Average Error: 13.6 → 13.6
Time: 5.3s
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 r420755 = 0.5;
        double r420756 = 1.0;
        double r420757 = x;
        double r420758 = 4.0;
        double r420759 = p;
        double r420760 = r420758 * r420759;
        double r420761 = r420760 * r420759;
        double r420762 = r420757 * r420757;
        double r420763 = r420761 + r420762;
        double r420764 = sqrt(r420763);
        double r420765 = r420757 / r420764;
        double r420766 = r420756 + r420765;
        double r420767 = r420755 * r420766;
        double r420768 = sqrt(r420767);
        return r420768;
}
↓
double f(double p, double x) {
        double r420769 = 0.5;
        double r420770 = 1.0;
        double r420771 = x;
        double r420772 = 4.0;
        double r420773 = p;
        double r420774 = r420772 * r420773;
        double r420775 = r420774 * r420773;
        double r420776 = r420771 * r420771;
        double r420777 = r420775 + r420776;
        double r420778 = sqrt(r420777);
        double r420779 = r420771 / r420778;
        double r420780 = r420770 + r420779;
        double r420781 = r420769 * r420780;
        double r420782 = sqrt(r420781);
        return r420782;
}

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))))))))