Average Error: 13.9 → 13.9
Time: 16.2s
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 r9821387 = 0.5;
        double r9821388 = 1.0;
        double r9821389 = x;
        double r9821390 = 4.0;
        double r9821391 = p;
        double r9821392 = r9821390 * r9821391;
        double r9821393 = r9821392 * r9821391;
        double r9821394 = r9821389 * r9821389;
        double r9821395 = r9821393 + r9821394;
        double r9821396 = sqrt(r9821395);
        double r9821397 = r9821389 / r9821396;
        double r9821398 = r9821388 + r9821397;
        double r9821399 = r9821387 * r9821398;
        double r9821400 = sqrt(r9821399);
        return r9821400;
}
↓
double f(double p, double x) {
        double r9821401 = x;
        double r9821402 = r9821401 * r9821401;
        double r9821403 = p;
        double r9821404 = 4.0;
        double r9821405 = r9821403 * r9821404;
        double r9821406 = r9821403 * r9821405;
        double r9821407 = r9821402 + r9821406;
        double r9821408 = sqrt(r9821407);
        double r9821409 = r9821401 / r9821408;
        double r9821410 = -0.9999964161259105;
        bool r9821411 = r9821409 <= r9821410;
        double r9821412 = 0.5;
        double r9821413 = 1.0;
        double r9821414 = r9821413 * r9821413;
        double r9821415 = r9821409 * r9821409;
        double r9821416 = r9821414 - r9821415;
        double r9821417 = r9821413 - r9821409;
        double r9821418 = r9821416 / r9821417;
        double r9821419 = r9821412 * r9821418;
        double r9821420 = sqrt(r9821419);
        double r9821421 = 1.0;
        double r9821422 = r9821421 / r9821408;
        double r9821423 = sqrt(r9821422);
        double r9821424 = r9821401 * r9821423;
        double r9821425 = r9821424 * r9821423;
        double r9821426 = r9821425 + r9821413;
        double r9821427 = r9821412 * r9821426;
        double r9821428 = sqrt(r9821427);
        double r9821429 = r9821411 ? r9821420 : r9821428;
        return r9821429;
}

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))))))))