Average Error: 13.3 → 13.3
Time: 12.7s
Precision: 64
\[1.000000000000000006295358232172963997211 \cdot 10^{-150} \lt \left|x\right| \lt 9.999999999999999808355961724373745905731 \cdot 10^{149}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\sqrt{\left(1 + \frac{x}{\sqrt{x \cdot x + \left(p \cdot p\right) \cdot 4}}\right) \cdot 0.5}\]
\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}
↓
\sqrt{\left(1 + \frac{x}{\sqrt{x \cdot x + \left(p \cdot p\right) \cdot 4}}\right) \cdot 0.5}
double f(double p, double x) {
        double r383025 = 0.5;
        double r383026 = 1.0;
        double r383027 = x;
        double r383028 = 4.0;
        double r383029 = p;
        double r383030 = r383028 * r383029;
        double r383031 = r383030 * r383029;
        double r383032 = r383027 * r383027;
        double r383033 = r383031 + r383032;
        double r383034 = sqrt(r383033);
        double r383035 = r383027 / r383034;
        double r383036 = r383026 + r383035;
        double r383037 = r383025 * r383036;
        double r383038 = sqrt(r383037);
        return r383038;
}
↓
double f(double p, double x) {
        double r383039 = 1.0;
        double r383040 = x;
        double r383041 = r383040 * r383040;
        double r383042 = p;
        double r383043 = r383042 * r383042;
        double r383044 = 4.0;
        double r383045 = r383043 * r383044;
        double r383046 = r383041 + r383045;
        double r383047 = sqrt(r383046);
        double r383048 = r383040 / r383047;
        double r383049 = r383039 + r383048;
        double r383050 = 0.5;
        double r383051 = r383049 * r383050;
        double r383052 = sqrt(r383051);
        return r383052;
}

Error

Bits error versus p

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.3
Target13.3
Herbie13.3
\[\sqrt{0.5 + \frac{\mathsf{copysign}\left(0.5, x\right)}{\mathsf{hypot}\left(1, \frac{2 \cdot p}{x}\right)}}\]

Derivation

  1. Initial program 13.3

    \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
  2. Using strategy rm

  3. Applied add-cbrt-cube13.3

    \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \cdot \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right) \cdot \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}}}\]

  4. Simplified13.3

    \[\leadsto \sqrt[3]{\color{blue}{{\left(\sqrt{0.5 \cdot \left(\frac{x}{\sqrt{x \cdot x + \left(p \cdot p\right) \cdot 4}} + 1\right)}\right)}^{3}}}\]
  5. Using strategy rm

  6. Applied rem-cbrt-cube13.3

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

  7. Final simplification13.3

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

Reproduce

herbie shell --seed 2019196 
(FPCore (p x)
  :name "Given's Rotation SVD example"
  :pre (< 1e-150 (fabs x) 1e+150)

  :herbie-target
  (sqrt (+ 0.5 (/ (copysign 0.5 x) (hypot 1.0 (/ (* 2.0 p) x)))))

  (sqrt (* 0.5 (+ 1.0 (/ x (sqrt (+ (* (* 4.0 p) p) (* x x))))))))