Average Error: 13.0 → 13.0
Time: 22.7s
Precision: 64
\[10^{-150} \lt \left|x\right| \lt 10^{+150}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\sqrt{\frac{0.5 \cdot \left(0.5 \cdot 0.5\right) + \frac{x \cdot 0.5}{\sqrt{x \cdot x + \left(p \cdot 4\right) \cdot p}} \cdot \left(\frac{x \cdot 0.5}{\sqrt{x \cdot x + \left(p \cdot 4\right) \cdot p}} \cdot \frac{x \cdot 0.5}{\sqrt{x \cdot x + \left(p \cdot 4\right) \cdot p}}\right)}{\left(0.5 - \frac{x \cdot 0.5}{\sqrt{x \cdot x + \left(p \cdot 4\right) \cdot p}}\right) \cdot 0.5 + \frac{x \cdot 0.5}{\sqrt{x \cdot x + \left(p \cdot 4\right) \cdot p}} \cdot \frac{x \cdot 0.5}{\sqrt{x \cdot x + \left(p \cdot 4\right) \cdot p}}}}\]
\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}
↓
\sqrt{\frac{0.5 \cdot \left(0.5 \cdot 0.5\right) + \frac{x \cdot 0.5}{\sqrt{x \cdot x + \left(p \cdot 4\right) \cdot p}} \cdot \left(\frac{x \cdot 0.5}{\sqrt{x \cdot x + \left(p \cdot 4\right) \cdot p}} \cdot \frac{x \cdot 0.5}{\sqrt{x \cdot x + \left(p \cdot 4\right) \cdot p}}\right)}{\left(0.5 - \frac{x \cdot 0.5}{\sqrt{x \cdot x + \left(p \cdot 4\right) \cdot p}}\right) \cdot 0.5 + \frac{x \cdot 0.5}{\sqrt{x \cdot x + \left(p \cdot 4\right) \cdot p}} \cdot \frac{x \cdot 0.5}{\sqrt{x \cdot x + \left(p \cdot 4\right) \cdot p}}}}
double f(double p, double x) {
        double r4076181 = 0.5;
        double r4076182 = 1.0;
        double r4076183 = x;
        double r4076184 = 4.0;
        double r4076185 = p;
        double r4076186 = r4076184 * r4076185;
        double r4076187 = r4076186 * r4076185;
        double r4076188 = r4076183 * r4076183;
        double r4076189 = r4076187 + r4076188;
        double r4076190 = sqrt(r4076189);
        double r4076191 = r4076183 / r4076190;
        double r4076192 = r4076182 + r4076191;
        double r4076193 = r4076181 * r4076192;
        double r4076194 = sqrt(r4076193);
        return r4076194;
}
↓
double f(double p, double x) {
        double r4076195 = 0.5;
        double r4076196 = r4076195 * r4076195;
        double r4076197 = r4076195 * r4076196;
        double r4076198 = x;
        double r4076199 = r4076198 * r4076195;
        double r4076200 = r4076198 * r4076198;
        double r4076201 = p;
        double r4076202 = 4.0;
        double r4076203 = r4076201 * r4076202;
        double r4076204 = r4076203 * r4076201;
        double r4076205 = r4076200 + r4076204;
        double r4076206 = sqrt(r4076205);
        double r4076207 = r4076199 / r4076206;
        double r4076208 = r4076207 * r4076207;
        double r4076209 = r4076207 * r4076208;
        double r4076210 = r4076197 + r4076209;
        double r4076211 = r4076195 - r4076207;
        double r4076212 = r4076211 * r4076195;
        double r4076213 = r4076212 + r4076208;
        double r4076214 = r4076210 / r4076213;
        double r4076215 = sqrt(r4076214);
        return r4076215;
}

Error

Bits error versus p

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.0
Target13.0
Herbie13.0
\[\sqrt{\frac{1}{2} + \frac{\mathsf{copysign}\left(\frac{1}{2}, x\right)}{\mathsf{hypot}\left(1, \frac{2 \cdot p}{x}\right)}}\]

Derivation

  1. Initial program 13.0

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

  2. Simplified13.0

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

  4. Applied flip3-+13.0

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

  5. Simplified13.0

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

  6. Simplified13.0

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

  7. Final simplification13.0

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

Reproduce

herbie shell --seed 2019152 
(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))))))))