Average Error: 13.2 → 13.2
Time: 27.7s
Precision: 64
\[10^{-150} \lt \left|x\right| \lt 10^{+150}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.0566411135844027 \cdot 10^{+32}:\\ \;\;\;\;\sqrt{e^{\log \left(\frac{\left(0.5 \cdot \frac{x}{\sqrt{\left(p \cdot 4\right) \cdot p + x \cdot x}}\right) \cdot \left(0.5 \cdot \frac{x}{\sqrt{\left(p \cdot 4\right) \cdot p + x \cdot x}}\right) - 0.5 \cdot 0.5}{0.5 \cdot \frac{x}{\sqrt{\left(p \cdot 4\right) \cdot p + x \cdot x}} - 0.5}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \sqrt[3]{\left(\left(0.5 \cdot \frac{x}{\sqrt{\left(p \cdot 4\right) \cdot p + x \cdot x}}\right) \cdot \left(0.5 \cdot \frac{x}{\sqrt{\left(p \cdot 4\right) \cdot p + x \cdot x}}\right)\right) \cdot \left(0.5 \cdot \frac{x}{\sqrt{\left(p \cdot 4\right) \cdot p + x \cdot x}}\right)}}\\ \end{array}\]
\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}\;x \le -1.0566411135844027 \cdot 10^{+32}:\\
\;\;\;\;\sqrt{e^{\log \left(\frac{\left(0.5 \cdot \frac{x}{\sqrt{\left(p \cdot 4\right) \cdot p + x \cdot x}}\right) \cdot \left(0.5 \cdot \frac{x}{\sqrt{\left(p \cdot 4\right) \cdot p + x \cdot x}}\right) - 0.5 \cdot 0.5}{0.5 \cdot \frac{x}{\sqrt{\left(p \cdot 4\right) \cdot p + x \cdot x}} - 0.5}\right)}}\\

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

\end{array}
double f(double p, double x) {
        double r74063924 = 0.5;
        double r74063925 = 1.0;
        double r74063926 = x;
        double r74063927 = 4.0;
        double r74063928 = p;
        double r74063929 = r74063927 * r74063928;
        double r74063930 = r74063929 * r74063928;
        double r74063931 = r74063926 * r74063926;
        double r74063932 = r74063930 + r74063931;
        double r74063933 = sqrt(r74063932);
        double r74063934 = r74063926 / r74063933;
        double r74063935 = r74063925 + r74063934;
        double r74063936 = r74063924 * r74063935;
        double r74063937 = sqrt(r74063936);
        return r74063937;
}
↓
double f(double p, double x) {
        double r74063938 = x;
        double r74063939 = -1.0566411135844027e+32;
        bool r74063940 = r74063938 <= r74063939;
        double r74063941 = 0.5;
        double r74063942 = p;
        double r74063943 = 4.0;
        double r74063944 = r74063942 * r74063943;
        double r74063945 = r74063944 * r74063942;
        double r74063946 = r74063938 * r74063938;
        double r74063947 = r74063945 + r74063946;
        double r74063948 = sqrt(r74063947);
        double r74063949 = r74063938 / r74063948;
        double r74063950 = r74063941 * r74063949;
        double r74063951 = r74063950 * r74063950;
        double r74063952 = r74063941 * r74063941;
        double r74063953 = r74063951 - r74063952;
        double r74063954 = r74063950 - r74063941;
        double r74063955 = r74063953 / r74063954;
        double r74063956 = log(r74063955);
        double r74063957 = exp(r74063956);
        double r74063958 = sqrt(r74063957);
        double r74063959 = r74063951 * r74063950;
        double r74063960 = cbrt(r74063959);
        double r74063961 = r74063941 + r74063960;
        double r74063962 = sqrt(r74063961);
        double r74063963 = r74063940 ? r74063958 : r74063962;
        return r74063963;
}

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

Derivation

  1. Split input into 2 regimes

  2. if x < -1.0566411135844027e+32

    1. Initial program 30.9

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

    2. Simplified30.9

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

    4. Applied add-exp-log30.9

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

    6. Applied flip-+30.9

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

    if -1.0566411135844027e+32 < x

    1. Initial program 8.8

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

    2. Simplified8.8

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

    4. Applied add-cbrt-cube8.8

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

  4. Final simplification13.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0566411135844027 \cdot 10^{+32}:\\ \;\;\;\;\sqrt{e^{\log \left(\frac{\left(0.5 \cdot \frac{x}{\sqrt{\left(p \cdot 4\right) \cdot p + x \cdot x}}\right) \cdot \left(0.5 \cdot \frac{x}{\sqrt{\left(p \cdot 4\right) \cdot p + x \cdot x}}\right) - 0.5 \cdot 0.5}{0.5 \cdot \frac{x}{\sqrt{\left(p \cdot 4\right) \cdot p + x \cdot x}} - 0.5}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \sqrt[3]{\left(\left(0.5 \cdot \frac{x}{\sqrt{\left(p \cdot 4\right) \cdot p + x \cdot x}}\right) \cdot \left(0.5 \cdot \frac{x}{\sqrt{\left(p \cdot 4\right) \cdot p + x \cdot x}}\right)\right) \cdot \left(0.5 \cdot \frac{x}{\sqrt{\left(p \cdot 4\right) \cdot p + x \cdot x}}\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019124 
(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))))))))