Average Error: 13.2 → 13.2
Time: 31.9s
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 r4167387 = 0.5;
        double r4167388 = 1.0;
        double r4167389 = x;
        double r4167390 = 4.0;
        double r4167391 = p;
        double r4167392 = r4167390 * r4167391;
        double r4167393 = r4167392 * r4167391;
        double r4167394 = r4167389 * r4167389;
        double r4167395 = r4167393 + r4167394;
        double r4167396 = sqrt(r4167395);
        double r4167397 = r4167389 / r4167396;
        double r4167398 = r4167388 + r4167397;
        double r4167399 = r4167387 * r4167398;
        double r4167400 = sqrt(r4167399);
        return r4167400;
}
↓
double f(double p, double x) {
        double r4167401 = 1.0;
        double r4167402 = x;
        double r4167403 = r4167402 * r4167402;
        double r4167404 = p;
        double r4167405 = 4.0;
        double r4167406 = r4167405 * r4167404;
        double r4167407 = r4167404 * r4167406;
        double r4167408 = r4167403 + r4167407;
        double r4167409 = sqrt(r4167408);
        double r4167410 = r4167402 / r4167409;
        double r4167411 = r4167401 + r4167410;
        double r4167412 = 0.5;
        double r4167413 = r4167411 * r4167412;
        double r4167414 = sqrt(r4167413);
        return r4167414;
}

Error

Bits error versus p

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 13.2

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

  2. Final simplification13.2

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

Reproduce

herbie shell --seed 2019153 +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))))))))