Average Error: 12.7 → 12.7
Time: 5.4s
Precision: 64
\[1.00000000000000001 \cdot 10^{-150} \lt \left|x\right| \lt 9.99999999999999981 \cdot 10^{149}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\sqrt{0.5 \cdot \frac{{1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}{1 \cdot \log \left(e^{1 - \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right) + \frac{x \cdot x}{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\]
\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{{1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}{1 \cdot \log \left(e^{1 - \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right) + \frac{x \cdot x}{\left(4 \cdot p\right) \cdot p + x \cdot x}}}
double f(double p, double x) {
        double r288450 = 0.5;
        double r288451 = 1.0;
        double r288452 = x;
        double r288453 = 4.0;
        double r288454 = p;
        double r288455 = r288453 * r288454;
        double r288456 = r288455 * r288454;
        double r288457 = r288452 * r288452;
        double r288458 = r288456 + r288457;
        double r288459 = sqrt(r288458);
        double r288460 = r288452 / r288459;
        double r288461 = r288451 + r288460;
        double r288462 = r288450 * r288461;
        double r288463 = sqrt(r288462);
        return r288463;
}
↓
double f(double p, double x) {
        double r288464 = 0.5;
        double r288465 = 1.0;
        double r288466 = 3.0;
        double r288467 = pow(r288465, r288466);
        double r288468 = x;
        double r288469 = 4.0;
        double r288470 = p;
        double r288471 = r288469 * r288470;
        double r288472 = r288471 * r288470;
        double r288473 = r288468 * r288468;
        double r288474 = r288472 + r288473;
        double r288475 = sqrt(r288474);
        double r288476 = r288468 / r288475;
        double r288477 = pow(r288476, r288466);
        double r288478 = r288467 + r288477;
        double r288479 = r288465 - r288476;
        double r288480 = exp(r288479);
        double r288481 = log(r288480);
        double r288482 = r288465 * r288481;
        double r288483 = r288473 / r288474;
        double r288484 = r288482 + r288483;
        double r288485 = r288478 / r288484;
        double r288486 = r288464 * r288485;
        double r288487 = sqrt(r288486);
        return r288487;
}

Error

Bits error versus p

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original12.7
Target12.7
Herbie12.7
\[\sqrt{0.5 + \frac{\mathsf{copysign}\left(0.5, x\right)}{\mathsf{hypot}\left(1, \frac{2 \cdot p}{x}\right)}}\]

Derivation

  1. Initial program 12.7

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

  3. Applied flip3-+12.7

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

  4. Simplified12.7

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

  6. Applied add-log-exp12.7

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

  7. Applied add-log-exp12.7

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

  8. Applied diff-log12.7

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

  9. Simplified12.7

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

  10. Final simplification12.7

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

Reproduce

herbie shell --seed 2020083 
(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))))))))