Average Error: 13.0 → 14.4
Time: 5.4s
Precision: 64
\[1.00000000000000001 \cdot 10^{-150} \lt \left|x\right| \lt 9.99999999999999981 \cdot 10^{149}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\sqrt{0.5 \cdot \left(1 + \frac{1}{\sqrt{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}} \cdot \left(\frac{\sqrt[3]{x}}{\frac{\left|\sqrt[3]{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right|}{\sqrt[3]{x}}} \cdot \frac{\sqrt[3]{x}}{\sqrt{\sqrt[3]{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}}\right)\right)}\]
\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}
↓
\sqrt{0.5 \cdot \left(1 + \frac{1}{\sqrt{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}} \cdot \left(\frac{\sqrt[3]{x}}{\frac{\left|\sqrt[3]{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right|}{\sqrt[3]{x}}} \cdot \frac{\sqrt[3]{x}}{\sqrt{\sqrt[3]{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}}\right)\right)}
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) (1.0 + ((double) (((double) (1.0 / ((double) sqrt(((double) sqrt(((double) (((double) (((double) (4.0 * p)) * p)) + ((double) (x * x)))))))))) * ((double) (((double) (((double) cbrt(x)) / ((double) (((double) fabs(((double) cbrt(((double) sqrt(((double) (((double) (((double) (4.0 * p)) * p)) + ((double) (x * x)))))))))) / ((double) cbrt(x)))))) * ((double) (((double) cbrt(x)) / ((double) sqrt(((double) cbrt(((double) sqrt(((double) (((double) (((double) (4.0 * p)) * p)) + ((double) (x * x))))))))))))))))))))));
}

Error

Bits error versus p

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.0
Target13.0
Herbie14.4
\[\sqrt{0.5 + \frac{\mathsf{copysign}\left(0.5, x\right)}{\mathsf{hypot}\left(1, \frac{2 \cdot p}{x}\right)}}\]

Derivation

  1. Initial program 13.0

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

  3. Applied add-sqr-sqrt13.0

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

  4. Applied sqrt-prod13.9

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

  5. Applied *-un-lft-identity13.9

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

  6. Applied times-frac13.9

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

  8. Applied add-cube-cbrt14.4

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

  9. Applied sqrt-prod14.4

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

  10. Applied add-cube-cbrt14.4

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

  11. Applied times-frac14.5

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

  12. Simplified14.4

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

  13. Final simplification14.4

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

Reproduce

herbie shell --seed 2020114 +o rules:numerics
(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 (/ (* 2 p) x)))))

  (sqrt (* 0.5 (+ 1 (/ x (sqrt (+ (* (* 4 p) p) (* x x))))))))