Average Error: 12.9 → 13.1
Time: 4.8s
Precision: 64
\[1.00000000000000001 \cdot 10^{-150} \lt \left|x\right| \lt 9.99999999999999981 \cdot 10^{149}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\sqrt{0.5 \cdot \left(1 + x \cdot \frac{\sqrt{1}}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
\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 + x \cdot \frac{\sqrt{1}}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}
double code(double p, double x) {
	return sqrt((0.5 * (1.0 + (x / sqrt((((4.0 * p) * p) + (x * x)))))));
}
↓
double code(double p, double x) {
	return sqrt((0.5 * (1.0 + (x * (sqrt(1.0) / sqrt((((4.0 * p) * p) + (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

Original12.9
Target12.9
Herbie13.1
\[\sqrt{0.5 + \frac{\mathsf{copysign}\left(0.5, x\right)}{\mathsf{hypot}\left(1, \frac{2 \cdot p}{x}\right)}}\]

Derivation

  1. Initial program 12.9

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

  3. Applied div-inv13.1

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

  5. Applied *-un-lft-identity13.1

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

  6. Applied sqrt-prod13.1

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

  7. Applied add-sqr-sqrt13.1

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

  8. Applied times-frac13.1

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

  9. Applied associate-*r*13.1

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

  10. Simplified13.1

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

  11. Final simplification13.1

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

Reproduce

herbie shell --seed 2020060 +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))))))))