Average Error: 12.9 → 12.9
Time: 10.8s
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 r246404 = 0.5;
        double r246405 = 1.0;
        double r246406 = x;
        double r246407 = 4.0;
        double r246408 = p;
        double r246409 = r246407 * r246408;
        double r246410 = r246409 * r246408;
        double r246411 = r246406 * r246406;
        double r246412 = r246410 + r246411;
        double r246413 = sqrt(r246412);
        double r246414 = r246406 / r246413;
        double r246415 = r246405 + r246414;
        double r246416 = r246404 * r246415;
        double r246417 = sqrt(r246416);
        return r246417;
}
↓
double f(double p, double x) {
        double r246418 = 0.5;
        double r246419 = 1.0;
        double r246420 = x;
        double r246421 = 4.0;
        double r246422 = p;
        double r246423 = 2.0;
        double r246424 = pow(r246422, r246423);
        double r246425 = r246420 * r246420;
        double r246426 = fma(r246421, r246424, r246425);
        double r246427 = sqrt(r246426);
        double r246428 = r246420 / r246427;
        double r246429 = r246419 + r246428;
        double r246430 = sqrt(r246429);
        double r246431 = r246430 * r246430;
        double r246432 = r246418 * r246431;
        double r246433 = sqrt(r246432);
        return r246433;
}

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))))))))