Average Error: 13.4 → 13.7
Time: 11.6s
Precision: 64
\[1.000000000000000006295358232172963997211 \cdot 10^{-150} \lt \left|x\right| \lt 9.999999999999999808355961724373745905731 \cdot 10^{149}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\sqrt{0.5 \cdot \frac{{1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}{1 \cdot \left(1 - \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right) + \frac{x}{\sqrt[3]{4 \cdot {p}^{2} + {x}^{2}} \cdot \sqrt[3]{4 \cdot {p}^{2} + {x}^{2}}} \cdot \frac{x}{\sqrt[3]{4 \cdot {p}^{2} + {x}^{2}}}}}\]
\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}
↓
\sqrt{0.5 \cdot \frac{{1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}{1 \cdot \left(1 - \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right) + \frac{x}{\sqrt[3]{4 \cdot {p}^{2} + {x}^{2}} \cdot \sqrt[3]{4 \cdot {p}^{2} + {x}^{2}}} \cdot \frac{x}{\sqrt[3]{4 \cdot {p}^{2} + {x}^{2}}}}}
double f(double p, double x) {
        double r226622 = 0.5;
        double r226623 = 1.0;
        double r226624 = x;
        double r226625 = 4.0;
        double r226626 = p;
        double r226627 = r226625 * r226626;
        double r226628 = r226627 * r226626;
        double r226629 = r226624 * r226624;
        double r226630 = r226628 + r226629;
        double r226631 = sqrt(r226630);
        double r226632 = r226624 / r226631;
        double r226633 = r226623 + r226632;
        double r226634 = r226622 * r226633;
        double r226635 = sqrt(r226634);
        return r226635;
}
↓
double f(double p, double x) {
        double r226636 = 0.5;
        double r226637 = 1.0;
        double r226638 = 3.0;
        double r226639 = pow(r226637, r226638);
        double r226640 = x;
        double r226641 = 4.0;
        double r226642 = p;
        double r226643 = r226641 * r226642;
        double r226644 = r226643 * r226642;
        double r226645 = r226640 * r226640;
        double r226646 = r226644 + r226645;
        double r226647 = sqrt(r226646);
        double r226648 = r226640 / r226647;
        double r226649 = pow(r226648, r226638);
        double r226650 = r226639 + r226649;
        double r226651 = r226637 - r226648;
        double r226652 = r226637 * r226651;
        double r226653 = 2.0;
        double r226654 = pow(r226642, r226653);
        double r226655 = r226641 * r226654;
        double r226656 = pow(r226640, r226653);
        double r226657 = r226655 + r226656;
        double r226658 = cbrt(r226657);
        double r226659 = r226658 * r226658;
        double r226660 = r226640 / r226659;
        double r226661 = r226640 / r226658;
        double r226662 = r226660 * r226661;
        double r226663 = r226652 + r226662;
        double r226664 = r226650 / r226663;
        double r226665 = r226636 * r226664;
        double r226666 = sqrt(r226665);
        return r226666;
}

Error

Bits error versus p

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.4
Target13.5
Herbie13.7
\[\sqrt{0.5 + \frac{\mathsf{copysign}\left(0.5, x\right)}{\mathsf{hypot}\left(1, \frac{2 \cdot p}{x}\right)}}\]

Derivation

  1. Initial program 13.4

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

  3. Applied flip3-+13.5

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

  4. Simplified13.5

    \[\leadsto \sqrt{0.5 \cdot \frac{{1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}{\color{blue}{1 \cdot \left(1 - \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right) + \frac{x \cdot x}{4 \cdot {p}^{2} + {x}^{2}}}}}\]
  5. Using strategy rm

  6. Applied add-cube-cbrt13.7

    \[\leadsto \sqrt{0.5 \cdot \frac{{1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}{1 \cdot \left(1 - \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right) + \frac{x \cdot x}{\color{blue}{\left(\sqrt[3]{4 \cdot {p}^{2} + {x}^{2}} \cdot \sqrt[3]{4 \cdot {p}^{2} + {x}^{2}}\right) \cdot \sqrt[3]{4 \cdot {p}^{2} + {x}^{2}}}}}}\]

  7. Applied times-frac13.7

    \[\leadsto \sqrt{0.5 \cdot \frac{{1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}{1 \cdot \left(1 - \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right) + \color{blue}{\frac{x}{\sqrt[3]{4 \cdot {p}^{2} + {x}^{2}} \cdot \sqrt[3]{4 \cdot {p}^{2} + {x}^{2}}} \cdot \frac{x}{\sqrt[3]{4 \cdot {p}^{2} + {x}^{2}}}}}}\]

  8. Final simplification13.7

    \[\leadsto \sqrt{0.5 \cdot \frac{{1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}{1 \cdot \left(1 - \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right) + \frac{x}{\sqrt[3]{4 \cdot {p}^{2} + {x}^{2}} \cdot \sqrt[3]{4 \cdot {p}^{2} + {x}^{2}}} \cdot \frac{x}{\sqrt[3]{4 \cdot {p}^{2} + {x}^{2}}}}}\]

Reproduce

herbie shell --seed 2019350 
(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))))))))