Average Error: 13.2 → 13.6
Time: 18.7s
Precision: 64
\[10^{-150} \lt \left|x\right| \lt 10^{+150}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;p \le 2.604086757605581 \cdot 10^{-117}:\\ \;\;\;\;\sqrt{{e}^{\left(\log \left(\mathsf{fma}\left(\left(\frac{x}{\sqrt{\mathsf{fma}\left(p, \left(4 \cdot p\right), \left(x \cdot x\right)\right)}}\right), 0.5, 0.5\right)\right)\right)}}\\ \mathbf{elif}\;p \le 3.078696592908661 \cdot 10^{-91}:\\ \;\;\;\;\sqrt{\left(\frac{\frac{-1}{x}}{\frac{-1}{p}} \cdot \frac{\frac{-1}{x}}{\frac{-1}{p}}\right) \cdot 1.0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{e}^{\left(\log \left(\mathsf{fma}\left(\left(\frac{x}{\sqrt{\mathsf{fma}\left(p, \left(4 \cdot p\right), \left(x \cdot x\right)\right)}}\right), 0.5, 0.5\right)\right)\right)}}\\ \end{array}\]
\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}
↓
\begin{array}{l}
\mathbf{if}\;p \le 2.604086757605581 \cdot 10^{-117}:\\
\;\;\;\;\sqrt{{e}^{\left(\log \left(\mathsf{fma}\left(\left(\frac{x}{\sqrt{\mathsf{fma}\left(p, \left(4 \cdot p\right), \left(x \cdot x\right)\right)}}\right), 0.5, 0.5\right)\right)\right)}}\\

\mathbf{elif}\;p \le 3.078696592908661 \cdot 10^{-91}:\\
\;\;\;\;\sqrt{\left(\frac{\frac{-1}{x}}{\frac{-1}{p}} \cdot \frac{\frac{-1}{x}}{\frac{-1}{p}}\right) \cdot 1.0}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{{e}^{\left(\log \left(\mathsf{fma}\left(\left(\frac{x}{\sqrt{\mathsf{fma}\left(p, \left(4 \cdot p\right), \left(x \cdot x\right)\right)}}\right), 0.5, 0.5\right)\right)\right)}}\\

\end{array}
double f(double p, double x) {
        double r68599885 = 0.5;
        double r68599886 = 1.0;
        double r68599887 = x;
        double r68599888 = 4.0;
        double r68599889 = p;
        double r68599890 = r68599888 * r68599889;
        double r68599891 = r68599890 * r68599889;
        double r68599892 = r68599887 * r68599887;
        double r68599893 = r68599891 + r68599892;
        double r68599894 = sqrt(r68599893);
        double r68599895 = r68599887 / r68599894;
        double r68599896 = r68599886 + r68599895;
        double r68599897 = r68599885 * r68599896;
        double r68599898 = sqrt(r68599897);
        return r68599898;
}
↓
double f(double p, double x) {
        double r68599899 = p;
        double r68599900 = 2.604086757605581e-117;
        bool r68599901 = r68599899 <= r68599900;
        double r68599902 = exp(1.0);
        double r68599903 = x;
        double r68599904 = 4.0;
        double r68599905 = r68599904 * r68599899;
        double r68599906 = r68599903 * r68599903;
        double r68599907 = fma(r68599899, r68599905, r68599906);
        double r68599908 = sqrt(r68599907);
        double r68599909 = r68599903 / r68599908;
        double r68599910 = 0.5;
        double r68599911 = fma(r68599909, r68599910, r68599910);
        double r68599912 = log(r68599911);
        double r68599913 = pow(r68599902, r68599912);
        double r68599914 = sqrt(r68599913);
        double r68599915 = 3.078696592908661e-91;
        bool r68599916 = r68599899 <= r68599915;
        double r68599917 = -1.0;
        double r68599918 = r68599917 / r68599903;
        double r68599919 = r68599917 / r68599899;
        double r68599920 = r68599918 / r68599919;
        double r68599921 = r68599920 * r68599920;
        double r68599922 = 1.0;
        double r68599923 = r68599921 * r68599922;
        double r68599924 = sqrt(r68599923);
        double r68599925 = r68599916 ? r68599924 : r68599914;
        double r68599926 = r68599901 ? r68599914 : r68599925;
        return r68599926;
}

Error

Bits error versus p

Bits error versus x

Target

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

Derivation

  1. Split input into 2 regimes

  2. if p < 2.604086757605581e-117 or 3.078696592908661e-91 < p

    1. Initial program 13.0

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

    2. Simplified13.0

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(\left(\frac{x}{\sqrt{\mathsf{fma}\left(p, \left(4 \cdot p\right), \left(x \cdot x\right)\right)}}\right), 0.5, 0.5\right)}}\]
    3. Using strategy rm

    4. Applied add-exp-log13.0

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

    6. Applied *-un-lft-identity13.0

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

    7. Applied exp-prod13.0

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

    8. Simplified13.0

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

    if 2.604086757605581e-117 < p < 3.078696592908661e-91

    1. Initial program 24.2

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

    2. Simplified24.2

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(\left(\frac{x}{\sqrt{\mathsf{fma}\left(p, \left(4 \cdot p\right), \left(x \cdot x\right)\right)}}\right), 0.5, 0.5\right)}}\]
    3. Using strategy rm

    4. Applied add-exp-log24.2

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

    6. Applied *-un-lft-identity24.2

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

    7. Applied exp-prod24.2

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

    8. Simplified24.2

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

    9. Taylor expanded around -inf 62.2

      \[\leadsto \sqrt{\color{blue}{e^{\left(\log 1.0 + 2 \cdot \log \left(\frac{-1}{x}\right)\right) - 2 \cdot \log \left(\frac{-1}{p}\right)}}}\]

    10. Simplified44.1

      \[\leadsto \sqrt{\color{blue}{1.0 \cdot \left(\frac{\frac{-1}{x}}{\frac{-1}{p}} \cdot \frac{\frac{-1}{x}}{\frac{-1}{p}}\right)}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification13.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;p \le 2.604086757605581 \cdot 10^{-117}:\\ \;\;\;\;\sqrt{{e}^{\left(\log \left(\mathsf{fma}\left(\left(\frac{x}{\sqrt{\mathsf{fma}\left(p, \left(4 \cdot p\right), \left(x \cdot x\right)\right)}}\right), 0.5, 0.5\right)\right)\right)}}\\ \mathbf{elif}\;p \le 3.078696592908661 \cdot 10^{-91}:\\ \;\;\;\;\sqrt{\left(\frac{\frac{-1}{x}}{\frac{-1}{p}} \cdot \frac{\frac{-1}{x}}{\frac{-1}{p}}\right) \cdot 1.0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{e}^{\left(\log \left(\mathsf{fma}\left(\left(\frac{x}{\sqrt{\mathsf{fma}\left(p, \left(4 \cdot p\right), \left(x \cdot x\right)\right)}}\right), 0.5, 0.5\right)\right)\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019124 +o rules:numerics
(FPCore (p x)
  :name "Given's Rotation SVD example"
  :pre (< 1e-150 (fabs x) 1e+150)

  :herbie-target
  (sqrt (+ 1/2 (/ (copysign 1/2 x) (hypot 1 (/ (* 2 p) x)))))

  (sqrt (* 0.5 (+ 1 (/ x (sqrt (+ (* (* 4 p) p) (* x x))))))))