Average Error: 31.0 → 0.4
Time: 3.2s
Precision: binary64
\[\sqrt{2 \cdot {x}^{2}}\]
↓
\[\sqrt{\sqrt{2}} \cdot \left(\sqrt{\sqrt{\sqrt{2}}} \cdot \left(\left|{x}^{\left(2 \cdot 0.5\right)}\right| \cdot \sqrt{\sqrt{\sqrt{2}}}\right)\right)\]
\sqrt{2 \cdot {x}^{2}}
↓
\sqrt{\sqrt{2}} \cdot \left(\sqrt{\sqrt{\sqrt{2}}} \cdot \left(\left|{x}^{\left(2 \cdot 0.5\right)}\right| \cdot \sqrt{\sqrt{\sqrt{2}}}\right)\right)
(FPCore (x) :precision binary64 (sqrt (* 2.0 (pow x 2.0))))
↓
(FPCore (x)
 :precision binary64
 (*
  (sqrt (sqrt 2.0))
  (*
   (sqrt (sqrt (sqrt 2.0)))
   (* (fabs (pow x (* 2.0 0.5))) (sqrt (sqrt (sqrt 2.0)))))))
double code(double x) {
	return ((double) sqrt(((double) (2.0 * ((double) pow(x, 2.0))))));
}
↓
double code(double x) {
	return ((double) (((double) sqrt(((double) sqrt(2.0)))) * ((double) (((double) sqrt(((double) sqrt(((double) sqrt(2.0)))))) * ((double) (((double) fabs(((double) pow(x, ((double) (2.0 * 0.5)))))) * ((double) sqrt(((double) sqrt(((double) sqrt(2.0))))))))))));
}

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

    \[\sqrt{2 \cdot {x}^{2}}\]
  2. Using strategy rm

  3. Applied sqrt-prodError: 31.1 bits

    \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{{x}^{2}}}\]
  4. Using strategy rm

  5. Applied sqr-powError: 31.1 bits

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

  6. Applied rem-sqrt-squareError: 0.4 bits

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

  8. Applied add-sqr-sqrtError: 0.4 bits

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

  9. Applied sqrt-prodError: 0.6 bits

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

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

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

  11. SimplifiedError: 0.4 bits

    \[\leadsto \sqrt{\sqrt{2}} \cdot \color{blue}{\left(\left|{x}^{\left(2 \cdot 0.5\right)}\right| \cdot \sqrt{\sqrt{2}}\right)}\]
  12. Using strategy rm

  13. Applied add-sqr-sqrtError: 0.4 bits

    \[\leadsto \sqrt{\sqrt{2}} \cdot \left(\left|{x}^{\left(2 \cdot 0.5\right)}\right| \cdot \sqrt{\sqrt{\color{blue}{\sqrt{2} \cdot \sqrt{2}}}}\right)\]

  14. Applied sqrt-prodError: 0.4 bits

    \[\leadsto \sqrt{\sqrt{2}} \cdot \left(\left|{x}^{\left(2 \cdot 0.5\right)}\right| \cdot \sqrt{\color{blue}{\sqrt{\sqrt{2}} \cdot \sqrt{\sqrt{2}}}}\right)\]

  15. Applied sqrt-prodError: 0.4 bits

    \[\leadsto \sqrt{\sqrt{2}} \cdot \left(\left|{x}^{\left(2 \cdot 0.5\right)}\right| \cdot \color{blue}{\left(\sqrt{\sqrt{\sqrt{2}}} \cdot \sqrt{\sqrt{\sqrt{2}}}\right)}\right)\]

  16. Applied associate-*r*Error: 0.4 bits

    \[\leadsto \sqrt{\sqrt{2}} \cdot \color{blue}{\left(\left(\left|{x}^{\left(2 \cdot 0.5\right)}\right| \cdot \sqrt{\sqrt{\sqrt{2}}}\right) \cdot \sqrt{\sqrt{\sqrt{2}}}\right)}\]

  17. Final simplificationError: 0.4 bits

    \[\leadsto \sqrt{\sqrt{2}} \cdot \left(\sqrt{\sqrt{\sqrt{2}}} \cdot \left(\left|{x}^{\left(2 \cdot 0.5\right)}\right| \cdot \sqrt{\sqrt{\sqrt{2}}}\right)\right)\]

Reproduce

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