Average Error: 13.2 → 13.2
Time: 46.5s
Precision: 64
\[10^{-150} \lt \left|x\right| \lt 10^{+150}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\sqrt[3]{\left(0.5 + \frac{x}{\frac{\sqrt{x \cdot x + p \cdot \left(p \cdot 4\right)}}{0.5}}\right) \cdot \sqrt{0.5 + \frac{x}{\frac{\sqrt{x \cdot x + p \cdot \left(p \cdot 4\right)}}{0.5}}}}\]
\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}
↓
\sqrt[3]{\left(0.5 + \frac{x}{\frac{\sqrt{x \cdot x + p \cdot \left(p \cdot 4\right)}}{0.5}}\right) \cdot \sqrt{0.5 + \frac{x}{\frac{\sqrt{x \cdot x + p \cdot \left(p \cdot 4\right)}}{0.5}}}}
double f(double p, double x) {
        double r10242862 = 0.5;
        double r10242863 = 1.0;
        double r10242864 = x;
        double r10242865 = 4.0;
        double r10242866 = p;
        double r10242867 = r10242865 * r10242866;
        double r10242868 = r10242867 * r10242866;
        double r10242869 = r10242864 * r10242864;
        double r10242870 = r10242868 + r10242869;
        double r10242871 = sqrt(r10242870);
        double r10242872 = r10242864 / r10242871;
        double r10242873 = r10242863 + r10242872;
        double r10242874 = r10242862 * r10242873;
        double r10242875 = sqrt(r10242874);
        return r10242875;
}
↓
double f(double p, double x) {
        double r10242876 = 0.5;
        double r10242877 = x;
        double r10242878 = r10242877 * r10242877;
        double r10242879 = p;
        double r10242880 = 4.0;
        double r10242881 = r10242879 * r10242880;
        double r10242882 = r10242879 * r10242881;
        double r10242883 = r10242878 + r10242882;
        double r10242884 = sqrt(r10242883);
        double r10242885 = r10242884 / r10242876;
        double r10242886 = r10242877 / r10242885;
        double r10242887 = r10242876 + r10242886;
        double r10242888 = sqrt(r10242887);
        double r10242889 = r10242887 * r10242888;
        double r10242890 = cbrt(r10242889);
        return r10242890;
}

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

Derivation

  1. Initial program 13.2

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

  2. Simplified13.2

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

  4. Applied add-cbrt-cube13.2

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

  5. Simplified13.2

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

  6. Final simplification13.2

    \[\leadsto \sqrt[3]{\left(0.5 + \frac{x}{\frac{\sqrt{x \cdot x + p \cdot \left(p \cdot 4\right)}}{0.5}}\right) \cdot \sqrt{0.5 + \frac{x}{\frac{\sqrt{x \cdot x + p \cdot \left(p \cdot 4\right)}}{0.5}}}}\]

Reproduce

herbie shell --seed 2019162 
(FPCore (p x)
  :name "Given's Rotation SVD example"
  :pre (< 1e-150 (fabs x) 1e+150)

  :herbie-target
  (sqrt (+ 1/2 (/ (copysign 1/2 x) (hypot 1 (/ (* 2 p) x)))))

  (sqrt (* 0.5 (+ 1 (/ x (sqrt (+ (* (* 4 p) p) (* x x))))))))