Average Error: 13.2 → 13.2
Time: 45.5s
Precision: 64
\[10^{-150} \lt \left|x\right| \lt 10^{+150}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\sqrt[3]{\left(0.5 + \frac{x}{\frac{\sqrt{x \cdot x + p \cdot \left(p \cdot 4\right)}}{0.5}}\right) \cdot \sqrt{0.5 + \frac{x}{\frac{\sqrt{x \cdot x + p \cdot \left(p \cdot 4\right)}}{0.5}}}}\]
\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}
↓
\sqrt[3]{\left(0.5 + \frac{x}{\frac{\sqrt{x \cdot x + p \cdot \left(p \cdot 4\right)}}{0.5}}\right) \cdot \sqrt{0.5 + \frac{x}{\frac{\sqrt{x \cdot x + p \cdot \left(p \cdot 4\right)}}{0.5}}}}
double f(double p, double x) {
        double r11002601 = 0.5;
        double r11002602 = 1.0;
        double r11002603 = x;
        double r11002604 = 4.0;
        double r11002605 = p;
        double r11002606 = r11002604 * r11002605;
        double r11002607 = r11002606 * r11002605;
        double r11002608 = r11002603 * r11002603;
        double r11002609 = r11002607 + r11002608;
        double r11002610 = sqrt(r11002609);
        double r11002611 = r11002603 / r11002610;
        double r11002612 = r11002602 + r11002611;
        double r11002613 = r11002601 * r11002612;
        double r11002614 = sqrt(r11002613);
        return r11002614;
}
↓
double f(double p, double x) {
        double r11002615 = 0.5;
        double r11002616 = x;
        double r11002617 = r11002616 * r11002616;
        double r11002618 = p;
        double r11002619 = 4.0;
        double r11002620 = r11002618 * r11002619;
        double r11002621 = r11002618 * r11002620;
        double r11002622 = r11002617 + r11002621;
        double r11002623 = sqrt(r11002622);
        double r11002624 = r11002623 / r11002615;
        double r11002625 = r11002616 / r11002624;
        double r11002626 = r11002615 + r11002625;
        double r11002627 = sqrt(r11002626);
        double r11002628 = r11002626 * r11002627;
        double r11002629 = cbrt(r11002628);
        return r11002629;
}

Error

Bits error versus p

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 13.2

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

  2. Simplified13.2

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

  4. Applied add-cbrt-cube13.2

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

  5. Simplified13.2

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

  6. Final simplification13.2

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

Reproduce

herbie shell --seed 2019162 
(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))))))))