Average Error: 13.3 → 13.6
Time: 21.9s
Precision: 64
\[1.000000000000000006295358232172963997211 \cdot 10^{-150} \lt \left|x\right| \lt 9.999999999999999808355961724373745905731 \cdot 10^{149}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\sqrt{0.5 \cdot \left(1 + \frac{1}{\sqrt{\mathsf{fma}\left(4, {p}^{2}, {x}^{2}\right)}} \cdot x\right)}\]
\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 + \frac{1}{\sqrt{\mathsf{fma}\left(4, {p}^{2}, {x}^{2}\right)}} \cdot x\right)}
double f(double p, double x) {
        double r256494 = 0.5;
        double r256495 = 1.0;
        double r256496 = x;
        double r256497 = 4.0;
        double r256498 = p;
        double r256499 = r256497 * r256498;
        double r256500 = r256499 * r256498;
        double r256501 = r256496 * r256496;
        double r256502 = r256500 + r256501;
        double r256503 = sqrt(r256502);
        double r256504 = r256496 / r256503;
        double r256505 = r256495 + r256504;
        double r256506 = r256494 * r256505;
        double r256507 = sqrt(r256506);
        return r256507;
}
↓
double f(double p, double x) {
        double r256508 = 0.5;
        double r256509 = 1.0;
        double r256510 = 1.0;
        double r256511 = 4.0;
        double r256512 = p;
        double r256513 = 2.0;
        double r256514 = pow(r256512, r256513);
        double r256515 = x;
        double r256516 = pow(r256515, r256513);
        double r256517 = fma(r256511, r256514, r256516);
        double r256518 = sqrt(r256517);
        double r256519 = r256510 / r256518;
        double r256520 = r256519 * r256515;
        double r256521 = r256509 + r256520;
        double r256522 = r256508 * r256521;
        double r256523 = sqrt(r256522);
        return r256523;
}

Error

Bits error versus p

Bits error versus x

Target

Original13.3
Target13.3
Herbie13.6
\[\sqrt{0.5 + \frac{\mathsf{copysign}\left(0.5, x\right)}{\mathsf{hypot}\left(1, \frac{2 \cdot p}{x}\right)}}\]

Derivation

  1. Initial program 13.3

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

  3. Applied add-log-exp13.3

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

  4. Simplified13.3

    \[\leadsto \sqrt{0.5 \cdot \left(1 + \log \color{blue}{\left(e^{\frac{x}{\sqrt{\mathsf{fma}\left(4, {p}^{2}, {x}^{2}\right)}}}\right)}\right)}\]
  5. Using strategy rm

  6. Applied div-inv13.7

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

  7. Applied exp-prod34.3

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

  8. Applied log-pow42.2

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

  9. Simplified13.6

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

  10. Final simplification13.6

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

Reproduce

herbie shell --seed 2019326 +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))))))))