Average Error: 12.9 → 12.9
Time: 5.5s
Precision: 64
\[1.000000000000000006295358232172963997211 \cdot 10^{-150} \lt \left|x\right| \lt 9.999999999999999808355961724373745905731 \cdot 10^{149}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\sqrt{0.5 \cdot \left(1 + \left(\sqrt[3]{\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}} \cdot \sqrt[3]{\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right) \cdot \sqrt[3]{\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right)}\]
\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 + \left(\sqrt[3]{\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}} \cdot \sqrt[3]{\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right) \cdot \sqrt[3]{\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right)}
double f(double p, double x) {
        double r265675 = 0.5;
        double r265676 = 1.0;
        double r265677 = x;
        double r265678 = 4.0;
        double r265679 = p;
        double r265680 = r265678 * r265679;
        double r265681 = r265680 * r265679;
        double r265682 = r265677 * r265677;
        double r265683 = r265681 + r265682;
        double r265684 = sqrt(r265683);
        double r265685 = r265677 / r265684;
        double r265686 = r265676 + r265685;
        double r265687 = r265675 * r265686;
        double r265688 = sqrt(r265687);
        return r265688;
}
↓
double f(double p, double x) {
        double r265689 = 0.5;
        double r265690 = 1.0;
        double r265691 = x;
        double r265692 = 4.0;
        double r265693 = p;
        double r265694 = r265692 * r265693;
        double r265695 = r265694 * r265693;
        double r265696 = r265691 * r265691;
        double r265697 = r265695 + r265696;
        double r265698 = sqrt(r265697);
        double r265699 = r265691 / r265698;
        double r265700 = cbrt(r265699);
        double r265701 = r265700 * r265700;
        double r265702 = r265701 * r265700;
        double r265703 = r265690 + r265702;
        double r265704 = r265689 * r265703;
        double r265705 = sqrt(r265704);
        return r265705;
}

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
Herbie12.9
\[\sqrt{0.5 + \frac{\mathsf{copysign}\left(0.5, x\right)}{\mathsf{hypot}\left(1, \frac{2 \cdot p}{x}\right)}}\]

Derivation

  1. Initial program 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 add-cube-cbrt12.9

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

  4. Final simplification12.9

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

Reproduce

herbie shell --seed 2020002 
(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))))))))