Average Error: 13.6 → 13.6
Time: 14.9s
Precision: 64
\[1.00000000000000001 \cdot 10^{-150} \lt \left|x\right| \lt 9.99999999999999981 \cdot 10^{149}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\frac{\sqrt{0.5 \cdot \log \left(e^{{1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}\right)}}{\sqrt{\frac{x}{\left|{\left(\left(4 \cdot p\right) \cdot p + x \cdot x\right)}^{\frac{1}{3}}\right| \cdot \sqrt{\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}}} \cdot \left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} - 1\right) + 1 \cdot 1}}\]
\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}
↓
\frac{\sqrt{0.5 \cdot \log \left(e^{{1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}\right)}}{\sqrt{\frac{x}{\left|{\left(\left(4 \cdot p\right) \cdot p + x \cdot x\right)}^{\frac{1}{3}}\right| \cdot \sqrt{\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}}} \cdot \left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} - 1\right) + 1 \cdot 1}}
double f(double p, double x) {
        double r175882 = 0.5;
        double r175883 = 1.0;
        double r175884 = x;
        double r175885 = 4.0;
        double r175886 = p;
        double r175887 = r175885 * r175886;
        double r175888 = r175887 * r175886;
        double r175889 = r175884 * r175884;
        double r175890 = r175888 + r175889;
        double r175891 = sqrt(r175890);
        double r175892 = r175884 / r175891;
        double r175893 = r175883 + r175892;
        double r175894 = r175882 * r175893;
        double r175895 = sqrt(r175894);
        return r175895;
}
↓
double f(double p, double x) {
        double r175896 = 0.5;
        double r175897 = 1.0;
        double r175898 = 3.0;
        double r175899 = pow(r175897, r175898);
        double r175900 = x;
        double r175901 = 4.0;
        double r175902 = p;
        double r175903 = r175901 * r175902;
        double r175904 = r175903 * r175902;
        double r175905 = r175900 * r175900;
        double r175906 = r175904 + r175905;
        double r175907 = sqrt(r175906);
        double r175908 = r175900 / r175907;
        double r175909 = pow(r175908, r175898);
        double r175910 = r175899 + r175909;
        double r175911 = exp(r175910);
        double r175912 = log(r175911);
        double r175913 = r175896 * r175912;
        double r175914 = sqrt(r175913);
        double r175915 = 0.3333333333333333;
        double r175916 = pow(r175906, r175915);
        double r175917 = fabs(r175916);
        double r175918 = cbrt(r175906);
        double r175919 = sqrt(r175918);
        double r175920 = r175917 * r175919;
        double r175921 = r175900 / r175920;
        double r175922 = r175908 - r175897;
        double r175923 = r175921 * r175922;
        double r175924 = r175897 * r175897;
        double r175925 = r175923 + r175924;
        double r175926 = sqrt(r175925);
        double r175927 = r175914 / r175926;
        return r175927;
}

Error

Bits error versus p

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.6
Target13.6
Herbie13.6
\[\sqrt{0.5 + \frac{\mathsf{copysign}\left(0.5, x\right)}{\mathsf{hypot}\left(1, \frac{2 \cdot p}{x}\right)}}\]

Derivation

  1. Initial program 13.6

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

  3. Applied flip3-+13.6

    \[\leadsto \sqrt{0.5 \cdot \color{blue}{\frac{{1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}{1 \cdot 1 + \left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} - 1 \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}}}\]

  4. Applied associate-*r/13.6

    \[\leadsto \sqrt{\color{blue}{\frac{0.5 \cdot \left({1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}\right)}{1 \cdot 1 + \left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} - 1 \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}}}\]

  5. Applied sqrt-div13.6

    \[\leadsto \color{blue}{\frac{\sqrt{0.5 \cdot \left({1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}\right)}}{\sqrt{1 \cdot 1 + \left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} - 1 \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}}}\]

  6. Simplified13.6

    \[\leadsto \frac{\sqrt{0.5 \cdot \left({1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}\right)}}{\color{blue}{\sqrt{\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} - 1\right) + 1 \cdot 1}}}\]
  7. Using strategy rm

  8. Applied add-log-exp13.6

    \[\leadsto \frac{\sqrt{0.5 \cdot \left({1}^{3} + \color{blue}{\log \left(e^{{\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}\right)}\right)}}{\sqrt{\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} - 1\right) + 1 \cdot 1}}\]

  9. Applied add-log-exp13.6

    \[\leadsto \frac{\sqrt{0.5 \cdot \left(\color{blue}{\log \left(e^{{1}^{3}}\right)} + \log \left(e^{{\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}\right)\right)}}{\sqrt{\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} - 1\right) + 1 \cdot 1}}\]

  10. Applied sum-log13.6

    \[\leadsto \frac{\sqrt{0.5 \cdot \color{blue}{\log \left(e^{{1}^{3}} \cdot e^{{\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}\right)}}}{\sqrt{\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} - 1\right) + 1 \cdot 1}}\]

  11. Simplified13.6

    \[\leadsto \frac{\sqrt{0.5 \cdot \log \color{blue}{\left(e^{{1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}\right)}}}{\sqrt{\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} - 1\right) + 1 \cdot 1}}\]
  12. Using strategy rm

  13. Applied add-cube-cbrt13.6

    \[\leadsto \frac{\sqrt{0.5 \cdot \log \left(e^{{1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}\right)}}{\sqrt{\frac{x}{\sqrt{\color{blue}{\left(\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x} \cdot \sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}\right) \cdot \sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}}}} \cdot \left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} - 1\right) + 1 \cdot 1}}\]

  14. Applied sqrt-prod13.6

    \[\leadsto \frac{\sqrt{0.5 \cdot \log \left(e^{{1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}\right)}}{\sqrt{\frac{x}{\color{blue}{\sqrt{\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x} \cdot \sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \sqrt{\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}}}} \cdot \left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} - 1\right) + 1 \cdot 1}}\]

  15. Simplified13.6

    \[\leadsto \frac{\sqrt{0.5 \cdot \log \left(e^{{1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}\right)}}{\sqrt{\frac{x}{\color{blue}{\left|\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}\right|} \cdot \sqrt{\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}}} \cdot \left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} - 1\right) + 1 \cdot 1}}\]
  16. Using strategy rm

  17. Applied pow1/313.6

    \[\leadsto \frac{\sqrt{0.5 \cdot \log \left(e^{{1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}\right)}}{\sqrt{\frac{x}{\left|\color{blue}{{\left(\left(4 \cdot p\right) \cdot p + x \cdot x\right)}^{\frac{1}{3}}}\right| \cdot \sqrt{\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}}} \cdot \left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} - 1\right) + 1 \cdot 1}}\]

  18. Final simplification13.6

    \[\leadsto \frac{\sqrt{0.5 \cdot \log \left(e^{{1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}\right)}}{\sqrt{\frac{x}{\left|{\left(\left(4 \cdot p\right) \cdot p + x \cdot x\right)}^{\frac{1}{3}}\right| \cdot \sqrt{\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}}} \cdot \left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} - 1\right) + 1 \cdot 1}}\]

Reproduce

herbie shell --seed 2019195 
(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))))))))