Average Error: 13.2 → 13.2
Time: 16.2s
Precision: 64
\[10^{-150} \lt \left|x\right| \lt 10^{+150}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\sqrt{\frac{x}{\frac{\sqrt{x \cdot x + \left(p \cdot 4\right) \cdot p}}{0.5}} + 0.5}\]
\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}
↓
\sqrt{\frac{x}{\frac{\sqrt{x \cdot x + \left(p \cdot 4\right) \cdot p}}{0.5}} + 0.5}
double f(double p, double x) {
        double r4338997 = 0.5;
        double r4338998 = 1.0;
        double r4338999 = x;
        double r4339000 = 4.0;
        double r4339001 = p;
        double r4339002 = r4339000 * r4339001;
        double r4339003 = r4339002 * r4339001;
        double r4339004 = r4338999 * r4338999;
        double r4339005 = r4339003 + r4339004;
        double r4339006 = sqrt(r4339005);
        double r4339007 = r4338999 / r4339006;
        double r4339008 = r4338998 + r4339007;
        double r4339009 = r4338997 * r4339008;
        double r4339010 = sqrt(r4339009);
        return r4339010;
}
↓
double f(double p, double x) {
        double r4339011 = x;
        double r4339012 = r4339011 * r4339011;
        double r4339013 = p;
        double r4339014 = 4.0;
        double r4339015 = r4339013 * r4339014;
        double r4339016 = r4339015 * r4339013;
        double r4339017 = r4339012 + r4339016;
        double r4339018 = sqrt(r4339017);
        double r4339019 = 0.5;
        double r4339020 = r4339018 / r4339019;
        double r4339021 = r4339011 / r4339020;
        double r4339022 = r4339021 + r4339019;
        double r4339023 = sqrt(r4339022);
        return r4339023;
}

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. Final simplification13.2

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

Reproduce

herbie shell --seed 2019153 
(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))))))))