Average Error: 13.6 → 13.6
Time: 17.7s
Precision: 64
\[10^{-150} \lt \left|x\right| \lt 10^{+150}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\sqrt{\left(\sqrt[3]{\log \left(e^{0.5 + \frac{0.5}{\frac{\sqrt{x \cdot x + p \cdot \left(4 \cdot p\right)}}{x}}}\right)} \cdot \sqrt[3]{\log \left(e^{0.5 + \frac{0.5}{\frac{\sqrt{x \cdot x + p \cdot \left(4 \cdot p\right)}}{x}}}\right)}\right) \cdot \sqrt[3]{\log \left(e^{0.5 + \frac{0.5}{\frac{\sqrt{x \cdot x + p \cdot \left(4 \cdot p\right)}}{x}}}\right)}}\]
\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}
↓
\sqrt{\left(\sqrt[3]{\log \left(e^{0.5 + \frac{0.5}{\frac{\sqrt{x \cdot x + p \cdot \left(4 \cdot p\right)}}{x}}}\right)} \cdot \sqrt[3]{\log \left(e^{0.5 + \frac{0.5}{\frac{\sqrt{x \cdot x + p \cdot \left(4 \cdot p\right)}}{x}}}\right)}\right) \cdot \sqrt[3]{\log \left(e^{0.5 + \frac{0.5}{\frac{\sqrt{x \cdot x + p \cdot \left(4 \cdot p\right)}}{x}}}\right)}}
double f(double p, double x) {
        double r3679796 = 0.5;
        double r3679797 = 1.0;
        double r3679798 = x;
        double r3679799 = 4.0;
        double r3679800 = p;
        double r3679801 = r3679799 * r3679800;
        double r3679802 = r3679801 * r3679800;
        double r3679803 = r3679798 * r3679798;
        double r3679804 = r3679802 + r3679803;
        double r3679805 = sqrt(r3679804);
        double r3679806 = r3679798 / r3679805;
        double r3679807 = r3679797 + r3679806;
        double r3679808 = r3679796 * r3679807;
        double r3679809 = sqrt(r3679808);
        return r3679809;
}
↓
double f(double p, double x) {
        double r3679810 = 0.5;
        double r3679811 = x;
        double r3679812 = r3679811 * r3679811;
        double r3679813 = p;
        double r3679814 = 4.0;
        double r3679815 = r3679814 * r3679813;
        double r3679816 = r3679813 * r3679815;
        double r3679817 = r3679812 + r3679816;
        double r3679818 = sqrt(r3679817);
        double r3679819 = r3679818 / r3679811;
        double r3679820 = r3679810 / r3679819;
        double r3679821 = r3679810 + r3679820;
        double r3679822 = exp(r3679821);
        double r3679823 = log(r3679822);
        double r3679824 = cbrt(r3679823);
        double r3679825 = r3679824 * r3679824;
        double r3679826 = r3679825 * r3679824;
        double r3679827 = sqrt(r3679826);
        return r3679827;
}

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

Derivation

  1. Initial program 13.6

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

  2. Simplified13.6

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

  4. Applied add-log-exp13.6

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

  5. Applied add-log-exp13.6

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

  6. Applied sum-log13.6

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

  7. Simplified13.6

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

  9. Applied add-cube-cbrt13.6

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

  10. Final simplification13.6

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

Reproduce

herbie shell --seed 2019144 
(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))))))))