Average Error: 30.4 → 0.3
Time: 3.0s
Precision: binary64
\[\sqrt{{x}^{2} + {x}^{2}}\]
↓
\[\left(\left|{x}^{\left(2 \cdot \frac{1}{2}\right)}\right| \cdot \left(\sqrt{\sqrt{2}} \cdot \left|\sqrt[3]{\sqrt{2}}\right|\right)\right) \cdot \sqrt{\sqrt[3]{\sqrt{2}}}\]
\sqrt{{x}^{2} + {x}^{2}}
↓
\left(\left|{x}^{\left(2 \cdot \frac{1}{2}\right)}\right| \cdot \left(\sqrt{\sqrt{2}} \cdot \left|\sqrt[3]{\sqrt{2}}\right|\right)\right) \cdot \sqrt{\sqrt[3]{\sqrt{2}}}
double code(double x) {
	return ((double) sqrt(((double) (((double) pow(x, 2.0)) + ((double) pow(x, 2.0))))));
}
↓
double code(double x) {
	return ((double) (((double) (((double) fabs(((double) pow(x, ((double) (2.0 * 0.5)))))) * ((double) (((double) sqrt(((double) sqrt(2.0)))) * ((double) fabs(((double) cbrt(((double) sqrt(2.0)))))))))) * ((double) sqrt(((double) cbrt(((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 30.4

    \[\sqrt{{x}^{2} + {x}^{2}}\]

  2. Simplified30.4

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

  4. Applied sqrt-prod30.5

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

  6. Applied sqr-pow30.5

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

  7. Applied rem-sqrt-square0.4

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

  9. Applied add-sqr-sqrt0.4

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

  10. Applied sqrt-prod0.6

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

  11. Applied associate-*r*0.4

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

  12. Simplified0.4

    \[\leadsto \color{blue}{\left(\left|{x}^{\left(2 \cdot \frac{1}{2}\right)}\right| \cdot \sqrt{\sqrt{2}}\right)} \cdot \sqrt{\sqrt{2}}\]
  13. Using strategy rm

  14. Applied add-cube-cbrt0.4

    \[\leadsto \left(\left|{x}^{\left(2 \cdot \frac{1}{2}\right)}\right| \cdot \sqrt{\sqrt{2}}\right) \cdot \sqrt{\color{blue}{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \sqrt[3]{\sqrt{2}}}}\]

  15. Applied sqrt-prod0.7

    \[\leadsto \left(\left|{x}^{\left(2 \cdot \frac{1}{2}\right)}\right| \cdot \sqrt{\sqrt{2}}\right) \cdot \color{blue}{\left(\sqrt{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}} \cdot \sqrt{\sqrt[3]{\sqrt{2}}}\right)}\]

  16. Applied associate-*r*0.4

    \[\leadsto \color{blue}{\left(\left(\left|{x}^{\left(2 \cdot \frac{1}{2}\right)}\right| \cdot \sqrt{\sqrt{2}}\right) \cdot \sqrt{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}\right) \cdot \sqrt{\sqrt[3]{\sqrt{2}}}}\]

  17. Simplified0.3

    \[\leadsto \color{blue}{\left(\left|{x}^{\left(2 \cdot \frac{1}{2}\right)}\right| \cdot \left(\sqrt{\sqrt{2}} \cdot \left|\sqrt[3]{\sqrt{2}}\right|\right)\right)} \cdot \sqrt{\sqrt[3]{\sqrt{2}}}\]

  18. Final simplification0.3

    \[\leadsto \left(\left|{x}^{\left(2 \cdot \frac{1}{2}\right)}\right| \cdot \left(\sqrt{\sqrt{2}} \cdot \left|\sqrt[3]{\sqrt{2}}\right|\right)\right) \cdot \sqrt{\sqrt[3]{\sqrt{2}}}\]

Reproduce

herbie shell --seed 2020185 
(FPCore (x)
  :name "sqrt E"
  :precision binary64
  (sqrt (+ (pow x 2.0) (pow x 2.0))))