Average Error: 30.5 → 0.4
Time: 3.6s
Precision: binary64
\[\sqrt{{x}^{2} + {x}^{2}}\]
↓
\[\sqrt{\sqrt{\sqrt{2}}} \cdot \left(\sqrt{\sqrt{\sqrt{2}}} \cdot \left(\sqrt{\sqrt{2}} \cdot \left|x\right|\right)\right)\]
\sqrt{{x}^{2} + {x}^{2}}
↓
\sqrt{\sqrt{\sqrt{2}}} \cdot \left(\sqrt{\sqrt{\sqrt{2}}} \cdot \left(\sqrt{\sqrt{2}} \cdot \left|x\right|\right)\right)
(FPCore (x) :precision binary64 (sqrt (+ (pow x 2.0) (pow x 2.0))))
↓
(FPCore (x)
 :precision binary64
 (*
  (sqrt (sqrt (sqrt 2.0)))
  (* (sqrt (sqrt (sqrt 2.0))) (* (sqrt (sqrt 2.0)) (fabs x)))))
double code(double x) {
	return sqrt(pow(x, 2.0) + pow(x, 2.0));
}
↓
double code(double x) {
	return sqrt(sqrt(sqrt(2.0))) * (sqrt(sqrt(sqrt(2.0))) * (sqrt(sqrt(2.0)) * fabs(x)));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 30.5

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

  2. Simplified30.5

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

  4. Applied add-sqr-sqrt_binary6430.7

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

  6. Applied sqrt-prod_binary6430.7

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

  7. Applied sqrt-prod_binary6430.6

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

  8. Applied associate-*l*_binary6430.6

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

  9. Simplified30.6

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

  11. Applied sqrt-prod_binary6430.6

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

  12. Applied sqrt-prod_binary6430.7

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

  13. Applied associate-*l*_binary6430.7

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

  14. Simplified0.4

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

  16. Applied add-sqr-sqrt_binary640.4

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

  17. Applied associate-*l*_binary640.4

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

  18. Simplified0.4

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

  19. Final simplification0.4

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

Reproduce

herbie shell --seed 2021045 
(FPCore (x)
  :name "sqrt E"
  :precision binary64
  (sqrt (+ (pow x 2.0) (pow x 2.0))))