Average Error: 13.2 → 13.2
Time: 32.4s
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 r3852452 = 0.5;
        double r3852453 = 1.0;
        double r3852454 = x;
        double r3852455 = 4.0;
        double r3852456 = p;
        double r3852457 = r3852455 * r3852456;
        double r3852458 = r3852457 * r3852456;
        double r3852459 = r3852454 * r3852454;
        double r3852460 = r3852458 + r3852459;
        double r3852461 = sqrt(r3852460);
        double r3852462 = r3852454 / r3852461;
        double r3852463 = r3852453 + r3852462;
        double r3852464 = r3852452 * r3852463;
        double r3852465 = sqrt(r3852464);
        return r3852465;
}
↓
double f(double p, double x) {
        double r3852466 = 1.0;
        double r3852467 = x;
        double r3852468 = r3852467 * r3852467;
        double r3852469 = p;
        double r3852470 = 4.0;
        double r3852471 = r3852470 * r3852469;
        double r3852472 = r3852469 * r3852471;
        double r3852473 = r3852468 + r3852472;
        double r3852474 = sqrt(r3852473);
        double r3852475 = r3852467 / r3852474;
        double r3852476 = r3852466 + r3852475;
        double r3852477 = 0.5;
        double r3852478 = r3852476 * r3852477;
        double r3852479 = sqrt(r3852478);
        return r3852479;
}

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))))))))