Average Error: 13.2 → 13.2
Time: 8.0s
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 r374997 = 0.5;
        double r374998 = 1.0;
        double r374999 = x;
        double r375000 = 4.0;
        double r375001 = p;
        double r375002 = r375000 * r375001;
        double r375003 = r375002 * r375001;
        double r375004 = r374999 * r374999;
        double r375005 = r375003 + r375004;
        double r375006 = sqrt(r375005);
        double r375007 = r374999 / r375006;
        double r375008 = r374998 + r375007;
        double r375009 = r374997 * r375008;
        double r375010 = sqrt(r375009);
        return r375010;
}
↓
double f(double p, double x) {
        double r375011 = x;
        double r375012 = 4.0;
        double r375013 = p;
        double r375014 = r375012 * r375013;
        double r375015 = r375014 * r375013;
        double r375016 = r375011 * r375011;
        double r375017 = r375015 + r375016;
        double r375018 = sqrt(r375017);
        double r375019 = r375011 / r375018;
        double r375020 = 1.1238977166951507e-10;
        bool r375021 = r375019 <= r375020;
        double r375022 = 0.5;
        double r375023 = 1.0;
        double r375024 = r375023 * r375023;
        double r375025 = r375019 * r375019;
        double r375026 = r375024 - r375025;
        double r375027 = r375022 * r375026;
        double r375028 = sqrt(r375027);
        double r375029 = r375023 - r375019;
        double r375030 = sqrt(r375029);
        double r375031 = r375028 / r375030;
        double r375032 = cbrt(r375017);
        double r375033 = r375032 * r375032;
        double r375034 = r375033 * r375032;
        double r375035 = sqrt(r375034);
        double r375036 = r375011 / r375035;
        double r375037 = r375023 + r375036;
        double r375038 = r375022 * r375037;
        double r375039 = sqrt(r375038);
        double r375040 = exp(r375039);
        double r375041 = log(r375040);
        double r375042 = r375021 ? r375031 : r375041;
        return r375042;
}

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))))))))