Average Error: 13.7 → 13.7
Time: 49.2s
Precision: 64
\[10^{-150} \lt \left|x\right| \lt 10^{+150}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\sqrt[3]{\sqrt{\log \left(e^{0.5 + x \cdot \frac{0.5}{\sqrt{x \cdot x + p \cdot \left(p \cdot 4\right)}}}\right)} \cdot \left(0.5 + x \cdot \frac{0.5}{\sqrt{x \cdot x + p \cdot \left(p \cdot 4\right)}}\right)}\]
\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}
↓
\sqrt[3]{\sqrt{\log \left(e^{0.5 + x \cdot \frac{0.5}{\sqrt{x \cdot x + p \cdot \left(p \cdot 4\right)}}}\right)} \cdot \left(0.5 + x \cdot \frac{0.5}{\sqrt{x \cdot x + p \cdot \left(p \cdot 4\right)}}\right)}
double f(double p, double x) {
        double r95242972 = 0.5;
        double r95242973 = 1.0;
        double r95242974 = x;
        double r95242975 = 4.0;
        double r95242976 = p;
        double r95242977 = r95242975 * r95242976;
        double r95242978 = r95242977 * r95242976;
        double r95242979 = r95242974 * r95242974;
        double r95242980 = r95242978 + r95242979;
        double r95242981 = sqrt(r95242980);
        double r95242982 = r95242974 / r95242981;
        double r95242983 = r95242973 + r95242982;
        double r95242984 = r95242972 * r95242983;
        double r95242985 = sqrt(r95242984);
        return r95242985;
}
↓
double f(double p, double x) {
        double r95242986 = 0.5;
        double r95242987 = x;
        double r95242988 = r95242987 * r95242987;
        double r95242989 = p;
        double r95242990 = 4.0;
        double r95242991 = r95242989 * r95242990;
        double r95242992 = r95242989 * r95242991;
        double r95242993 = r95242988 + r95242992;
        double r95242994 = sqrt(r95242993);
        double r95242995 = r95242986 / r95242994;
        double r95242996 = r95242987 * r95242995;
        double r95242997 = r95242986 + r95242996;
        double r95242998 = exp(r95242997);
        double r95242999 = log(r95242998);
        double r95243000 = sqrt(r95242999);
        double r95243001 = r95243000 * r95242997;
        double r95243002 = cbrt(r95243001);
        return r95243002;
}

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
Target13.7
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-cbrt-cube13.7

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

  5. Simplified14.0

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

  7. Applied add-log-exp13.7

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

  8. Final simplification13.7

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

Reproduce

herbie shell --seed 2019125 
(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))))))))