Average Error: 13.4 → 13.4
Time: 13.2s
Precision: 64
\[1.000000000000000006295358232172963997211 \cdot 10^{-150} \lt \left|x\right| \lt 9.999999999999999808355961724373745905731 \cdot 10^{149}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\sqrt{0.5 \cdot \frac{\frac{{\left({1}^{3}\right)}^{3} + {\left({\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}\right)}^{3}}{{\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3} \cdot \left({\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3} - {1}^{3}\right) + {1}^{6}}}{1 \cdot 1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} - 1\right)}}\]
\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}
↓
\sqrt{0.5 \cdot \frac{\frac{{\left({1}^{3}\right)}^{3} + {\left({\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}\right)}^{3}}{{\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3} \cdot \left({\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3} - {1}^{3}\right) + {1}^{6}}}{1 \cdot 1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} - 1\right)}}
double f(double p, double x) {
        double r195353 = 0.5;
        double r195354 = 1.0;
        double r195355 = x;
        double r195356 = 4.0;
        double r195357 = p;
        double r195358 = r195356 * r195357;
        double r195359 = r195358 * r195357;
        double r195360 = r195355 * r195355;
        double r195361 = r195359 + r195360;
        double r195362 = sqrt(r195361);
        double r195363 = r195355 / r195362;
        double r195364 = r195354 + r195363;
        double r195365 = r195353 * r195364;
        double r195366 = sqrt(r195365);
        return r195366;
}
↓
double f(double p, double x) {
        double r195367 = 0.5;
        double r195368 = 1.0;
        double r195369 = 3.0;
        double r195370 = pow(r195368, r195369);
        double r195371 = pow(r195370, r195369);
        double r195372 = x;
        double r195373 = 4.0;
        double r195374 = p;
        double r195375 = r195373 * r195374;
        double r195376 = r195375 * r195374;
        double r195377 = r195372 * r195372;
        double r195378 = r195376 + r195377;
        double r195379 = sqrt(r195378);
        double r195380 = r195372 / r195379;
        double r195381 = pow(r195380, r195369);
        double r195382 = pow(r195381, r195369);
        double r195383 = r195371 + r195382;
        double r195384 = r195381 - r195370;
        double r195385 = r195381 * r195384;
        double r195386 = 6.0;
        double r195387 = pow(r195368, r195386);
        double r195388 = r195385 + r195387;
        double r195389 = r195383 / r195388;
        double r195390 = r195368 * r195368;
        double r195391 = r195380 - r195368;
        double r195392 = r195380 * r195391;
        double r195393 = r195390 + r195392;
        double r195394 = r195389 / r195393;
        double r195395 = r195367 * r195394;
        double r195396 = sqrt(r195395);
        return r195396;
}

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
Herbie13.4
\[\sqrt{0.5 + \frac{\mathsf{copysign}\left(0.5, x\right)}{\mathsf{hypot}\left(1, \frac{2 \cdot p}{x}\right)}}\]

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 flip3-+13.4

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

  4. Simplified13.4

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

  6. Applied flip3-+13.4

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

  7. Simplified13.4

    \[\leadsto \sqrt{0.5 \cdot \frac{\frac{{\left({1}^{3}\right)}^{3} + {\left({\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}\right)}^{3}}{\color{blue}{{\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3} \cdot \left({\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3} - {1}^{3}\right) + {1}^{6}}}}{1 \cdot 1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} - 1\right)}}\]

  8. Final simplification13.4

    \[\leadsto \sqrt{0.5 \cdot \frac{\frac{{\left({1}^{3}\right)}^{3} + {\left({\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}\right)}^{3}}{{\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3} \cdot \left({\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3} - {1}^{3}\right) + {1}^{6}}}{1 \cdot 1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} - 1\right)}}\]

Reproduce

herbie shell --seed 2019235 
(FPCore (p x)
  :name "Given's Rotation SVD example"
  :precision binary64
  :pre (< 1.00000000000000001e-150 (fabs x) 9.99999999999999981e149)

  :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))))))))