Average Error: 13.3 → 13.7
Time: 14.3s
Precision: 64
\[1.000000000000000006295358232172963997211 \cdot 10^{-150} \lt \left|x\right| \lt 9.999999999999999808355961724373745905731 \cdot 10^{149}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\sqrt{0.5 \cdot e^{\left(\sqrt[3]{\log \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \cdot \sqrt[3]{\log \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right) \cdot \sqrt[3]{\log \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}}}\]
\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}
↓
\sqrt{0.5 \cdot e^{\left(\sqrt[3]{\log \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \cdot \sqrt[3]{\log \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right) \cdot \sqrt[3]{\log \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}}}
double f(double p, double x) {
        double r158153 = 0.5;
        double r158154 = 1.0;
        double r158155 = x;
        double r158156 = 4.0;
        double r158157 = p;
        double r158158 = r158156 * r158157;
        double r158159 = r158158 * r158157;
        double r158160 = r158155 * r158155;
        double r158161 = r158159 + r158160;
        double r158162 = sqrt(r158161);
        double r158163 = r158155 / r158162;
        double r158164 = r158154 + r158163;
        double r158165 = r158153 * r158164;
        double r158166 = sqrt(r158165);
        return r158166;
}
↓
double f(double p, double x) {
        double r158167 = 0.5;
        double r158168 = 1.0;
        double r158169 = x;
        double r158170 = 4.0;
        double r158171 = p;
        double r158172 = r158170 * r158171;
        double r158173 = r158172 * r158171;
        double r158174 = r158169 * r158169;
        double r158175 = r158173 + r158174;
        double r158176 = sqrt(r158175);
        double r158177 = r158169 / r158176;
        double r158178 = r158168 + r158177;
        double r158179 = log(r158178);
        double r158180 = cbrt(r158179);
        double r158181 = r158180 * r158180;
        double r158182 = r158181 * r158180;
        double r158183 = exp(r158182);
        double r158184 = r158167 * r158183;
        double r158185 = sqrt(r158184);
        return r158185;
}

Error

Bits error versus p

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.3
Target13.3
Herbie13.7
\[\sqrt{0.5 + \frac{\mathsf{copysign}\left(0.5, x\right)}{\mathsf{hypot}\left(1, \frac{2 \cdot p}{x}\right)}}\]

Derivation

  1. Initial program 13.3

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

  3. Applied add-exp-log13.3

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

  5. Applied add-cube-cbrt13.7

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

  6. Final simplification13.7

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

Reproduce

herbie shell --seed 2019326 
(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))))))))