Average Error: 13.1 → 8.6
Time: 7.4s
Precision: 64
\[1.000000000000000006295358232172963997211 \cdot 10^{-150} \lt \left|x\right| \lt 9.999999999999999808355961724373745905731 \cdot 10^{149}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \le -1:\\ \;\;\;\;e^{\left(\log \left(\sqrt{2} \cdot \sqrt{0.5}\right) + \log \left(\frac{-1}{x}\right)\right) - \log \left(\frac{-1}{p}\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\sqrt{0.5 \cdot \left({1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}\right)}\right) - \log \left(\sqrt{1 \cdot 1 + \left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} - 1 \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right)}\\ \end{array}\]
\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}:\\
\;\;\;\;e^{\log \left(\sqrt{0.5 \cdot \left({1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}\right)}\right) - \log \left(\sqrt{1 \cdot 1 + \left(\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 \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right)}\\

\end{array}
double f(double p, double x) {
        double r308252 = 0.5;
        double r308253 = 1.0;
        double r308254 = x;
        double r308255 = 4.0;
        double r308256 = p;
        double r308257 = r308255 * r308256;
        double r308258 = r308257 * r308256;
        double r308259 = r308254 * r308254;
        double r308260 = r308258 + r308259;
        double r308261 = sqrt(r308260);
        double r308262 = r308254 / r308261;
        double r308263 = r308253 + r308262;
        double r308264 = r308252 * r308263;
        double r308265 = sqrt(r308264);
        return r308265;
}
↓
double f(double p, double x) {
        double r308266 = x;
        double r308267 = 4.0;
        double r308268 = p;
        double r308269 = r308267 * r308268;
        double r308270 = r308269 * r308268;
        double r308271 = r308266 * r308266;
        double r308272 = r308270 + r308271;
        double r308273 = sqrt(r308272);
        double r308274 = r308266 / r308273;
        double r308275 = -1.0;
        bool r308276 = r308274 <= r308275;
        double r308277 = 2.0;
        double r308278 = sqrt(r308277);
        double r308279 = 0.5;
        double r308280 = sqrt(r308279);
        double r308281 = r308278 * r308280;
        double r308282 = log(r308281);
        double r308283 = -1.0;
        double r308284 = r308283 / r308266;
        double r308285 = log(r308284);
        double r308286 = r308282 + r308285;
        double r308287 = r308283 / r308268;
        double r308288 = log(r308287);
        double r308289 = r308286 - r308288;
        double r308290 = exp(r308289);
        double r308291 = 1.0;
        double r308292 = 3.0;
        double r308293 = pow(r308291, r308292);
        double r308294 = pow(r308274, r308292);
        double r308295 = r308293 + r308294;
        double r308296 = r308279 * r308295;
        double r308297 = sqrt(r308296);
        double r308298 = log(r308297);
        double r308299 = r308291 * r308291;
        double r308300 = r308274 * r308274;
        double r308301 = r308291 * r308274;
        double r308302 = r308300 - r308301;
        double r308303 = r308299 + r308302;
        double r308304 = sqrt(r308303);
        double r308305 = log(r308304);
        double r308306 = r308298 - r308305;
        double r308307 = exp(r308306);
        double r308308 = r308276 ? r308290 : r308307;
        return r308308;
}

Error

Bits error versus p

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Split input into 2 regimes

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

    1. Initial program 53.9

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

    3. Applied add-exp-log53.9

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

    4. Taylor expanded around -inf 35.1

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

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

    1. Initial program 0.3

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

    3. Applied add-exp-log0.3

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

    5. Applied flip3-+0.3

      \[\leadsto e^{\log \left(\sqrt{0.5 \cdot \color{blue}{\frac{{1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}{1 \cdot 1 + \left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} - 1 \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}}}\right)}\]

    6. Applied associate-*r/0.3

      \[\leadsto e^{\log \left(\sqrt{\color{blue}{\frac{0.5 \cdot \left({1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}\right)}{1 \cdot 1 + \left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} - 1 \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}}}\right)}\]

    7. Applied sqrt-div0.3

      \[\leadsto e^{\log \color{blue}{\left(\frac{\sqrt{0.5 \cdot \left({1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}\right)}}{\sqrt{1 \cdot 1 + \left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} - 1 \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}}\right)}}\]

    8. Applied log-div0.3

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

  4. Final simplification8.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \le -1:\\ \;\;\;\;e^{\left(\log \left(\sqrt{2} \cdot \sqrt{0.5}\right) + \log \left(\frac{-1}{x}\right)\right) - \log \left(\frac{-1}{p}\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\sqrt{0.5 \cdot \left({1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}\right)}\right) - \log \left(\sqrt{1 \cdot 1 + \left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} - 1 \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019344 +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))))))))