Average Error: 13.1 → 13.1
Time: 16.2s
Precision: 64
\[1.000000000000000006295358232172963997211 \cdot 10^{-150} \lt \left|x\right| \lt 9.999999999999999808355961724373745905731 \cdot 10^{149}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\sqrt{0.5 \cdot \sqrt[3]{e^{\left(\sqrt[3]{\log \left({\left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}\right)} \cdot \sqrt[3]{\log \left({\left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}\right)}\right) \cdot \sqrt[3]{\left(\sqrt[3]{\log \left({\left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}\right)} \cdot \sqrt[3]{\log \left({\left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}\right)}\right) \cdot \sqrt[3]{\log \left({\left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}\right)}}}}}\]
\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}
↓
\sqrt{0.5 \cdot \sqrt[3]{e^{\left(\sqrt[3]{\log \left({\left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}\right)} \cdot \sqrt[3]{\log \left({\left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}\right)}\right) \cdot \sqrt[3]{\left(\sqrt[3]{\log \left({\left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}\right)} \cdot \sqrt[3]{\log \left({\left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}\right)}\right) \cdot \sqrt[3]{\log \left({\left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}\right)}}}}}
double f(double p, double x) {
        double r227591 = 0.5;
        double r227592 = 1.0;
        double r227593 = x;
        double r227594 = 4.0;
        double r227595 = p;
        double r227596 = r227594 * r227595;
        double r227597 = r227596 * r227595;
        double r227598 = r227593 * r227593;
        double r227599 = r227597 + r227598;
        double r227600 = sqrt(r227599);
        double r227601 = r227593 / r227600;
        double r227602 = r227592 + r227601;
        double r227603 = r227591 * r227602;
        double r227604 = sqrt(r227603);
        return r227604;
}
↓
double f(double p, double x) {
        double r227605 = 0.5;
        double r227606 = 1.0;
        double r227607 = x;
        double r227608 = 4.0;
        double r227609 = p;
        double r227610 = r227608 * r227609;
        double r227611 = r227610 * r227609;
        double r227612 = r227607 * r227607;
        double r227613 = r227611 + r227612;
        double r227614 = sqrt(r227613);
        double r227615 = r227607 / r227614;
        double r227616 = r227606 + r227615;
        double r227617 = 3.0;
        double r227618 = pow(r227616, r227617);
        double r227619 = log(r227618);
        double r227620 = cbrt(r227619);
        double r227621 = r227620 * r227620;
        double r227622 = r227621 * r227620;
        double r227623 = cbrt(r227622);
        double r227624 = r227621 * r227623;
        double r227625 = exp(r227624);
        double r227626 = cbrt(r227625);
        double r227627 = r227605 * r227626;
        double r227628 = sqrt(r227627);
        return r227628;
}

Error

Bits error versus p

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 13.1

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

  3. Applied add-cbrt-cube13.1

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

  4. Simplified13.1

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

  6. Applied add-exp-log13.1

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

  7. Applied pow-exp13.1

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

  8. Simplified13.1

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

  10. Applied add-cube-cbrt13.1

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

  12. Applied add-cube-cbrt13.1

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

  13. Final simplification13.1

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

Reproduce

herbie shell --seed 2019298 
(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))))))))