Average Error: 13.7 → 13.7
Time: 18.1s
Precision: 64
\[10^{-150} \lt \left|x\right| \lt 10^{+150}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\sqrt{\log \left(e^{0.5 + \frac{x}{\frac{\sqrt{x \cdot x + \left(p \cdot p\right) \cdot 4}}{0.5}}}\right)}\]
\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}
↓
\sqrt{\log \left(e^{0.5 + \frac{x}{\frac{\sqrt{x \cdot x + \left(p \cdot p\right) \cdot 4}}{0.5}}}\right)}
double f(double p, double x) {
        double r8348675 = 0.5;
        double r8348676 = 1.0;
        double r8348677 = x;
        double r8348678 = 4.0;
        double r8348679 = p;
        double r8348680 = r8348678 * r8348679;
        double r8348681 = r8348680 * r8348679;
        double r8348682 = r8348677 * r8348677;
        double r8348683 = r8348681 + r8348682;
        double r8348684 = sqrt(r8348683);
        double r8348685 = r8348677 / r8348684;
        double r8348686 = r8348676 + r8348685;
        double r8348687 = r8348675 * r8348686;
        double r8348688 = sqrt(r8348687);
        return r8348688;
}
↓
double f(double p, double x) {
        double r8348689 = 0.5;
        double r8348690 = x;
        double r8348691 = r8348690 * r8348690;
        double r8348692 = p;
        double r8348693 = r8348692 * r8348692;
        double r8348694 = 4.0;
        double r8348695 = r8348693 * r8348694;
        double r8348696 = r8348691 + r8348695;
        double r8348697 = sqrt(r8348696);
        double r8348698 = r8348697 / r8348689;
        double r8348699 = r8348690 / r8348698;
        double r8348700 = r8348689 + r8348699;
        double r8348701 = exp(r8348700);
        double r8348702 = log(r8348701);
        double r8348703 = sqrt(r8348702);
        return r8348703;
}

Error

Bits error versus p

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.7
Target13.6
Herbie13.7
\[\sqrt{\frac{1}{2} + \frac{\mathsf{copysign}\left(\frac{1}{2}, x\right)}{\mathsf{hypot}\left(1, \left(\frac{2 \cdot p}{x}\right)\right)}}\]

Derivation

  1. Initial program 13.7

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

  2. Simplified13.7

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

  4. Applied add-log-exp13.7

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

  5. Applied add-log-exp13.7

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

  6. Applied sum-log13.7

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

  7. Simplified13.7

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

  8. Final simplification13.7

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

Reproduce

herbie shell --seed 2019130 
(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))))))))