Average Error: 13.6 → 13.8
Time: 5.3s
Precision: 64
\[1.00000000000000001 \cdot 10^{-150} \lt \left|x\right| \lt 9.99999999999999981 \cdot 10^{149}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\sqrt{0.5 \cdot \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \frac{1}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right)\right)}\]
\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 + \mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \frac{1}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right)\right)}
double f(double p, double x) {
        double r208480 = 0.5;
        double r208481 = 1.0;
        double r208482 = x;
        double r208483 = 4.0;
        double r208484 = p;
        double r208485 = r208483 * r208484;
        double r208486 = r208485 * r208484;
        double r208487 = r208482 * r208482;
        double r208488 = r208486 + r208487;
        double r208489 = sqrt(r208488);
        double r208490 = r208482 / r208489;
        double r208491 = r208481 + r208490;
        double r208492 = r208480 * r208491;
        double r208493 = sqrt(r208492);
        return r208493;
}
↓
double f(double p, double x) {
        double r208494 = 0.5;
        double r208495 = 1.0;
        double r208496 = x;
        double r208497 = 1.0;
        double r208498 = 4.0;
        double r208499 = p;
        double r208500 = r208498 * r208499;
        double r208501 = r208500 * r208499;
        double r208502 = r208496 * r208496;
        double r208503 = r208501 + r208502;
        double r208504 = sqrt(r208503);
        double r208505 = r208497 / r208504;
        double r208506 = r208496 * r208505;
        double r208507 = log1p(r208506);
        double r208508 = expm1(r208507);
        double r208509 = r208495 + r208508;
        double r208510 = r208494 * r208509;
        double r208511 = sqrt(r208510);
        return r208511;
}

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
Herbie13.8
\[\sqrt{0.5 + \frac{\mathsf{copysign}\left(0.5, x\right)}{\mathsf{hypot}\left(1, \frac{2 \cdot p}{x}\right)}}\]

Derivation

  1. Initial program 13.6

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

  3. Applied div-inv13.8

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

  5. Applied expm1-log1p-u13.8

    \[\leadsto \sqrt{0.5 \cdot \left(1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \frac{1}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right)}\right)}\]

  6. Final simplification13.8

    \[\leadsto \sqrt{0.5 \cdot \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \frac{1}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right)\right)}\]

Reproduce

herbie shell --seed 2020035 +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))))))))