Average Error: 13.6 → 8.4
Time: 5.9s
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 r293529 = 0.5;
        double r293530 = 1.0;
        double r293531 = x;
        double r293532 = 4.0;
        double r293533 = p;
        double r293534 = r293532 * r293533;
        double r293535 = r293534 * r293533;
        double r293536 = r293531 * r293531;
        double r293537 = r293535 + r293536;
        double r293538 = sqrt(r293537);
        double r293539 = r293531 / r293538;
        double r293540 = r293530 + r293539;
        double r293541 = r293529 * r293540;
        double r293542 = sqrt(r293541);
        return r293542;
}
↓
double f(double p, double x) {
        double r293543 = x;
        double r293544 = 4.0;
        double r293545 = p;
        double r293546 = r293544 * r293545;
        double r293547 = r293546 * r293545;
        double r293548 = r293543 * r293543;
        double r293549 = r293547 + r293548;
        double r293550 = sqrt(r293549);
        double r293551 = r293543 / r293550;
        double r293552 = -1.0;
        bool r293553 = r293551 <= r293552;
        double r293554 = 2.0;
        double r293555 = sqrt(r293554);
        double r293556 = 0.5;
        double r293557 = sqrt(r293556);
        double r293558 = r293555 * r293557;
        double r293559 = log(r293558);
        double r293560 = -1.0;
        double r293561 = r293560 / r293543;
        double r293562 = log(r293561);
        double r293563 = r293559 + r293562;
        double r293564 = r293560 / r293545;
        double r293565 = log(r293564);
        double r293566 = r293563 - r293565;
        double r293567 = exp(r293566);
        double r293568 = 1.0;
        double r293569 = exp(r293551);
        double r293570 = log(r293569);
        double r293571 = r293568 + r293570;
        double r293572 = r293556 * r293571;
        double r293573 = sqrt(r293572);
        double r293574 = r293553 ? r293567 : r293573;
        return r293574;
}

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 +o rules:numerics
(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))))))))