\sqrt{2}}} \cdot \color{blue}{\left({\left({\left(\sqrt{2}\right)}^{3}\right)}^{0.25} \cdot \left|x\right|\right)}\]

  17. SimplifiedError: 0.3 bits

    \[\leadsto \sqrt{\sqrt{\sqrt{2}}} \cdot \color{blue}{\left(\left|x\right| \cdot {\left(2 \cdot \sqrt{2}\right)}^{0.25}\right)}\]

  18. Final simplificationError: 0.3 bits

    \[\leadsto \sqrt{\sqrt{\sqrt{2}}} \cdot \left(\left|x\right| \cdot {\left(2 \cdot \sqrt{2}\right)}^{0.25}\right)\]

Reproduce

herbie shell --seed 2020203 
(FPCore (x)
  :name "sqrt C"
  :precision binary64
  (sqrt (* 2.0 (* x x))))