Average Error: 12.9 → 12.9
Time: 12.9s
Precision: 64
\[1.00000000000000001 \cdot 10^{-150} \lt \left|x\right| \lt 9.99999999999999981 \cdot 10^{149}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\sqrt{0.5 \cdot \left(\sqrt{1 + \frac{x}{\sqrt{\mathsf{fma}\left(4, {p}^{2}, x \cdot x\right)}}} \cdot \sqrt{1 + \frac{x}{\sqrt{\mathsf{fma}\left(4, {p}^{2}, x \cdot x\right)}}}\right)}\]
\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(\sqrt{1 + \frac{x}{\sqrt{\mathsf{fma}\left(4, {p}^{2}, x \cdot x\right)}}} \cdot \sqrt{1 + \frac{x}{\sqrt{\mathsf{fma}\left(4, {p}^{2}, x \cdot x\right)}}}\right)}
double f(double p, double x) {
        double r338450 = 0.5;
        double r338451 = 1.0;
        double r338452 = x;
        double r338453 = 4.0;
        double r338454 = p;
        double r338455 = r338453 * r338454;
        double r338456 = r338455 * r338454;
        double r338457 = r338452 * r338452;
        double r338458 = r338456 + r338457;
        double r338459 = sqrt(r338458);
        double r338460 = r338452 / r338459;
        double r338461 = r338451 + r338460;
        double r338462 = r338450 * r338461;
        double r338463 = sqrt(r338462);
        return r338463;
}
↓
double f(double p, double x) {
        double r338464 = 0.5;
        double r338465 = 1.0;
        double r338466 = x;
        double r338467 = 4.0;
        double r338468 = p;
        double r338469 = 2.0;
        double r338470 = pow(r338468, r338469);
        double r338471 = r338466 * r338466;
        double r338472 = fma(r338467, r338470, r338471);
        double r338473 = sqrt(r338472);
        double r338474 = r338466 / r338473;
        double r338475 = r338465 + r338474;
        double r338476 = sqrt(r338475);
        double r338477 = r338476 * r338476;
        double r338478 = r338464 * r338477;
        double r338479 = sqrt(r338478);
        return r338479;
}

Error

Bits error versus p

Bits error versus x

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 add-sqr-sqrt12.9

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

  4. Simplified12.9

    \[\leadsto \sqrt{0.5 \cdot \left(\color{blue}{\sqrt{1 + \frac{x}{\sqrt{\mathsf{fma}\left(4, {p}^{2}, x \cdot x\right)}}}} \cdot \sqrt{1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right)}\]

  5. Simplified12.9

    \[\leadsto \sqrt{0.5 \cdot \left(\sqrt{1 + \frac{x}{\sqrt{\mathsf{fma}\left(4, {p}^{2}, x \cdot x\right)}}} \cdot \color{blue}{\sqrt{1 + \frac{x}{\sqrt{\mathsf{fma}\left(4, {p}^{2}, x \cdot x\right)}}}}\right)}\]

  6. Final simplification12.9

    \[\leadsto \sqrt{0.5 \cdot \left(\sqrt{1 + \frac{x}{\sqrt{\mathsf{fma}\left(4, {p}^{2}, x \cdot x\right)}}} \cdot \sqrt{1 + \frac{x}{\sqrt{\mathsf{fma}\left(4, {p}^{2}, x \cdot x\right)}}}\right)}\]

Reproduce

herbie shell --seed 2020047 +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))))))))