Average Error: 12.7 → 12.9
Time: 12.1s
Precision: 64
\[1.000000000000000006295358232172963997211 \cdot 10^{-150} \lt \left|x\right| \lt 9.999999999999999808355961724373745905731 \cdot 10^{149}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\sqrt{0.5 \cdot \left(1 + x \cdot \frac{1}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
\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 + x \cdot \frac{1}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}
double f(double p, double x) {
        double r205600 = 0.5;
        double r205601 = 1.0;
        double r205602 = x;
        double r205603 = 4.0;
        double r205604 = p;
        double r205605 = r205603 * r205604;
        double r205606 = r205605 * r205604;
        double r205607 = r205602 * r205602;
        double r205608 = r205606 + r205607;
        double r205609 = sqrt(r205608);
        double r205610 = r205602 / r205609;
        double r205611 = r205601 + r205610;
        double r205612 = r205600 * r205611;
        double r205613 = sqrt(r205612);
        return r205613;
}
↓
double f(double p, double x) {
        double r205614 = 0.5;
        double r205615 = 1.0;
        double r205616 = x;
        double r205617 = 1.0;
        double r205618 = 4.0;
        double r205619 = p;
        double r205620 = r205618 * r205619;
        double r205621 = r205620 * r205619;
        double r205622 = r205616 * r205616;
        double r205623 = r205621 + r205622;
        double r205624 = sqrt(r205623);
        double r205625 = r205617 / r205624;
        double r205626 = r205616 * r205625;
        double r205627 = r205615 + r205626;
        double r205628 = r205614 * r205627;
        double r205629 = sqrt(r205628);
        return r205629;
}

Error

Bits error versus p

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original12.7
Target12.7
Herbie12.9
\[\sqrt{0.5 + \frac{\mathsf{copysign}\left(0.5, x\right)}{\mathsf{hypot}\left(1, \frac{2 \cdot p}{x}\right)}}\]

Derivation

  1. Initial program 12.7

    \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
  2. Using strategy rm

  3. Applied div-inv12.9

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

  4. Final simplification12.9

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

Reproduce

herbie shell --seed 2019209 
(FPCore (p x)
  :name "Given's Rotation SVD example"
  :precision binary64
  :pre (< 1.00000000000000001e-150 (fabs x) 9.99999999999999981e149)

  :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))))))))