Average Error: 12.9 → 12.9
Time: 18.4s
Precision: 64
\[10^{-150} \lt \left|x\right| \lt 10^{+150}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\sqrt{\log \left(e^{0.5 + \frac{x \cdot 0.5}{\sqrt{x \cdot x + \left(p \cdot 4\right) \cdot p}}}\right)}\]
\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 \cdot 0.5}{\sqrt{x \cdot x + \left(p \cdot 4\right) \cdot p}}}\right)}
double f(double p, double x) {
        double r3608675 = 0.5;
        double r3608676 = 1.0;
        double r3608677 = x;
        double r3608678 = 4.0;
        double r3608679 = p;
        double r3608680 = r3608678 * r3608679;
        double r3608681 = r3608680 * r3608679;
        double r3608682 = r3608677 * r3608677;
        double r3608683 = r3608681 + r3608682;
        double r3608684 = sqrt(r3608683);
        double r3608685 = r3608677 / r3608684;
        double r3608686 = r3608676 + r3608685;
        double r3608687 = r3608675 * r3608686;
        double r3608688 = sqrt(r3608687);
        return r3608688;
}
↓
double f(double p, double x) {
        double r3608689 = 0.5;
        double r3608690 = x;
        double r3608691 = r3608690 * r3608689;
        double r3608692 = r3608690 * r3608690;
        double r3608693 = p;
        double r3608694 = 4.0;
        double r3608695 = r3608693 * r3608694;
        double r3608696 = r3608695 * r3608693;
        double r3608697 = r3608692 + r3608696;
        double r3608698 = sqrt(r3608697);
        double r3608699 = r3608691 / r3608698;
        double r3608700 = r3608689 + r3608699;
        double r3608701 = exp(r3608700);
        double r3608702 = log(r3608701);
        double r3608703 = sqrt(r3608702);
        return r3608703;
}

Error

Bits error versus p

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original12.9
Target12.9
Herbie12.9
\[\sqrt{\frac{1}{2} + \frac{\mathsf{copysign}\left(\frac{1}{2}, x\right)}{\mathsf{hypot}\left(1, \frac{2 \cdot p}{x}\right)}}\]

Derivation

  1. Initial program 12.9

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

  2. Simplified12.9

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

  4. Applied add-sqr-sqrt12.9

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

  5. Applied sqrt-prod13.9

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

  7. Applied add-log-exp13.9

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

  8. Applied add-log-exp13.6

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

  9. Applied sum-log13.6

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

  10. Simplified12.9

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

  11. Final simplification12.9

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

Reproduce

herbie shell --seed 2019155 
(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))))))))