Average Error: 13.0 → 13.0
Time: 4.8s
Precision: 64
\[1.00000000000000001 \cdot 10^{-150} \lt \left|x\right| \lt 9.99999999999999981 \cdot 10^{149}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{1 \cdot \left(\left(4 \cdot p\right) \cdot p + x \cdot x\right)}}\right)}\]
\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}
↓
\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{1 \cdot \left(\left(4 \cdot p\right) \cdot p + x \cdot x\right)}}\right)}
double f(double p, double x) {
        double r231807 = 0.5;
        double r231808 = 1.0;
        double r231809 = x;
        double r231810 = 4.0;
        double r231811 = p;
        double r231812 = r231810 * r231811;
        double r231813 = r231812 * r231811;
        double r231814 = r231809 * r231809;
        double r231815 = r231813 + r231814;
        double r231816 = sqrt(r231815);
        double r231817 = r231809 / r231816;
        double r231818 = r231808 + r231817;
        double r231819 = r231807 * r231818;
        double r231820 = sqrt(r231819);
        return r231820;
}
↓
double f(double p, double x) {
        double r231821 = 0.5;
        double r231822 = 1.0;
        double r231823 = x;
        double r231824 = 1.0;
        double r231825 = 4.0;
        double r231826 = p;
        double r231827 = r231825 * r231826;
        double r231828 = r231827 * r231826;
        double r231829 = r231823 * r231823;
        double r231830 = r231828 + r231829;
        double r231831 = r231824 * r231830;
        double r231832 = sqrt(r231831);
        double r231833 = r231823 / r231832;
        double r231834 = r231822 + r231833;
        double r231835 = r231821 * r231834;
        double r231836 = sqrt(r231835);
        return r231836;
}

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
Herbie13.0
\[\sqrt{0.5 + \frac{\mathsf{copysign}\left(0.5, x\right)}{\mathsf{hypot}\left(1, \frac{2 \cdot p}{x}\right)}}\]

Derivation

  1. Initial program 13.0

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

  3. Applied *-un-lft-identity13.0

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

  4. Final simplification13.0

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

Reproduce

herbie shell --seed 2020089 +o rules:numerics
(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))))))))