Given's Rotation SVD example

Average Error: 13.0 → 13.0

Time: 18.4s

Precision: 64

\[10^{-150} \lt \left|x\right| \lt 10^{+150}\]

Math

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

↓

\[\sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(p, 4 \cdot p, x \cdot x\right)}}, 0.5, 0.5\right)}\]

C

double f(double p, double x) {
        double r6456411 = 0.5;
        double r6456412 = 1.0;
        double r6456413 = x;
        double r6456414 = 4.0;
        double r6456415 = p;
        double r6456416 = r6456414 * r6456415;
        double r6456417 = r6456416 * r6456415;
        double r6456418 = r6456413 * r6456413;
        double r6456419 = r6456417 + r6456418;
        double r6456420 = sqrt(r6456419);
        double r6456421 = r6456413 / r6456420;
        double r6456422 = r6456412 + r6456421;
        double r6456423 = r6456411 * r6456422;
        double r6456424 = sqrt(r6456423);
        return r6456424;
}

↓

double f(double p, double x) {
        double r6456425 = x;
        double r6456426 = p;
        double r6456427 = 4.0;
        double r6456428 = r6456427 * r6456426;
        double r6456429 = r6456425 * r6456425;
        double r6456430 = fma(r6456426, r6456428, r6456429);
        double r6456431 = sqrt(r6456430);
        double r6456432 = r6456425 / r6456431;
        double r6456433 = 0.5;
        double r6456434 = fma(r6456432, r6456433, r6456433);
        double r6456435 = sqrt(r6456434);
        return r6456435;
}

Error

Bits error versus p

Bits error versus x

Target

Original	13.0
Target	13.0
Herbie	13.0

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

Derivation

1. Initial program 13.0

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

2. Simplified 13.0

   \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(p, 4 \cdot p, x \cdot x\right)}}, 0.5, 0.5\right)}}\]

3. Final simplification 13.0

   \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(p, 4 \cdot p, x \cdot x\right)}}, 0.5, 0.5\right)}\]

Reproduce

herbie shell --seed 2019152 +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))))))))