Average Error: 13.6 → 13.6
Time: 31.2s
Precision: 64
\[10^{-150} \lt \left|x\right| \lt 10^{+150}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\sqrt{\left(1 + \frac{x}{\sqrt{x \cdot x + p \cdot \left(4 \cdot p\right)}}\right) \cdot 0.5}\]
\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{x \cdot x + p \cdot \left(4 \cdot p\right)}}\right) \cdot 0.5}
double f(double p, double x) {
        double r3817160 = 0.5;
        double r3817161 = 1.0;
        double r3817162 = x;
        double r3817163 = 4.0;
        double r3817164 = p;
        double r3817165 = r3817163 * r3817164;
        double r3817166 = r3817165 * r3817164;
        double r3817167 = r3817162 * r3817162;
        double r3817168 = r3817166 + r3817167;
        double r3817169 = sqrt(r3817168);
        double r3817170 = r3817162 / r3817169;
        double r3817171 = r3817161 + r3817170;
        double r3817172 = r3817160 * r3817171;
        double r3817173 = sqrt(r3817172);
        return r3817173;
}
↓
double f(double p, double x) {
        double r3817174 = 1.0;
        double r3817175 = x;
        double r3817176 = r3817175 * r3817175;
        double r3817177 = p;
        double r3817178 = 4.0;
        double r3817179 = r3817178 * r3817177;
        double r3817180 = r3817177 * r3817179;
        double r3817181 = r3817176 + r3817180;
        double r3817182 = sqrt(r3817181);
        double r3817183 = r3817175 / r3817182;
        double r3817184 = r3817174 + r3817183;
        double r3817185 = 0.5;
        double r3817186 = r3817184 * r3817185;
        double r3817187 = sqrt(r3817186);
        return r3817187;
}

Error

Bits error versus p

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.6
Target13.6
Herbie13.6
\[\sqrt{\frac{1}{2} + \frac{\mathsf{copysign}\left(\frac{1}{2}, x\right)}{\mathsf{hypot}\left(1, \frac{2 \cdot p}{x}\right)}}\]

Derivation

  1. Initial program 13.6

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

  2. Final simplification13.6

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

Reproduce

herbie shell --seed 2019156 +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))))))))