Average Error: 30.6 → 0.8
Time: 7.9s
Precision: binary64
\[\sqrt{{x}^{2} + {x}^{2}}\]
↓
\[\sqrt{\sqrt[3]{\left|x \cdot \sqrt{2}\right|}} \cdot \left(\sqrt{\left|x \cdot \sqrt{2}\right|} \cdot \left|\sqrt[3]{\left|\sqrt{\sqrt{2}} \cdot \left(x \cdot \sqrt{\sqrt{2}}\right)\right|}\right|\right)\]
\sqrt{{x}^{2} + {x}^{2}}
↓
\sqrt{\sqrt[3]{\left|x \cdot \sqrt{2}\right|}} \cdot \left(\sqrt{\left|x \cdot \sqrt{2}\right|} \cdot \left|\sqrt[3]{\left|\sqrt{\sqrt{2}} \cdot \left(x \cdot \sqrt{\sqrt{2}}\right)\right|}\right|\right)
(FPCore (x) :precision binary64 (sqrt (+ (pow x 2.0) (pow x 2.0))))
↓
(FPCore (x)
 :precision binary64
 (*
  (sqrt (cbrt (fabs (* x (sqrt 2.0)))))
  (*
   (sqrt (fabs (* x (sqrt 2.0))))
   (fabs (cbrt (fabs (* (sqrt (sqrt 2.0)) (* x (sqrt (sqrt 2.0))))))))))
double code(double x) {
	return sqrt(pow(x, 2.0) + pow(x, 2.0));
}
↓
double code(double x) {
	return sqrt(cbrt(fabs(x * sqrt(2.0)))) * (sqrt(fabs(x * sqrt(2.0))) * fabs(cbrt(fabs(sqrt(sqrt(2.0)) * (x * sqrt(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 30.6

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

  2. Simplified30.6

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

  4. Applied add-sqr-sqrt_binary6430.8

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

  5. Simplified30.8

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

  7. Applied add-cube-cbrt_binary6431.0

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

  8. Applied sqrt-prod_binary6431.0

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

  9. Simplified31.0

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

  11. Applied pow1_binary6431.0

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

  12. Applied pow1_binary6431.0

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

  13. Applied pow1_binary6431.0

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

  14. Applied pow-prod-down_binary6431.0

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

  15. Applied pow-prod-down_binary6431.0

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

  16. Simplified0.9

    \[\leadsto {\color{blue}{\left(\sqrt{\sqrt[3]{\left|x \cdot \sqrt{2}\right|}} \cdot \left(\sqrt{\left|x \cdot \sqrt{2}\right|} \cdot \left|\sqrt[3]{\left|x \cdot \sqrt{2}\right|}\right|\right)\right)}}^{1}\]
  17. Using strategy rm

  18. Applied add-sqr-sqrt_binary640.8

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

  19. Applied associate-*r*_binary640.8

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

  20. Final simplification0.8

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

Reproduce

herbie shell --seed 2021176 
(FPCore (x)
  :name "sqrt E"
  :precision binary64
  (sqrt (+ (pow x 2.0) (pow x 2.0))))