Average Error: 13.6 → 8.9
Time: 5.5s
Precision: 64
\[1.00000000000000001 \cdot 10^{-150} \lt \left|x\right| \lt 9.99999999999999981 \cdot 10^{149}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \le -1:\\ \;\;\;\;e^{\left(\log \left(\sqrt{2} \cdot \sqrt{0.5}\right) + \log \left(\frac{-1}{x}\right)\right) - \log \left(\frac{-1}{p}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\\ \end{array}\]
\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:\\
\;\;\;\;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 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\\

\end{array}
double f(double p, double x) {
        double r272678 = 0.5;
        double r272679 = 1.0;
        double r272680 = x;
        double r272681 = 4.0;
        double r272682 = p;
        double r272683 = r272681 * r272682;
        double r272684 = r272683 * r272682;
        double r272685 = r272680 * r272680;
        double r272686 = r272684 + r272685;
        double r272687 = sqrt(r272686);
        double r272688 = r272680 / r272687;
        double r272689 = r272679 + r272688;
        double r272690 = r272678 * r272689;
        double r272691 = sqrt(r272690);
        return r272691;
}
↓
double f(double p, double x) {
        double r272692 = x;
        double r272693 = 4.0;
        double r272694 = p;
        double r272695 = r272693 * r272694;
        double r272696 = r272695 * r272694;
        double r272697 = r272692 * r272692;
        double r272698 = r272696 + r272697;
        double r272699 = sqrt(r272698);
        double r272700 = r272692 / r272699;
        double r272701 = -1.0;
        bool r272702 = r272700 <= r272701;
        double r272703 = 2.0;
        double r272704 = sqrt(r272703);
        double r272705 = 0.5;
        double r272706 = sqrt(r272705);
        double r272707 = r272704 * r272706;
        double r272708 = log(r272707);
        double r272709 = -1.0;
        double r272710 = r272709 / r272692;
        double r272711 = log(r272710);
        double r272712 = r272708 + r272711;
        double r272713 = r272709 / r272694;
        double r272714 = log(r272713);
        double r272715 = r272712 - r272714;
        double r272716 = exp(r272715);
        double r272717 = 1.0;
        double r272718 = r272717 + r272700;
        double r272719 = r272705 * r272718;
        double r272720 = sqrt(r272719);
        double r272721 = r272702 ? r272716 : r272720;
        return r272721;
}

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
Herbie8.9
\[\sqrt{0.5 + \frac{\mathsf{copysign}\left(0.5, x\right)}{\mathsf{hypot}\left(1, \frac{2 \cdot p}{x}\right)}}\]

Derivation

  1. Split input into 2 regimes

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

    1. Initial program 54.5

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

    3. Applied div-inv55.4

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

    5. Applied add-log-exp55.4

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

    7. Applied add-exp-log55.4

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

    8. Taylor expanded around -inf 35.5

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

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

    1. Initial program 0.2

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

  4. Final simplification8.9

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

Reproduce

herbie shell --seed 2020062 
(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))))))))