Average Error: 30.3 → 0.4
Time: 3.1s
Precision: binary64
\[\sqrt{2 \cdot {x}^{2}}\]
↓
\[\sqrt{\sqrt{\sqrt{2}}} \cdot \left(\sqrt{\sqrt{2}} \cdot \left(\sqrt{\sqrt{\sqrt{2}}} \cdot \left|{x}^{\left(\frac{2}{2}\right)}\right|\right)\right)\]
\sqrt{2 \cdot {x}^{2}}
↓
\sqrt{\sqrt{\sqrt{2}}} \cdot \left(\sqrt{\sqrt{2}} \cdot \left(\sqrt{\sqrt{\sqrt{2}}} \cdot \left|{x}^{\left(\frac{2}{2}\right)}\right|\right)\right)
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(((double) sqrt(2.0)))))) * ((double) (((double) sqrt(((double) sqrt(2.0)))) * ((double) (((double) sqrt(((double) sqrt(((double) sqrt(2.0)))))) * ((double) fabs(((double) pow(x, ((double) (2.0 / 2.0))))))))))));
}

Error

Bits error versus x

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Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 30.3

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

  3. Applied sqrt-prod30.5

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

  5. Applied sqr-pow30.5

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

  6. Applied rem-sqrt-square0.4

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

  8. Applied add-sqr-sqrt0.4

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

  9. Applied sqrt-prod0.6

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

  10. Applied associate-*l*0.4

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

  11. Simplified0.4

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

  13. Applied add-sqr-sqrt0.4

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

  14. Applied sqrt-prod0.4

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

  15. Applied sqrt-prod0.4

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

  16. Applied associate-*l*0.4

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

  17. Simplified0.4

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

  18. Final simplification0.4

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

Reproduce

herbie shell --seed 2020178 
(FPCore (x)
  :name "sqrt D"
  :precision binary64
  (sqrt (* 2.0 (pow x 2.0))))