Average Error: 13.5 → 14.0
Time: 43.8s
Precision: 64
\[10^{-150} \lt \left|x\right| \lt 10^{+150}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\sqrt{\sqrt[3]{\left(\frac{x}{\frac{\sqrt{x \cdot x + \left(p \cdot 4\right) \cdot p}}{0.5}} + 0.5\right) \cdot \left(\left(\frac{x}{\frac{\sqrt{x \cdot x + \left(p \cdot 4\right) \cdot p}}{0.5}} + 0.5\right) \cdot \left(\frac{x}{\frac{\sqrt{x \cdot x + \left(p \cdot 4\right) \cdot p}}{0.5}} + 0.5\right)\right)}}\]
\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}
↓
\sqrt{\sqrt[3]{\left(\frac{x}{\frac{\sqrt{x \cdot x + \left(p \cdot 4\right) \cdot p}}{0.5}} + 0.5\right) \cdot \left(\left(\frac{x}{\frac{\sqrt{x \cdot x + \left(p \cdot 4\right) \cdot p}}{0.5}} + 0.5\right) \cdot \left(\frac{x}{\frac{\sqrt{x \cdot x + \left(p \cdot 4\right) \cdot p}}{0.5}} + 0.5\right)\right)}}
double f(double p, double x) {
        double r10315251 = 0.5;
        double r10315252 = 1.0;
        double r10315253 = x;
        double r10315254 = 4.0;
        double r10315255 = p;
        double r10315256 = r10315254 * r10315255;
        double r10315257 = r10315256 * r10315255;
        double r10315258 = r10315253 * r10315253;
        double r10315259 = r10315257 + r10315258;
        double r10315260 = sqrt(r10315259);
        double r10315261 = r10315253 / r10315260;
        double r10315262 = r10315252 + r10315261;
        double r10315263 = r10315251 * r10315262;
        double r10315264 = sqrt(r10315263);
        return r10315264;
}
↓
double f(double p, double x) {
        double r10315265 = x;
        double r10315266 = r10315265 * r10315265;
        double r10315267 = p;
        double r10315268 = 4.0;
        double r10315269 = r10315267 * r10315268;
        double r10315270 = r10315269 * r10315267;
        double r10315271 = r10315266 + r10315270;
        double r10315272 = sqrt(r10315271);
        double r10315273 = 0.5;
        double r10315274 = r10315272 / r10315273;
        double r10315275 = r10315265 / r10315274;
        double r10315276 = r10315275 + r10315273;
        double r10315277 = r10315276 * r10315276;
        double r10315278 = r10315276 * r10315277;
        double r10315279 = cbrt(r10315278);
        double r10315280 = sqrt(r10315279);
        return r10315280;
}

Error

Bits error versus p

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.5
Target13.5
Herbie14.0
\[\sqrt{\frac{1}{2} + \frac{\mathsf{copysign}\left(\frac{1}{2}, x\right)}{\mathsf{hypot}\left(1, \frac{2 \cdot p}{x}\right)}}\]

Derivation

  1. Initial program 13.5

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

  2. Simplified13.5

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

  4. Applied add-cbrt-cube14.0

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

  5. Final simplification14.0

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

Reproduce

herbie shell --seed 2019158 
(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))))))))