Average Error: 13.2 → 11.2
Time: 6.5s
Precision: 64
\[1.00000000000000001 \cdot 10^{-150} \lt \left|x\right| \lt 9.99999999999999981 \cdot 10^{149}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \le -1:\\ \;\;\;\;\sqrt{0.5 \cdot e^{2 \cdot \left(\log \left(\frac{-1}{x}\right) - \log \left(\frac{-1}{p}\right)\right) + \log 2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot e^{\log \left(\sqrt{1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right) + \log \left(\sqrt{1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right)}}\\ \end{array}\]
\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}\;\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \le -1:\\
\;\;\;\;\sqrt{0.5 \cdot e^{2 \cdot \left(\log \left(\frac{-1}{x}\right) - \log \left(\frac{-1}{p}\right)\right) + \log 2}}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{0.5 \cdot e^{\log \left(\sqrt{1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right) + \log \left(\sqrt{1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right)}}\\

\end{array}
double f(double p, double x) {
        double r270803 = 0.5;
        double r270804 = 1.0;
        double r270805 = x;
        double r270806 = 4.0;
        double r270807 = p;
        double r270808 = r270806 * r270807;
        double r270809 = r270808 * r270807;
        double r270810 = r270805 * r270805;
        double r270811 = r270809 + r270810;
        double r270812 = sqrt(r270811);
        double r270813 = r270805 / r270812;
        double r270814 = r270804 + r270813;
        double r270815 = r270803 * r270814;
        double r270816 = sqrt(r270815);
        return r270816;
}
↓
double f(double p, double x) {
        double r270817 = x;
        double r270818 = 4.0;
        double r270819 = p;
        double r270820 = r270818 * r270819;
        double r270821 = r270820 * r270819;
        double r270822 = r270817 * r270817;
        double r270823 = r270821 + r270822;
        double r270824 = sqrt(r270823);
        double r270825 = r270817 / r270824;
        double r270826 = -1.0;
        bool r270827 = r270825 <= r270826;
        double r270828 = 0.5;
        double r270829 = 2.0;
        double r270830 = -1.0;
        double r270831 = r270830 / r270817;
        double r270832 = log(r270831);
        double r270833 = r270830 / r270819;
        double r270834 = log(r270833);
        double r270835 = r270832 - r270834;
        double r270836 = r270829 * r270835;
        double r270837 = 2.0;
        double r270838 = log(r270837);
        double r270839 = r270836 + r270838;
        double r270840 = exp(r270839);
        double r270841 = r270828 * r270840;
        double r270842 = sqrt(r270841);
        double r270843 = 1.0;
        double r270844 = r270843 + r270825;
        double r270845 = sqrt(r270844);
        double r270846 = log(r270845);
        double r270847 = r270846 + r270846;
        double r270848 = exp(r270847);
        double r270849 = r270828 * r270848;
        double r270850 = sqrt(r270849);
        double r270851 = r270827 ? r270842 : r270850;
        return r270851;
}

Error

Bits error versus p

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.2
Target13.2
Herbie11.2
\[\sqrt{0.5 + \frac{\mathsf{copysign}\left(0.5, x\right)}{\mathsf{hypot}\left(1, \frac{2 \cdot p}{x}\right)}}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ x (sqrt (+ (* (* 4.0 p) p) (* x x)))) < -1.0

    1. Initial program 53.6

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

    3. Applied add-exp-log53.6

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

    4. Taylor expanded around -inf 45.4

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

    5. Simplified45.4

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

    if -1.0 < (/ x (sqrt (+ (* (* 4.0 p) p) (* x x))))

    1. Initial program 0.2

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

    3. Applied add-exp-log0.2

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

    5. Applied add-sqr-sqrt0.2

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

    6. Applied log-prod0.2

      \[\leadsto \sqrt{0.5 \cdot e^{\color{blue}{\log \left(\sqrt{1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right) + \log \left(\sqrt{1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right)}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification11.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \le -1:\\ \;\;\;\;\sqrt{0.5 \cdot e^{2 \cdot \left(\log \left(\frac{-1}{x}\right) - \log \left(\frac{-1}{p}\right)\right) + \log 2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot e^{\log \left(\sqrt{1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right) + \log \left(\sqrt{1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020081 
(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))))))))