Average Error: 13.7 → 14.0
Time: 4.5s
Precision: 64
\[1.00000000000000001 \cdot 10^{-150} \lt \left|x\right| \lt 9.99999999999999981 \cdot 10^{149}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\sqrt{0.5 \cdot \sqrt[3]{{\left(1 + x \cdot {\left(\left(4 \cdot p\right) \cdot p + x \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)}^{3}}}\]
\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 + x \cdot {\left(\left(4 \cdot p\right) \cdot p + x \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)}^{3}}}
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 * cbrt(pow((1.0 + (x * pow((((4.0 * p) * p) + (x * x)), -0.5))), 3.0))));
}

Error

Bits error versus p

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.7
Target13.7
Herbie14.0
\[\sqrt{0.5 + \frac{\mathsf{copysign}\left(0.5, x\right)}{\mathsf{hypot}\left(1, \frac{2 \cdot p}{x}\right)}}\]

Derivation

  1. Initial program 13.7

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

  3. Applied div-inv14.0

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

  5. Applied pow1/214.0

    \[\leadsto \sqrt{0.5 \cdot \left(1 + x \cdot \frac{1}{\color{blue}{{\left(\left(4 \cdot p\right) \cdot p + x \cdot x\right)}^{\frac{1}{2}}}}\right)}\]

  6. Applied pow-flip14.0

    \[\leadsto \sqrt{0.5 \cdot \left(1 + x \cdot \color{blue}{{\left(\left(4 \cdot p\right) \cdot p + x \cdot x\right)}^{\left(-\frac{1}{2}\right)}}\right)}\]
  7. Using strategy rm

  8. Applied add-cbrt-cube14.0

    \[\leadsto \sqrt{0.5 \cdot \color{blue}{\sqrt[3]{\left(\left(1 + x \cdot {\left(\left(4 \cdot p\right) \cdot p + x \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right) \cdot \left(1 + x \cdot {\left(\left(4 \cdot p\right) \cdot p + x \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right) \cdot \left(1 + x \cdot {\left(\left(4 \cdot p\right) \cdot p + x \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)}}}\]

  9. Simplified14.0

    \[\leadsto \sqrt{0.5 \cdot \sqrt[3]{\color{blue}{{\left(1 + x \cdot {\left(\left(4 \cdot p\right) \cdot p + x \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)}^{3}}}}\]

  10. Final simplification14.0

    \[\leadsto \sqrt{0.5 \cdot \sqrt[3]{{\left(1 + x \cdot {\left(\left(4 \cdot p\right) \cdot p + x \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)}^{3}}}\]

Reproduce

herbie shell --seed 2020058 
(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))))))))