Average Error: 13.9 → 13.9
Time: 16.8s
Precision: 64
\[1.000000000000000006295358232172963997211 \cdot 10^{-150} \lt \left|x\right| \lt 9.999999999999999808355961724373745905731 \cdot 10^{149}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{x \cdot x + p \cdot \left(p \cdot 4\right)}} \le -0.9999964161259105033252581051783636212349:\\ \;\;\;\;\sqrt{0.5 \cdot \frac{1 \cdot 1 - \frac{x}{\sqrt{x \cdot x + p \cdot \left(p \cdot 4\right)}} \cdot \frac{x}{\sqrt{x \cdot x + p \cdot \left(p \cdot 4\right)}}}{1 - \frac{x}{\sqrt{x \cdot x + p \cdot \left(p \cdot 4\right)}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(\left(x \cdot \sqrt{\frac{1}{\sqrt{x \cdot x + p \cdot \left(p \cdot 4\right)}}}\right) \cdot \sqrt{\frac{1}{\sqrt{x \cdot x + p \cdot \left(p \cdot 4\right)}}} + 1\right)}\\ \end{array}\]
\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{x \cdot x + p \cdot \left(p \cdot 4\right)}} \le -0.9999964161259105033252581051783636212349:\\
\;\;\;\;\sqrt{0.5 \cdot \frac{1 \cdot 1 - \frac{x}{\sqrt{x \cdot x + p \cdot \left(p \cdot 4\right)}} \cdot \frac{x}{\sqrt{x \cdot x + p \cdot \left(p \cdot 4\right)}}}{1 - \frac{x}{\sqrt{x \cdot x + p \cdot \left(p \cdot 4\right)}}}}\\

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

\end{array}
double f(double p, double x) {
        double r8633785 = 0.5;
        double r8633786 = 1.0;
        double r8633787 = x;
        double r8633788 = 4.0;
        double r8633789 = p;
        double r8633790 = r8633788 * r8633789;
        double r8633791 = r8633790 * r8633789;
        double r8633792 = r8633787 * r8633787;
        double r8633793 = r8633791 + r8633792;
        double r8633794 = sqrt(r8633793);
        double r8633795 = r8633787 / r8633794;
        double r8633796 = r8633786 + r8633795;
        double r8633797 = r8633785 * r8633796;
        double r8633798 = sqrt(r8633797);
        return r8633798;
}
↓
double f(double p, double x) {
        double r8633799 = x;
        double r8633800 = r8633799 * r8633799;
        double r8633801 = p;
        double r8633802 = 4.0;
        double r8633803 = r8633801 * r8633802;
        double r8633804 = r8633801 * r8633803;
        double r8633805 = r8633800 + r8633804;
        double r8633806 = sqrt(r8633805);
        double r8633807 = r8633799 / r8633806;
        double r8633808 = -0.9999964161259105;
        bool r8633809 = r8633807 <= r8633808;
        double r8633810 = 0.5;
        double r8633811 = 1.0;
        double r8633812 = r8633811 * r8633811;
        double r8633813 = r8633807 * r8633807;
        double r8633814 = r8633812 - r8633813;
        double r8633815 = r8633811 - r8633807;
        double r8633816 = r8633814 / r8633815;
        double r8633817 = r8633810 * r8633816;
        double r8633818 = sqrt(r8633817);
        double r8633819 = 1.0;
        double r8633820 = r8633819 / r8633806;
        double r8633821 = sqrt(r8633820);
        double r8633822 = r8633799 * r8633821;
        double r8633823 = r8633822 * r8633821;
        double r8633824 = r8633823 + r8633811;
        double r8633825 = r8633810 * r8633824;
        double r8633826 = sqrt(r8633825);
        double r8633827 = r8633809 ? r8633818 : r8633826;
        return r8633827;
}

Error

Bits error versus p

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.9
Target13.9
Herbie13.9
\[\sqrt{0.5 + \frac{\mathsf{copysign}\left(0.5, x\right)}{\mathsf{hypot}\left(1, \frac{2 \cdot p}{x}\right)}}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ x (sqrt (+ (* (* 4.0 p) p) (* x x)))) < -0.9999964161259105

    1. Initial program 53.7

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

    3. Applied flip-+53.7

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

    if -0.9999964161259105 < (/ x (sqrt (+ (* (* 4.0 p) p) (* x x))))

    1. Initial program 0.0

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

    3. Applied div-inv0.0

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

    5. Applied add-sqr-sqrt0.1

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

    6. Applied associate-*r*0.1

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

  4. Final simplification13.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{x \cdot x + p \cdot \left(p \cdot 4\right)}} \le -0.9999964161259105033252581051783636212349:\\ \;\;\;\;\sqrt{0.5 \cdot \frac{1 \cdot 1 - \frac{x}{\sqrt{x \cdot x + p \cdot \left(p \cdot 4\right)}} \cdot \frac{x}{\sqrt{x \cdot x + p \cdot \left(p \cdot 4\right)}}}{1 - \frac{x}{\sqrt{x \cdot x + p \cdot \left(p \cdot 4\right)}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(\left(x \cdot \sqrt{\frac{1}{\sqrt{x \cdot x + p \cdot \left(p \cdot 4\right)}}}\right) \cdot \sqrt{\frac{1}{\sqrt{x \cdot x + p \cdot \left(p \cdot 4\right)}}} + 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(FPCore (p x)
  :name "Given's Rotation SVD example"
  :pre (< 1e-150 (fabs x) 1e+150)

  :herbie-target
  (sqrt (+ 0.5 (/ (copysign 0.5 x) (hypot 1.0 (/ (* 2.0 p) x)))))

  (sqrt (* 0.5 (+ 1.0 (/ x (sqrt (+ (* (* 4.0 p) p) (* x x))))))))