Average Error: 13.8 → 13.8
Time: 6.0s
Precision: 64
\[1.000000000000000006295358232172963997211 \cdot 10^{-150} \lt \left|x\right| \lt 9.999999999999999808355961724373745905731 \cdot 10^{149}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
\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 r319773 = 0.5;
        double r319774 = 1.0;
        double r319775 = x;
        double r319776 = 4.0;
        double r319777 = p;
        double r319778 = r319776 * r319777;
        double r319779 = r319778 * r319777;
        double r319780 = r319775 * r319775;
        double r319781 = r319779 + r319780;
        double r319782 = sqrt(r319781);
        double r319783 = r319775 / r319782;
        double r319784 = r319774 + r319783;
        double r319785 = r319773 * r319784;
        double r319786 = sqrt(r319785);
        return r319786;
}
↓
double f(double p, double x) {
        double r319787 = 0.5;
        double r319788 = 1.0;
        double r319789 = x;
        double r319790 = 4.0;
        double r319791 = p;
        double r319792 = r319790 * r319791;
        double r319793 = r319792 * r319791;
        double r319794 = r319789 * r319789;
        double r319795 = r319793 + r319794;
        double r319796 = sqrt(r319795);
        double r319797 = r319789 / r319796;
        double r319798 = r319788 + r319797;
        double r319799 = r319787 * r319798;
        double r319800 = sqrt(r319799);
        return r319800;
}

Error

Bits error versus p

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.8
Target13.8
Herbie13.8
\[\sqrt{0.5 + \frac{\mathsf{copysign}\left(0.5, x\right)}{\mathsf{hypot}\left(1, \frac{2 \cdot p}{x}\right)}}\]

Derivation

  1. Initial program 13.8

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

  2. Final simplification13.8

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

Reproduce

herbie shell --seed 2019346 +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))))))))