Average Error: 12.9 → 12.9
Time: 5.5s
Precision: 64
\[1.00000000000000001 \cdot 10^{-150} \lt \left|x\right| \lt 9.99999999999999981 \cdot 10^{149}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\sqrt{0.5 \cdot \sqrt[3]{{\left(\log \left(e^{1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right)\right)}^{3}}}\]
\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]{{\left(\log \left(e^{1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right)\right)}^{3}}}
double f(double p, double x) {
        double r318780 = 0.5;
        double r318781 = 1.0;
        double r318782 = x;
        double r318783 = 4.0;
        double r318784 = p;
        double r318785 = r318783 * r318784;
        double r318786 = r318785 * r318784;
        double r318787 = r318782 * r318782;
        double r318788 = r318786 + r318787;
        double r318789 = sqrt(r318788);
        double r318790 = r318782 / r318789;
        double r318791 = r318781 + r318790;
        double r318792 = r318780 * r318791;
        double r318793 = sqrt(r318792);
        return r318793;
}
↓
double f(double p, double x) {
        double r318794 = 0.5;
        double r318795 = 1.0;
        double r318796 = x;
        double r318797 = 4.0;
        double r318798 = p;
        double r318799 = r318797 * r318798;
        double r318800 = r318799 * r318798;
        double r318801 = r318796 * r318796;
        double r318802 = r318800 + r318801;
        double r318803 = sqrt(r318802);
        double r318804 = r318796 / r318803;
        double r318805 = r318795 + r318804;
        double r318806 = exp(r318805);
        double r318807 = log(r318806);
        double r318808 = 3.0;
        double r318809 = pow(r318807, r318808);
        double r318810 = cbrt(r318809);
        double r318811 = r318794 * r318810;
        double r318812 = sqrt(r318811);
        return r318812;
}

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 add-cbrt-cube12.9

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

  4. Simplified12.9

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

  6. Applied add-log-exp12.9

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

  7. Applied add-log-exp12.9

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

  8. Applied sum-log12.9

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

  9. Simplified12.9

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

  10. Final simplification12.9

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

Reproduce

herbie shell --seed 2020021 
(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))))))))