Average Error: 13.6 → 8.9
Time: 5.4s
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 r278581 = 0.5;
        double r278582 = 1.0;
        double r278583 = x;
        double r278584 = 4.0;
        double r278585 = p;
        double r278586 = r278584 * r278585;
        double r278587 = r278586 * r278585;
        double r278588 = r278583 * r278583;
        double r278589 = r278587 + r278588;
        double r278590 = sqrt(r278589);
        double r278591 = r278583 / r278590;
        double r278592 = r278582 + r278591;
        double r278593 = r278581 * r278592;
        double r278594 = sqrt(r278593);
        return r278594;
}
↓
double f(double p, double x) {
        double r278595 = x;
        double r278596 = 4.0;
        double r278597 = p;
        double r278598 = r278596 * r278597;
        double r278599 = r278598 * r278597;
        double r278600 = r278595 * r278595;
        double r278601 = r278599 + r278600;
        double r278602 = sqrt(r278601);
        double r278603 = r278595 / r278602;
        double r278604 = -1.0;
        bool r278605 = r278603 <= r278604;
        double r278606 = 2.0;
        double r278607 = sqrt(r278606);
        double r278608 = 0.5;
        double r278609 = sqrt(r278608);
        double r278610 = r278607 * r278609;
        double r278611 = log(r278610);
        double r278612 = -1.0;
        double r278613 = r278612 / r278595;
        double r278614 = log(r278613);
        double r278615 = r278611 + r278614;
        double r278616 = r278612 / r278597;
        double r278617 = log(r278616);
        double r278618 = r278615 - r278617;
        double r278619 = exp(r278618);
        double r278620 = 1.0;
        double r278621 = r278620 + r278603;
        double r278622 = r278608 * r278621;
        double r278623 = sqrt(r278622);
        double r278624 = r278605 ? r278619 : r278623;
        return r278624;
}

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))))))))