Average Error: 13.2 → 13.2
Time: 8.2s
Precision: 64
\[1.00000000000000001 \cdot 10^{-150} \lt \left|x\right| \lt 9.99999999999999981 \cdot 10^{149}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \le 1.1238977166951507 \cdot 10^{-10}:\\ \;\;\;\;\frac{\sqrt{0.5 \cdot \left(1 \cdot 1 - \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}}{\sqrt{1 - \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x} \cdot \sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}\right) \cdot \sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right)}}\right)\\ \end{array}\]
\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 1.1238977166951507 \cdot 10^{-10}:\\
\;\;\;\;\frac{\sqrt{0.5 \cdot \left(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}}\right)}}{\sqrt{1 - \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}}\\

\mathbf{else}:\\
\;\;\;\;\log \left(e^{\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x} \cdot \sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}\right) \cdot \sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right)}}\right)\\

\end{array}
double f(double p, double x) {
        double r319435 = 0.5;
        double r319436 = 1.0;
        double r319437 = x;
        double r319438 = 4.0;
        double r319439 = p;
        double r319440 = r319438 * r319439;
        double r319441 = r319440 * r319439;
        double r319442 = r319437 * r319437;
        double r319443 = r319441 + r319442;
        double r319444 = sqrt(r319443);
        double r319445 = r319437 / r319444;
        double r319446 = r319436 + r319445;
        double r319447 = r319435 * r319446;
        double r319448 = sqrt(r319447);
        return r319448;
}
↓
double f(double p, double x) {
        double r319449 = x;
        double r319450 = 4.0;
        double r319451 = p;
        double r319452 = r319450 * r319451;
        double r319453 = r319452 * r319451;
        double r319454 = r319449 * r319449;
        double r319455 = r319453 + r319454;
        double r319456 = sqrt(r319455);
        double r319457 = r319449 / r319456;
        double r319458 = 1.1238977166951507e-10;
        bool r319459 = r319457 <= r319458;
        double r319460 = 0.5;
        double r319461 = 1.0;
        double r319462 = r319461 * r319461;
        double r319463 = r319457 * r319457;
        double r319464 = r319462 - r319463;
        double r319465 = r319460 * r319464;
        double r319466 = sqrt(r319465);
        double r319467 = r319461 - r319457;
        double r319468 = sqrt(r319467);
        double r319469 = r319466 / r319468;
        double r319470 = cbrt(r319455);
        double r319471 = r319470 * r319470;
        double r319472 = r319471 * r319470;
        double r319473 = sqrt(r319472);
        double r319474 = r319449 / r319473;
        double r319475 = r319461 + r319474;
        double r319476 = r319460 * r319475;
        double r319477 = sqrt(r319476);
        double r319478 = exp(r319477);
        double r319479 = log(r319478);
        double r319480 = r319459 ? r319469 : r319479;
        return r319480;
}

Error

Bits error versus p

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.2
Target13.2
Herbie13.2
\[\sqrt{0.5 + \frac{\mathsf{copysign}\left(0.5, x\right)}{\mathsf{hypot}\left(1, \frac{2 \cdot p}{x}\right)}}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ x (sqrt (+ (* (* 4.0 p) p) (* x x)))) < 1.1238977166951507e-10

    1. Initial program 18.0

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

    3. Applied flip-+18.0

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

    4. Applied associate-*r/18.0

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

    5. Applied sqrt-div18.0

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

    if 1.1238977166951507e-10 < (/ 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 add-log-exp0.0

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

    5. Applied add-cube-cbrt0.0

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

  4. Final simplification13.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \le 1.1238977166951507 \cdot 10^{-10}:\\ \;\;\;\;\frac{\sqrt{0.5 \cdot \left(1 \cdot 1 - \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}}{\sqrt{1 - \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x} \cdot \sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}\right) \cdot \sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right)}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 +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))))))))