Average Error: 13.0 → 14.2
Time: 28.5s
Precision: 64
\[10^{-150} \lt \left|x\right| \lt 10^{+150}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\sqrt{\left(1 + \frac{x}{\sqrt{\sqrt{\sqrt{x \cdot x + p \cdot \left(4 \cdot p\right)}}} \cdot \left(\sqrt{\sqrt{x \cdot x + p \cdot \left(4 \cdot p\right)}} \cdot \sqrt{\sqrt{\sqrt{x \cdot x + p \cdot \left(4 \cdot p\right)}}}\right)}\right) \cdot 0.5}\]
\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}
↓
\sqrt{\left(1 + \frac{x}{\sqrt{\sqrt{\sqrt{x \cdot x + p \cdot \left(4 \cdot p\right)}}} \cdot \left(\sqrt{\sqrt{x \cdot x + p \cdot \left(4 \cdot p\right)}} \cdot \sqrt{\sqrt{\sqrt{x \cdot x + p \cdot \left(4 \cdot p\right)}}}\right)}\right) \cdot 0.5}
double f(double p, double x) {
        double r4107656 = 0.5;
        double r4107657 = 1.0;
        double r4107658 = x;
        double r4107659 = 4.0;
        double r4107660 = p;
        double r4107661 = r4107659 * r4107660;
        double r4107662 = r4107661 * r4107660;
        double r4107663 = r4107658 * r4107658;
        double r4107664 = r4107662 + r4107663;
        double r4107665 = sqrt(r4107664);
        double r4107666 = r4107658 / r4107665;
        double r4107667 = r4107657 + r4107666;
        double r4107668 = r4107656 * r4107667;
        double r4107669 = sqrt(r4107668);
        return r4107669;
}
↓
double f(double p, double x) {
        double r4107670 = 1.0;
        double r4107671 = x;
        double r4107672 = r4107671 * r4107671;
        double r4107673 = p;
        double r4107674 = 4.0;
        double r4107675 = r4107674 * r4107673;
        double r4107676 = r4107673 * r4107675;
        double r4107677 = r4107672 + r4107676;
        double r4107678 = sqrt(r4107677);
        double r4107679 = sqrt(r4107678);
        double r4107680 = sqrt(r4107679);
        double r4107681 = r4107679 * r4107680;
        double r4107682 = r4107680 * r4107681;
        double r4107683 = r4107671 / r4107682;
        double r4107684 = r4107670 + r4107683;
        double r4107685 = 0.5;
        double r4107686 = r4107684 * r4107685;
        double r4107687 = sqrt(r4107686);
        return r4107687;
}

Error

Bits error versus p

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.0
Target13.0
Herbie14.2
\[\sqrt{\frac{1}{2} + \frac{\mathsf{copysign}\left(\frac{1}{2}, x\right)}{\mathsf{hypot}\left(1, \frac{2 \cdot p}{x}\right)}}\]

Derivation

  1. Initial program 13.0

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

  3. Applied add-sqr-sqrt13.0

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

  4. Applied sqrt-prod13.9

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

  6. Applied add-sqr-sqrt13.9

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

  7. Applied sqrt-prod14.2

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

  8. Applied associate-*r*14.2

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

  9. Final simplification14.2

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

Reproduce

herbie shell --seed 2019152 +o rules:numerics
(FPCore (p x)
  :name "Given's Rotation SVD example"
  :pre (< 1e-150 (fabs x) 1e+150)

  :herbie-target
  (sqrt (+ 1/2 (/ (copysign 1/2 x) (hypot 1 (/ (* 2 p) x)))))

  (sqrt (* 0.5 (+ 1 (/ x (sqrt (+ (* (* 4 p) p) (* x x))))))))