Average Error: 13.4 → 14.8
Time: 5.4s
Precision: 64
\[1.000000000000000006295358232172963997211 \cdot 10^{-150} \lt \left|x\right| \lt 9.999999999999999808355961724373745905731 \cdot 10^{149}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\sqrt{0.5 \cdot \left(1 + x \cdot \frac{1}{\sqrt{\left(\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x} \cdot \sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}\right) \cdot \sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right)}\]
\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}
↓
\sqrt{0.5 \cdot \left(1 + x \cdot \frac{1}{\sqrt{\left(\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x} \cdot \sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}\right) \cdot \sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right)}
double f(double p, double x) {
        double r291245 = 0.5;
        double r291246 = 1.0;
        double r291247 = x;
        double r291248 = 4.0;
        double r291249 = p;
        double r291250 = r291248 * r291249;
        double r291251 = r291250 * r291249;
        double r291252 = r291247 * r291247;
        double r291253 = r291251 + r291252;
        double r291254 = sqrt(r291253);
        double r291255 = r291247 / r291254;
        double r291256 = r291246 + r291255;
        double r291257 = r291245 * r291256;
        double r291258 = sqrt(r291257);
        return r291258;
}
↓
double f(double p, double x) {
        double r291259 = 0.5;
        double r291260 = 1.0;
        double r291261 = x;
        double r291262 = 1.0;
        double r291263 = 4.0;
        double r291264 = p;
        double r291265 = r291263 * r291264;
        double r291266 = r291265 * r291264;
        double r291267 = r291261 * r291261;
        double r291268 = r291266 + r291267;
        double r291269 = cbrt(r291268);
        double r291270 = r291269 * r291269;
        double r291271 = r291270 * r291269;
        double r291272 = sqrt(r291271);
        double r291273 = r291262 / r291272;
        double r291274 = r291261 * r291273;
        double r291275 = r291260 + r291274;
        double r291276 = r291259 * r291275;
        double r291277 = sqrt(r291276);
        return r291277;
}

Error

Bits error versus p

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.4
Target13.4
Herbie14.8
\[\sqrt{0.5 + \frac{\mathsf{copysign}\left(0.5, x\right)}{\mathsf{hypot}\left(1, \frac{2 \cdot p}{x}\right)}}\]

Derivation

  1. Initial program 13.4

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

  3. Applied div-inv13.7

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

  5. Applied add-cube-cbrt14.8

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

  6. Final simplification14.8

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

Reproduce

herbie shell --seed 2020001 
(FPCore (p x)
  :name "Given's Rotation SVD example"
  :precision binary64
  :pre (< 1e-150 (fabs x) 1e+150)

  :herbie-target
  (sqrt (+ 0.5 (/ (copysign 0.5 x) (hypot 1 (/ (* 2 p) x)))))

  (sqrt (* 0.5 (+ 1 (/ x (sqrt (+ (* (* 4 p) p) (* x x))))))))