Average Error: 13.7 → 14.6
Time: 8.2s
Precision: 64
\[1.000000000000000006295358232172963997211 \cdot 10^{-150} \lt \left|x\right| \lt 9.999999999999999808355961724373745905731 \cdot 10^{149}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\sqrt{0.5 \cdot \sqrt[3]{{\left(1 + \frac{x}{\sqrt{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}} \cdot \frac{1}{\sqrt{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right)}^{3}}}\]
\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(1 + \frac{x}{\sqrt{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}} \cdot \frac{1}{\sqrt{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right)}^{3}}}
double f(double p, double x) {
        double r191671 = 0.5;
        double r191672 = 1.0;
        double r191673 = x;
        double r191674 = 4.0;
        double r191675 = p;
        double r191676 = r191674 * r191675;
        double r191677 = r191676 * r191675;
        double r191678 = r191673 * r191673;
        double r191679 = r191677 + r191678;
        double r191680 = sqrt(r191679);
        double r191681 = r191673 / r191680;
        double r191682 = r191672 + r191681;
        double r191683 = r191671 * r191682;
        double r191684 = sqrt(r191683);
        return r191684;
}
↓
double f(double p, double x) {
        double r191685 = 0.5;
        double r191686 = 1.0;
        double r191687 = x;
        double r191688 = 4.0;
        double r191689 = p;
        double r191690 = r191688 * r191689;
        double r191691 = r191690 * r191689;
        double r191692 = r191687 * r191687;
        double r191693 = r191691 + r191692;
        double r191694 = sqrt(r191693);
        double r191695 = sqrt(r191694);
        double r191696 = r191687 / r191695;
        double r191697 = 1.0;
        double r191698 = r191697 / r191695;
        double r191699 = r191696 * r191698;
        double r191700 = r191686 + r191699;
        double r191701 = 3.0;
        double r191702 = pow(r191700, r191701);
        double r191703 = cbrt(r191702);
        double r191704 = r191685 * r191703;
        double r191705 = sqrt(r191704);
        return r191705;
}

Error

Bits error versus p

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.7
Target13.7
Herbie14.6
\[\sqrt{0.5 + \frac{\mathsf{copysign}\left(0.5, x\right)}{\mathsf{hypot}\left(1, \frac{2 \cdot p}{x}\right)}}\]

Derivation

  1. Initial program 13.7

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

  3. Applied div-inv14.0

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

  5. Applied add-sqr-sqrt14.0

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

  6. Applied sqrt-prod14.6

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

  7. Applied *-un-lft-identity14.6

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

  8. Applied times-frac14.8

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

  9. Applied associate-*r*14.8

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

  10. Simplified14.6

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

  12. Applied add-cbrt-cube14.6

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

  13. Simplified14.6

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

  14. Final simplification14.6

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

Reproduce

herbie shell --seed 2019308 
(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))))))))