Average Error: 13.8 → 13.8
Time: 5.3s
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 r359739 = 0.5;
        double r359740 = 1.0;
        double r359741 = x;
        double r359742 = 4.0;
        double r359743 = p;
        double r359744 = r359742 * r359743;
        double r359745 = r359744 * r359743;
        double r359746 = r359741 * r359741;
        double r359747 = r359745 + r359746;
        double r359748 = sqrt(r359747);
        double r359749 = r359741 / r359748;
        double r359750 = r359740 + r359749;
        double r359751 = r359739 * r359750;
        double r359752 = sqrt(r359751);
        return r359752;
}
↓
double f(double p, double x) {
        double r359753 = 0.5;
        double r359754 = 1.0;
        double r359755 = x;
        double r359756 = 4.0;
        double r359757 = p;
        double r359758 = r359756 * r359757;
        double r359759 = r359758 * r359757;
        double r359760 = r359755 * r359755;
        double r359761 = r359759 + r359760;
        double r359762 = sqrt(r359761);
        double r359763 = r359755 / r359762;
        double r359764 = r359754 + r359763;
        double r359765 = r359753 * r359764;
        double r359766 = sqrt(r359765);
        return r359766;
}

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 
(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))))))))