Average Error: 13.4 → 14.9
Time: 5.0s
Precision: binary64
\[1.00000000000000001 \cdot 10^{-150} \lt \left|x\right| \lt 9.99999999999999981 \cdot 10^{149}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\sqrt{0.5 \cdot \sqrt[3]{{\left(1 + \frac{1}{\sqrt{\sqrt[3]{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}} \cdot \left|\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}\right|} \cdot \frac{x}{\sqrt{\sqrt[3]{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}}\right)}^{3}}}\]
\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}
↓
\sqrt{0.5 \cdot \sqrt[3]{{\left(1 + \frac{1}{\sqrt{\sqrt[3]{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}} \cdot \left|\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}\right|} \cdot \frac{x}{\sqrt{\sqrt[3]{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}}\right)}^{3}}}
double code(double p, double x) {
	return ((double) sqrt(((double) (0.5 * ((double) (1.0 + ((double) (x / ((double) sqrt(((double) (((double) (((double) (4.0 * p)) * p)) + ((double) (x * x))))))))))))));
}
↓
double code(double p, double x) {
	return ((double) sqrt(((double) (0.5 * ((double) cbrt(((double) pow(((double) (1.0 + ((double) (((double) (1.0 / ((double) (((double) sqrt(((double) cbrt(((double) sqrt(((double) (((double) (((double) (4.0 * p)) * p)) + ((double) (x * x)))))))))) * ((double) fabs(((double) cbrt(((double) (((double) (((double) (4.0 * p)) * p)) + ((double) (x * x)))))))))))) * ((double) (x / ((double) sqrt(((double) cbrt(((double) sqrt(((double) (((double) (((double) (4.0 * p)) * p)) + ((double) (x * x)))))))))))))))), 3.0))))))));
}

Error

Bits error versus p

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.4
Target13.4
Herbie14.9
\[\sqrt{0.5 + \frac{\mathsf{copysign}\left(0.5, x\right)}{\mathsf{hypot}\left(1, \frac{2 \cdot p}{x}\right)}}\]

Derivation

  1. Initial program 13.4

    \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
  2. Using strategy rm

  3. Applied add-cube-cbrt14.8

    \[\leadsto \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\color{blue}{\left(\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x} \cdot \sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}\right) \cdot \sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}}}}\right)}\]

  4. Applied sqrt-prod14.9

    \[\leadsto \sqrt{0.5 \cdot \left(1 + \frac{x}{\color{blue}{\sqrt{\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x} \cdot \sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \sqrt{\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}}}}\right)}\]

  5. Applied *-un-lft-identity14.9

    \[\leadsto \sqrt{0.5 \cdot \left(1 + \frac{\color{blue}{1 \cdot x}}{\sqrt{\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x} \cdot \sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \sqrt{\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right)}\]

  6. Applied times-frac14.8

    \[\leadsto \sqrt{0.5 \cdot \left(1 + \color{blue}{\frac{1}{\sqrt{\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x} \cdot \sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}}} \cdot \frac{x}{\sqrt{\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}}}}\right)}\]

  7. Simplified14.8

    \[\leadsto \sqrt{0.5 \cdot \left(1 + \color{blue}{\frac{1}{\left|\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}\right| \cdot 1}} \cdot \frac{x}{\sqrt{\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right)}\]
  8. Using strategy rm

  9. Applied add-sqr-sqrt14.8

    \[\leadsto \sqrt{0.5 \cdot \left(1 + \frac{1}{\left|\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}\right| \cdot 1} \cdot \frac{x}{\sqrt{\sqrt[3]{\color{blue}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x} \cdot \sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}}}\right)}\]

  10. Applied cbrt-prod14.8

    \[\leadsto \sqrt{0.5 \cdot \left(1 + \frac{1}{\left|\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}\right| \cdot 1} \cdot \frac{x}{\sqrt{\color{blue}{\sqrt[3]{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \sqrt[3]{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}}}\right)}\]

  11. Applied sqrt-prod14.9

    \[\leadsto \sqrt{0.5 \cdot \left(1 + \frac{1}{\left|\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}\right| \cdot 1} \cdot \frac{x}{\color{blue}{\sqrt{\sqrt[3]{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}} \cdot \sqrt{\sqrt[3]{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}}}\right)}\]

  12. Applied *-un-lft-identity14.9

    \[\leadsto \sqrt{0.5 \cdot \left(1 + \frac{1}{\left|\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}\right| \cdot 1} \cdot \frac{\color{blue}{1 \cdot x}}{\sqrt{\sqrt[3]{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}} \cdot \sqrt{\sqrt[3]{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}}\right)}\]

  13. Applied times-frac14.9

    \[\leadsto \sqrt{0.5 \cdot \left(1 + \frac{1}{\left|\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}\right| \cdot 1} \cdot \color{blue}{\left(\frac{1}{\sqrt{\sqrt[3]{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}} \cdot \frac{x}{\sqrt{\sqrt[3]{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}}\right)}\right)}\]

  14. Applied associate-*r*15.0

    \[\leadsto \sqrt{0.5 \cdot \left(1 + \color{blue}{\left(\frac{1}{\left|\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}\right| \cdot 1} \cdot \frac{1}{\sqrt{\sqrt[3]{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}}\right) \cdot \frac{x}{\sqrt{\sqrt[3]{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}}}\right)}\]

  15. Simplified14.9

    \[\leadsto \sqrt{0.5 \cdot \left(1 + \color{blue}{\frac{1}{\sqrt{\sqrt[3]{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}} \cdot \left|\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}\right|}} \cdot \frac{x}{\sqrt{\sqrt[3]{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}}\right)}\]
  16. Using strategy rm

  17. Applied add-cbrt-cube14.9

    \[\leadsto \sqrt{0.5 \cdot \color{blue}{\sqrt[3]{\left(\left(1 + \frac{1}{\sqrt{\sqrt[3]{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}} \cdot \left|\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}\right|} \cdot \frac{x}{\sqrt{\sqrt[3]{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}}\right) \cdot \left(1 + \frac{1}{\sqrt{\sqrt[3]{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}} \cdot \left|\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}\right|} \cdot \frac{x}{\sqrt{\sqrt[3]{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}}\right)\right) \cdot \left(1 + \frac{1}{\sqrt{\sqrt[3]{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}} \cdot \left|\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}\right|} \cdot \frac{x}{\sqrt{\sqrt[3]{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}}\right)}}}\]

  18. Simplified14.9

    \[\leadsto \sqrt{0.5 \cdot \sqrt[3]{\color{blue}{{\left(1 + \frac{1}{\sqrt{\sqrt[3]{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}} \cdot \left|\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}\right|} \cdot \frac{x}{\sqrt{\sqrt[3]{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}}\right)}^{3}}}}\]

  19. Final simplification14.9

    \[\leadsto \sqrt{0.5 \cdot \sqrt[3]{{\left(1 + \frac{1}{\sqrt{\sqrt[3]{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}} \cdot \left|\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}\right|} \cdot \frac{x}{\sqrt{\sqrt[3]{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}}\right)}^{3}}}\]

Reproduce

herbie shell --seed 2020153 
(FPCore (p x)
  :name "Given's Rotation SVD example"
  :precision binary64
  :pre (< 1e-150 (fabs x) 1e+150)

  :herbie-target
  (sqrt (+ 0.5 (/ (copysign 0.5 x) (hypot 1.0 (/ (* 2.0 p) x)))))

  (sqrt (* 0.5 (+ 1.0 (/ x (sqrt (+ (* (* 4.0 p) p) (* x x))))))))