Given's Rotation SVD example

Average Error: 13.1 → 13.3

Time: 7.1s

Precision: 64

\[1.000000000000000006295358232172963997211 \cdot 10^{-150} \lt \left|x\right| \lt 9.999999999999999808355961724373745905731 \cdot 10^{149}\]

Math

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

↓

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

C

double f(double p, double x) {
        double r389343 = 0.5;
        double r389344 = 1.0;
        double r389345 = x;
        double r389346 = 4.0;
        double r389347 = p;
        double r389348 = r389346 * r389347;
        double r389349 = r389348 * r389347;
        double r389350 = r389345 * r389345;
        double r389351 = r389349 + r389350;
        double r389352 = sqrt(r389351);
        double r389353 = r389345 / r389352;
        double r389354 = r389344 + r389353;
        double r389355 = r389343 * r389354;
        double r389356 = sqrt(r389355);
        return r389356;
}

↓

double f(double p, double x) {
        double r389357 = 0.5;
        double r389358 = 1.0;
        double r389359 = x;
        double r389360 = 1.0;
        double r389361 = 4.0;
        double r389362 = p;
        double r389363 = r389361 * r389362;
        double r389364 = r389363 * r389362;
        double r389365 = r389359 * r389359;
        double r389366 = r389364 + r389365;
        double r389367 = sqrt(r389366);
        double r389368 = r389360 / r389367;
        double r389369 = r389359 * r389368;
        double r389370 = pow(r389369, r389360);
        double r389371 = r389358 + r389370;
        double r389372 = r389357 * r389371;
        double r389373 = sqrt(r389372);
        return r389373;
}

Error

Bits error versus p

Bits error versus x

Target

Original	13.1
Target	13.1
Herbie	13.3

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

Derivation

1. Initial program 13.1

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

3. Applied div-inv 13.3

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

5. Applied pow1 13.3

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

6. Final simplification 13.3

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

Reproduce

herbie shell --seed 2019322 
(FPCore (p x)
  :name "Given's Rotation SVD example"
  :precision binary64
  :pre (< 1.00000000000000001e-150 (fabs x) 9.99999999999999981e149)

  :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))))))))