Average Error: 13.6 → 13.6
Time: 5.2s
Precision: 64
\[1.00000000000000001 \cdot 10^{-150} \lt \left|x\right| \lt 9.99999999999999981 \cdot 10^{149}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
\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 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}
double f(double p, double x) {
        double r329687 = 0.5;
        double r329688 = 1.0;
        double r329689 = x;
        double r329690 = 4.0;
        double r329691 = p;
        double r329692 = r329690 * r329691;
        double r329693 = r329692 * r329691;
        double r329694 = r329689 * r329689;
        double r329695 = r329693 + r329694;
        double r329696 = sqrt(r329695);
        double r329697 = r329689 / r329696;
        double r329698 = r329688 + r329697;
        double r329699 = r329687 * r329698;
        double r329700 = sqrt(r329699);
        return r329700;
}
↓
double f(double p, double x) {
        double r329701 = 0.5;
        double r329702 = 1.0;
        double r329703 = x;
        double r329704 = 4.0;
        double r329705 = p;
        double r329706 = r329704 * r329705;
        double r329707 = r329706 * r329705;
        double r329708 = r329703 * r329703;
        double r329709 = r329707 + r329708;
        double r329710 = sqrt(r329709);
        double r329711 = r329703 / r329710;
        double r329712 = r329702 + r329711;
        double r329713 = r329701 * r329712;
        double r329714 = sqrt(r329713);
        return r329714;
}

Error

Bits error versus p

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.6
Target13.6
Herbie13.6
\[\sqrt{0.5 + \frac{\mathsf{copysign}\left(0.5, x\right)}{\mathsf{hypot}\left(1, \frac{2 \cdot p}{x}\right)}}\]

Derivation

  1. Initial program 13.6

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

  2. Final simplification13.6

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

Reproduce

herbie shell --seed 2020062 +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))))))))