Average Error: 12.8 → 12.8
Time: 20.7s
Precision: 64
\[10^{-150} \lt \left|x\right| \lt 10^{+150}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\sqrt{\left(\sqrt[3]{0.5 + \frac{x}{\sqrt{x \cdot x + \left(p \cdot p\right) \cdot 4}} \cdot 0.5} \cdot \sqrt[3]{0.5 + \frac{x}{\sqrt{x \cdot x + \left(p \cdot p\right) \cdot 4}} \cdot 0.5}\right) \cdot \sqrt[3]{0.5 + \frac{x}{\sqrt{x \cdot x + \left(p \cdot p\right) \cdot 4}} \cdot 0.5}}\]
\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}
↓
\sqrt{\left(\sqrt[3]{0.5 + \frac{x}{\sqrt{x \cdot x + \left(p \cdot p\right) \cdot 4}} \cdot 0.5} \cdot \sqrt[3]{0.5 + \frac{x}{\sqrt{x \cdot x + \left(p \cdot p\right) \cdot 4}} \cdot 0.5}\right) \cdot \sqrt[3]{0.5 + \frac{x}{\sqrt{x \cdot x + \left(p \cdot p\right) \cdot 4}} \cdot 0.5}}
double f(double p, double x) {
        double r5803635 = 0.5;
        double r5803636 = 1.0;
        double r5803637 = x;
        double r5803638 = 4.0;
        double r5803639 = p;
        double r5803640 = r5803638 * r5803639;
        double r5803641 = r5803640 * r5803639;
        double r5803642 = r5803637 * r5803637;
        double r5803643 = r5803641 + r5803642;
        double r5803644 = sqrt(r5803643);
        double r5803645 = r5803637 / r5803644;
        double r5803646 = r5803636 + r5803645;
        double r5803647 = r5803635 * r5803646;
        double r5803648 = sqrt(r5803647);
        return r5803648;
}
↓
double f(double p, double x) {
        double r5803649 = 0.5;
        double r5803650 = x;
        double r5803651 = r5803650 * r5803650;
        double r5803652 = p;
        double r5803653 = r5803652 * r5803652;
        double r5803654 = 4.0;
        double r5803655 = r5803653 * r5803654;
        double r5803656 = r5803651 + r5803655;
        double r5803657 = sqrt(r5803656);
        double r5803658 = r5803650 / r5803657;
        double r5803659 = r5803658 * r5803649;
        double r5803660 = r5803649 + r5803659;
        double r5803661 = cbrt(r5803660);
        double r5803662 = r5803661 * r5803661;
        double r5803663 = r5803662 * r5803661;
        double r5803664 = sqrt(r5803663);
        return r5803664;
}

Error

Bits error versus p

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original12.8
Target12.8
Herbie12.8
\[\sqrt{\frac{1}{2} + \frac{\mathsf{copysign}\left(\frac{1}{2}, x\right)}{\mathsf{hypot}\left(1, \frac{2 \cdot p}{x}\right)}}\]

Derivation

  1. Initial program 12.8

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

  2. Simplified12.8

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

  4. Applied add-cube-cbrt12.8

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

  5. Final simplification12.8

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

Reproduce

herbie shell --seed 2019142 
(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))))))))