Average Error: 13.6 → 8.4
Time: 5.8s
Precision: 64
\[1.00000000000000001 \cdot 10^{-150} \lt \left|x\right| \lt 9.99999999999999981 \cdot 10^{149}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \le -1:\\ \;\;\;\;e^{\left(\log \left(\sqrt{2} \cdot \sqrt{0.5}\right) + \log \left(\frac{-1}{x}\right)\right) - \log \left(\frac{-1}{p}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \log \left(e^{\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right)\right)}\\ \end{array}\]
\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}\;\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \le -1:\\
\;\;\;\;e^{\left(\log \left(\sqrt{2} \cdot \sqrt{0.5}\right) + \log \left(\frac{-1}{x}\right)\right) - \log \left(\frac{-1}{p}\right)}\\

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

\end{array}
double f(double p, double x) {
        double r267833 = 0.5;
        double r267834 = 1.0;
        double r267835 = x;
        double r267836 = 4.0;
        double r267837 = p;
        double r267838 = r267836 * r267837;
        double r267839 = r267838 * r267837;
        double r267840 = r267835 * r267835;
        double r267841 = r267839 + r267840;
        double r267842 = sqrt(r267841);
        double r267843 = r267835 / r267842;
        double r267844 = r267834 + r267843;
        double r267845 = r267833 * r267844;
        double r267846 = sqrt(r267845);
        return r267846;
}
↓
double f(double p, double x) {
        double r267847 = x;
        double r267848 = 4.0;
        double r267849 = p;
        double r267850 = r267848 * r267849;
        double r267851 = r267850 * r267849;
        double r267852 = r267847 * r267847;
        double r267853 = r267851 + r267852;
        double r267854 = sqrt(r267853);
        double r267855 = r267847 / r267854;
        double r267856 = -1.0;
        bool r267857 = r267855 <= r267856;
        double r267858 = 2.0;
        double r267859 = sqrt(r267858);
        double r267860 = 0.5;
        double r267861 = sqrt(r267860);
        double r267862 = r267859 * r267861;
        double r267863 = log(r267862);
        double r267864 = -1.0;
        double r267865 = r267864 / r267847;
        double r267866 = log(r267865);
        double r267867 = r267863 + r267866;
        double r267868 = r267864 / r267849;
        double r267869 = log(r267868);
        double r267870 = r267867 - r267869;
        double r267871 = exp(r267870);
        double r267872 = 1.0;
        double r267873 = exp(r267855);
        double r267874 = log(r267873);
        double r267875 = r267872 + r267874;
        double r267876 = r267860 * r267875;
        double r267877 = sqrt(r267876);
        double r267878 = r267857 ? r267871 : r267877;
        return r267878;
}

Error

Bits error versus p

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.6
Target13.6
Herbie8.4
\[\sqrt{0.5 + \frac{\mathsf{copysign}\left(0.5, x\right)}{\mathsf{hypot}\left(1, \frac{2 \cdot p}{x}\right)}}\]

Derivation

  1. Split input into 2 regimes

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

    1. Initial program 54.4

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

    3. Applied div-inv55.2

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

    5. Applied add-log-exp55.2

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

    7. Applied add-exp-log55.2

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

    8. Taylor expanded around -inf 33.5

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

    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-log-exp0.2

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

  4. Final simplification8.4

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

Reproduce

herbie shell --seed 2020057 
(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))))))))