Average Error: 13.0 → 13.3
Time: 15.0s
Precision: 64
\[10^{-150} \lt \left|x\right| \lt 10^{+150}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\sqrt{\frac{\left(0.5 \cdot 0.5\right) \cdot \left(\frac{0.5 \cdot \left(\left(x \cdot x\right) \cdot \frac{x}{x \cdot x + p \cdot \left(4 \cdot p\right)}\right)}{\sqrt{x \cdot x + p \cdot \left(4 \cdot p\right)}} + 0.5\right)}{0.5 \cdot 0.5 - \left(0.5 - \frac{0.5 \cdot x}{\sqrt{x \cdot x + p \cdot \left(4 \cdot p\right)}}\right) \cdot \frac{0.5 \cdot x}{\sqrt{x \cdot x + p \cdot \left(4 \cdot p\right)}}}}\]
\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}
↓
\sqrt{\frac{\left(0.5 \cdot 0.5\right) \cdot \left(\frac{0.5 \cdot \left(\left(x \cdot x\right) \cdot \frac{x}{x \cdot x + p \cdot \left(4 \cdot p\right)}\right)}{\sqrt{x \cdot x + p \cdot \left(4 \cdot p\right)}} + 0.5\right)}{0.5 \cdot 0.5 - \left(0.5 - \frac{0.5 \cdot x}{\sqrt{x \cdot x + p \cdot \left(4 \cdot p\right)}}\right) \cdot \frac{0.5 \cdot x}{\sqrt{x \cdot x + p \cdot \left(4 \cdot p\right)}}}}
double f(double p, double x) {
        double r7715913 = 0.5;
        double r7715914 = 1.0;
        double r7715915 = x;
        double r7715916 = 4.0;
        double r7715917 = p;
        double r7715918 = r7715916 * r7715917;
        double r7715919 = r7715918 * r7715917;
        double r7715920 = r7715915 * r7715915;
        double r7715921 = r7715919 + r7715920;
        double r7715922 = sqrt(r7715921);
        double r7715923 = r7715915 / r7715922;
        double r7715924 = r7715914 + r7715923;
        double r7715925 = r7715913 * r7715924;
        double r7715926 = sqrt(r7715925);
        return r7715926;
}
↓
double f(double p, double x) {
        double r7715927 = 0.5;
        double r7715928 = r7715927 * r7715927;
        double r7715929 = x;
        double r7715930 = r7715929 * r7715929;
        double r7715931 = p;
        double r7715932 = 4.0;
        double r7715933 = r7715932 * r7715931;
        double r7715934 = r7715931 * r7715933;
        double r7715935 = r7715930 + r7715934;
        double r7715936 = r7715929 / r7715935;
        double r7715937 = r7715930 * r7715936;
        double r7715938 = r7715927 * r7715937;
        double r7715939 = sqrt(r7715935);
        double r7715940 = r7715938 / r7715939;
        double r7715941 = r7715940 + r7715927;
        double r7715942 = r7715928 * r7715941;
        double r7715943 = r7715927 * r7715929;
        double r7715944 = r7715943 / r7715939;
        double r7715945 = r7715927 - r7715944;
        double r7715946 = r7715945 * r7715944;
        double r7715947 = r7715928 - r7715946;
        double r7715948 = r7715942 / r7715947;
        double r7715949 = sqrt(r7715948);
        return r7715949;
}

Error

Bits error versus p

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.0
Target13.0
Herbie13.3
\[\sqrt{\frac{1}{2} + \frac{\mathsf{copysign}\left(\frac{1}{2}, x\right)}{\mathsf{hypot}\left(1, \frac{2 \cdot p}{x}\right)}}\]

Derivation

  1. Initial program 13.0

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

  2. Simplified13.0

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

  4. Applied flip3-+13.0

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

  5. Simplified21.5

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

  6. Simplified21.5

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

  8. Applied *-un-lft-identity21.5

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

  9. Applied times-frac13.3

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

  10. Simplified13.3

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

  11. Final simplification13.3

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

Reproduce

herbie shell --seed 2019152 
(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))))))))