Average Error: 13.3 → 13.3
Time: 16.3s
Precision: 64
\[1.000000000000000006295358232172963997211 \cdot 10^{-150} \lt \left|x\right| \lt 9.999999999999999808355961724373745905731 \cdot 10^{149}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\frac{\sqrt{0.5 \cdot \left(\left(1 \cdot 1\right) \cdot 1 + \frac{x}{\sqrt{x \cdot x + p \cdot \left(p \cdot 4\right)}} \cdot \sqrt[3]{\left(\frac{x \cdot x}{x \cdot x + p \cdot \left(p \cdot 4\right)} \cdot \frac{x \cdot x}{x \cdot x + p \cdot \left(p \cdot 4\right)}\right) \cdot \frac{x \cdot x}{x \cdot x + p \cdot \left(p \cdot 4\right)}}\right)}}{\sqrt{1 \cdot 1 + \left(\frac{x}{\sqrt{x \cdot x + p \cdot \left(p \cdot 4\right)}} - 1\right) \cdot \frac{x}{\sqrt{\sqrt{x \cdot x + p \cdot \left(p \cdot 4\right)}} \cdot \sqrt{\sqrt{x \cdot x + p \cdot \left(p \cdot 4\right)}}}}}\]
\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}
↓
\frac{\sqrt{0.5 \cdot \left(\left(1 \cdot 1\right) \cdot 1 + \frac{x}{\sqrt{x \cdot x + p \cdot \left(p \cdot 4\right)}} \cdot \sqrt[3]{\left(\frac{x \cdot x}{x \cdot x + p \cdot \left(p \cdot 4\right)} \cdot \frac{x \cdot x}{x \cdot x + p \cdot \left(p \cdot 4\right)}\right) \cdot \frac{x \cdot x}{x \cdot x + p \cdot \left(p \cdot 4\right)}}\right)}}{\sqrt{1 \cdot 1 + \left(\frac{x}{\sqrt{x \cdot x + p \cdot \left(p \cdot 4\right)}} - 1\right) \cdot \frac{x}{\sqrt{\sqrt{x \cdot x + p \cdot \left(p \cdot 4\right)}} \cdot \sqrt{\sqrt{x \cdot x + p \cdot \left(p \cdot 4\right)}}}}}
double f(double p, double x) {
        double r10221748 = 0.5;
        double r10221749 = 1.0;
        double r10221750 = x;
        double r10221751 = 4.0;
        double r10221752 = p;
        double r10221753 = r10221751 * r10221752;
        double r10221754 = r10221753 * r10221752;
        double r10221755 = r10221750 * r10221750;
        double r10221756 = r10221754 + r10221755;
        double r10221757 = sqrt(r10221756);
        double r10221758 = r10221750 / r10221757;
        double r10221759 = r10221749 + r10221758;
        double r10221760 = r10221748 * r10221759;
        double r10221761 = sqrt(r10221760);
        return r10221761;
}
↓
double f(double p, double x) {
        double r10221762 = 0.5;
        double r10221763 = 1.0;
        double r10221764 = r10221763 * r10221763;
        double r10221765 = r10221764 * r10221763;
        double r10221766 = x;
        double r10221767 = r10221766 * r10221766;
        double r10221768 = p;
        double r10221769 = 4.0;
        double r10221770 = r10221768 * r10221769;
        double r10221771 = r10221768 * r10221770;
        double r10221772 = r10221767 + r10221771;
        double r10221773 = sqrt(r10221772);
        double r10221774 = r10221766 / r10221773;
        double r10221775 = r10221767 / r10221772;
        double r10221776 = r10221775 * r10221775;
        double r10221777 = r10221776 * r10221775;
        double r10221778 = cbrt(r10221777);
        double r10221779 = r10221774 * r10221778;
        double r10221780 = r10221765 + r10221779;
        double r10221781 = r10221762 * r10221780;
        double r10221782 = sqrt(r10221781);
        double r10221783 = r10221774 - r10221763;
        double r10221784 = sqrt(r10221773);
        double r10221785 = r10221784 * r10221784;
        double r10221786 = r10221766 / r10221785;
        double r10221787 = r10221783 * r10221786;
        double r10221788 = r10221764 + r10221787;
        double r10221789 = sqrt(r10221788);
        double r10221790 = r10221782 / r10221789;
        return r10221790;
}

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 flip3-+13.3

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

  4. Applied associate-*r/13.3

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

  5. Applied sqrt-div13.3

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

  6. Simplified13.3

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

  7. Simplified13.3

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

  9. Applied add-cbrt-cube13.3

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

  11. Applied add-sqr-sqrt13.3

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

  12. Applied sqrt-prod13.3

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

  13. Final simplification13.3

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

Reproduce

herbie shell --seed 2019174 
(FPCore (p x)
  :name "Given's Rotation SVD example"
  :pre (< 1e-150 (fabs x) 1e+150)

  :herbie-target
  (sqrt (+ 0.5 (/ (copysign 0.5 x) (hypot 1.0 (/ (* 2.0 p) x)))))

  (sqrt (* 0.5 (+ 1.0 (/ x (sqrt (+ (* (* 4.0 p) p) (* x x))))))))