Average Error: 30.7 → 0.4
Time: 8.3s
Precision: binary64
\[\sqrt{{x}^{2} + {x}^{2}}\]
↓
\[\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \left|{x}^{\left(\frac{2}{2}\right)}\right|\right)\]
\sqrt{{x}^{2} + {x}^{2}}
↓
\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \left|{x}^{\left(\frac{2}{2}\right)}\right|\right)
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) cbrt(((double) sqrt(2.0)))) * ((double) cbrt(((double) sqrt(2.0)))))) * ((double) (((double) cbrt(((double) sqrt(2.0)))) * ((double) fabs(((double) pow(x, (2.0 / 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 30.7

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

  2. Simplified30.7

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

  4. Applied sqrt-prod30.9

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

  6. Applied sqr-pow30.9

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

  7. Applied rem-sqrt-square0.4

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

  9. Applied add-cube-cbrt0.4

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

  10. Applied associate-*l*0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2020182 
(FPCore (x)
  :name "sqrt E"
  :precision binary64
  (sqrt (+ (pow x 2.0) (pow x 2.0))))