Average Error: 13.6 → 13.6
Time: 17.8s
Precision: 64
\[10^{-150} \lt \left|x\right| \lt 10^{+150}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\sqrt{\frac{x}{\frac{\sqrt{x \cdot x + \left(p \cdot 4\right) \cdot p}}{0.5}} + 0.5}\]
\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}
↓
\sqrt{\frac{x}{\frac{\sqrt{x \cdot x + \left(p \cdot 4\right) \cdot p}}{0.5}} + 0.5}
double f(double p, double x) {
        double r3490497 = 0.5;
        double r3490498 = 1.0;
        double r3490499 = x;
        double r3490500 = 4.0;
        double r3490501 = p;
        double r3490502 = r3490500 * r3490501;
        double r3490503 = r3490502 * r3490501;
        double r3490504 = r3490499 * r3490499;
        double r3490505 = r3490503 + r3490504;
        double r3490506 = sqrt(r3490505);
        double r3490507 = r3490499 / r3490506;
        double r3490508 = r3490498 + r3490507;
        double r3490509 = r3490497 * r3490508;
        double r3490510 = sqrt(r3490509);
        return r3490510;
}
↓
double f(double p, double x) {
        double r3490511 = x;
        double r3490512 = r3490511 * r3490511;
        double r3490513 = p;
        double r3490514 = 4.0;
        double r3490515 = r3490513 * r3490514;
        double r3490516 = r3490515 * r3490513;
        double r3490517 = r3490512 + r3490516;
        double r3490518 = sqrt(r3490517);
        double r3490519 = 0.5;
        double r3490520 = r3490518 / r3490519;
        double r3490521 = r3490511 / r3490520;
        double r3490522 = r3490521 + r3490519;
        double r3490523 = sqrt(r3490522);
        return r3490523;
}

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

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. Simplified13.6

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

  3. Final simplification13.6

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

Reproduce

herbie shell --seed 2019156 
(FPCore (p x)
  :name "Given's Rotation SVD example"
  :pre (< 1e-150 (fabs x) 1e+150)

  :herbie-target
  (sqrt (+ 1/2 (/ (copysign 1/2 x) (hypot 1 (/ (* 2 p) x)))))

  (sqrt (* 0.5 (+ 1 (/ x (sqrt (+ (* (* 4 p) p) (* x x))))))))