Average Error: 13.3 → 13.3
Time: 16.4s
Precision: 64
\[1.000000000000000006295358232172963997211 \cdot 10^{-150} \lt \left|x\right| \lt 9.999999999999999808355961724373745905731 \cdot 10^{149}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\sqrt{0.5 \cdot e^{\log \left(\frac{\sqrt{{1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}}{\sqrt{\frac{x \cdot x}{\left(4 \cdot p\right) \cdot p + x \cdot x} - 1 \cdot \left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} - 1\right)}}\right) + \log \left(\frac{\sqrt{{1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}}{\sqrt{\frac{x \cdot x}{\left(4 \cdot p\right) \cdot p + x \cdot x} - 1 \cdot \left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} - 1\right)}}\right)}}\]
\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}
↓
\sqrt{0.5 \cdot e^{\log \left(\frac{\sqrt{{1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}}{\sqrt{\frac{x \cdot x}{\left(4 \cdot p\right) \cdot p + x \cdot x} - 1 \cdot \left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} - 1\right)}}\right) + \log \left(\frac{\sqrt{{1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}}{\sqrt{\frac{x \cdot x}{\left(4 \cdot p\right) \cdot p + x \cdot x} - 1 \cdot \left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} - 1\right)}}\right)}}
double f(double p, double x) {
        double r212576 = 0.5;
        double r212577 = 1.0;
        double r212578 = x;
        double r212579 = 4.0;
        double r212580 = p;
        double r212581 = r212579 * r212580;
        double r212582 = r212581 * r212580;
        double r212583 = r212578 * r212578;
        double r212584 = r212582 + r212583;
        double r212585 = sqrt(r212584);
        double r212586 = r212578 / r212585;
        double r212587 = r212577 + r212586;
        double r212588 = r212576 * r212587;
        double r212589 = sqrt(r212588);
        return r212589;
}
↓
double f(double p, double x) {
        double r212590 = 0.5;
        double r212591 = 1.0;
        double r212592 = 3.0;
        double r212593 = pow(r212591, r212592);
        double r212594 = x;
        double r212595 = 4.0;
        double r212596 = p;
        double r212597 = r212595 * r212596;
        double r212598 = r212597 * r212596;
        double r212599 = r212594 * r212594;
        double r212600 = r212598 + r212599;
        double r212601 = sqrt(r212600);
        double r212602 = r212594 / r212601;
        double r212603 = pow(r212602, r212592);
        double r212604 = r212593 + r212603;
        double r212605 = sqrt(r212604);
        double r212606 = r212599 / r212600;
        double r212607 = r212602 - r212591;
        double r212608 = r212591 * r212607;
        double r212609 = r212606 - r212608;
        double r212610 = sqrt(r212609);
        double r212611 = r212605 / r212610;
        double r212612 = log(r212611);
        double r212613 = r212612 + r212612;
        double r212614 = exp(r212613);
        double r212615 = r212590 * r212614;
        double r212616 = sqrt(r212615);
        return r212616;
}

Error

Bits error versus p

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.3
Target13.3
Herbie13.3
\[\sqrt{0.5 + \frac{\mathsf{copysign}\left(0.5, x\right)}{\mathsf{hypot}\left(1, \frac{2 \cdot p}{x}\right)}}\]

Derivation

  1. Initial program 13.3

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

  3. Applied add-exp-log13.3

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

  5. Applied flip3-+13.3

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

  6. Simplified13.3

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

  8. Applied add-sqr-sqrt13.3

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

  9. Applied add-sqr-sqrt13.3

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

  10. Applied times-frac13.3

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

  11. Applied log-prod13.3

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

  12. Final simplification13.3

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

Reproduce

herbie shell --seed 2019326 
(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))))))))