Average Error: 12.9 → 12.9
Time: 14.7s
Precision: 64
\[1.000000000000000006295358232172963997211 \cdot 10^{-150} \lt \left|x\right| \lt 9.999999999999999808355961724373745905731 \cdot 10^{149}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\sqrt{0.5 \cdot \frac{\log \left(e^{{1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}\right)}{1 \cdot \left(1 - \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right) + \frac{x \cdot x}{4 \cdot {p}^{2} + {x}^{2}}}}\]
\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{\log \left(e^{{1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}\right)}{1 \cdot \left(1 - \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right) + \frac{x \cdot x}{4 \cdot {p}^{2} + {x}^{2}}}}
double f(double p, double x) {
        double r289376 = 0.5;
        double r289377 = 1.0;
        double r289378 = x;
        double r289379 = 4.0;
        double r289380 = p;
        double r289381 = r289379 * r289380;
        double r289382 = r289381 * r289380;
        double r289383 = r289378 * r289378;
        double r289384 = r289382 + r289383;
        double r289385 = sqrt(r289384);
        double r289386 = r289378 / r289385;
        double r289387 = r289377 + r289386;
        double r289388 = r289376 * r289387;
        double r289389 = sqrt(r289388);
        return r289389;
}
↓
double f(double p, double x) {
        double r289390 = 0.5;
        double r289391 = 1.0;
        double r289392 = 3.0;
        double r289393 = pow(r289391, r289392);
        double r289394 = x;
        double r289395 = 4.0;
        double r289396 = p;
        double r289397 = r289395 * r289396;
        double r289398 = r289397 * r289396;
        double r289399 = r289394 * r289394;
        double r289400 = r289398 + r289399;
        double r289401 = sqrt(r289400);
        double r289402 = r289394 / r289401;
        double r289403 = pow(r289402, r289392);
        double r289404 = r289393 + r289403;
        double r289405 = exp(r289404);
        double r289406 = log(r289405);
        double r289407 = r289391 - r289402;
        double r289408 = r289391 * r289407;
        double r289409 = 2.0;
        double r289410 = pow(r289396, r289409);
        double r289411 = r289395 * r289410;
        double r289412 = pow(r289394, r289409);
        double r289413 = r289411 + r289412;
        double r289414 = r289399 / r289413;
        double r289415 = r289408 + r289414;
        double r289416 = r289406 / r289415;
        double r289417 = r289390 * r289416;
        double r289418 = sqrt(r289417);
        return r289418;
}

Error

Bits error versus p

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original12.9
Target12.9
Herbie12.9
\[\sqrt{0.5 + \frac{\mathsf{copysign}\left(0.5, x\right)}{\mathsf{hypot}\left(1, \frac{2 \cdot p}{x}\right)}}\]

Derivation

  1. Initial program 12.9

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

  3. Applied flip3-+12.9

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

  4. Simplified12.9

    \[\leadsto \sqrt{0.5 \cdot \frac{{1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}{\color{blue}{1 \cdot \left(1 - \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right) + \frac{x \cdot x}{4 \cdot {p}^{2} + {x}^{2}}}}}\]
  5. Using strategy rm

  6. Applied add-log-exp12.9

    \[\leadsto \sqrt{0.5 \cdot \frac{{1}^{3} + \color{blue}{\log \left(e^{{\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}\right)}}{1 \cdot \left(1 - \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right) + \frac{x \cdot x}{4 \cdot {p}^{2} + {x}^{2}}}}\]

  7. Applied add-log-exp12.9

    \[\leadsto \sqrt{0.5 \cdot \frac{\color{blue}{\log \left(e^{{1}^{3}}\right)} + \log \left(e^{{\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}\right)}{1 \cdot \left(1 - \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right) + \frac{x \cdot x}{4 \cdot {p}^{2} + {x}^{2}}}}\]

  8. Applied sum-log12.9

    \[\leadsto \sqrt{0.5 \cdot \frac{\color{blue}{\log \left(e^{{1}^{3}} \cdot e^{{\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}\right)}}{1 \cdot \left(1 - \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right) + \frac{x \cdot x}{4 \cdot {p}^{2} + {x}^{2}}}}\]

  9. Simplified12.9

    \[\leadsto \sqrt{0.5 \cdot \frac{\log \color{blue}{\left(e^{{1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}\right)}}{1 \cdot \left(1 - \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right) + \frac{x \cdot x}{4 \cdot {p}^{2} + {x}^{2}}}}\]

  10. Final simplification12.9

    \[\leadsto \sqrt{0.5 \cdot \frac{\log \left(e^{{1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}\right)}{1 \cdot \left(1 - \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right) + \frac{x \cdot x}{4 \cdot {p}^{2} + {x}^{2}}}}\]

Reproduce

herbie shell --seed 2019303 
(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))))))))