Average Error: 30.4 → 0.4
Time: 2.7s
Precision: binary64
\[\sqrt{{x}^{2} + {x}^{2}}\]
↓
\[\sqrt[3]{\sqrt{2}} \cdot \left(\left|{x}^{\left(\frac{2}{2}\right)}\right| \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)\right)\]
\sqrt{{x}^{2} + {x}^{2}}
↓
\sqrt[3]{\sqrt{2}} \cdot \left(\left|{x}^{\left(\frac{2}{2}\right)}\right| \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)\right)
(FPCore (x) :precision binary64 (sqrt (+ (pow x 2.0) (pow x 2.0))))
↓
(FPCore (x)
 :precision binary64
 (*
  (cbrt (sqrt 2.0))
  (* (fabs (pow x (/ 2.0 2.0))) (* (cbrt (sqrt 2.0)) (cbrt (sqrt 2.0))))))
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) cbrt(((double) sqrt(2.0)))) * ((double) (((double) fabs(((double) pow(x, (2.0 / 2.0))))) * ((double) (((double) cbrt(((double) sqrt(2.0)))) * ((double) cbrt(((double) sqrt(2.0))))))))));
}

Error

Bits error versus x

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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-prod_binary6430.5

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

  6. Applied sqr-pow_binary6430.5

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

  7. Applied rem-sqrt-square_binary640.4

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

  9. Applied add-cube-cbrt_binary640.4

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

  10. Applied associate-*r*_binary640.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2020204 
(FPCore (x)
  :name "sqrt E"
  :precision binary64
  (sqrt (+ (pow x 2.0) (pow x 2.0))))