Average Error: 12.9 → 12.9
Time: 4.1s
Precision: 64
\[1.00000000000000001 \cdot 10^{-150} \lt \left|x\right| \lt 9.99999999999999981 \cdot 10^{149}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\sqrt{0.5 \cdot \left(1 + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{1}\right)}\]
\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 + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{1}\right)}
double f(double p, double x) {
        double r240471 = 0.5;
        double r240472 = 1.0;
        double r240473 = x;
        double r240474 = 4.0;
        double r240475 = p;
        double r240476 = r240474 * r240475;
        double r240477 = r240476 * r240475;
        double r240478 = r240473 * r240473;
        double r240479 = r240477 + r240478;
        double r240480 = sqrt(r240479);
        double r240481 = r240473 / r240480;
        double r240482 = r240472 + r240481;
        double r240483 = r240471 * r240482;
        double r240484 = sqrt(r240483);
        return r240484;
}
↓
double f(double p, double x) {
        double r240485 = 0.5;
        double r240486 = 1.0;
        double r240487 = x;
        double r240488 = 4.0;
        double r240489 = p;
        double r240490 = r240488 * r240489;
        double r240491 = r240490 * r240489;
        double r240492 = r240487 * r240487;
        double r240493 = r240491 + r240492;
        double r240494 = sqrt(r240493);
        double r240495 = r240487 / r240494;
        double r240496 = 1.0;
        double r240497 = pow(r240495, r240496);
        double r240498 = r240486 + r240497;
        double r240499 = r240485 * r240498;
        double r240500 = sqrt(r240499);
        return r240500;
}

Error

Bits error versus p

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original12.9
Target12.9
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 12.9

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

  3. Applied div-inv13.1

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

  5. Applied *-un-lft-identity13.1

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

  7. Applied pow113.1

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

  8. Applied pow113.1

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

  9. Applied pow-prod-down13.1

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

  10. Simplified12.9

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

  11. Final simplification12.9

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

Reproduce

herbie shell --seed 2020060 +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))))))))