Given's Rotation SVD example

Average Error: 12.9 → 13.1

Time: 10.7s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

\[1.00000000000000001 \cdot 10^{-150} \lt \left|x\right| \lt 9.99999999999999981 \cdot 10^{149}\]

Math

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

↓

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

C

double f(double p, double x) {
        double r298413 = 0.5;
        double r298414 = 1.0;
        double r298415 = x;
        double r298416 = 4.0;
        double r298417 = p;
        double r298418 = r298416 * r298417;
        double r298419 = r298418 * r298417;
        double r298420 = r298415 * r298415;
        double r298421 = r298419 + r298420;
        double r298422 = sqrt(r298421);
        double r298423 = r298415 / r298422;
        double r298424 = r298414 + r298423;
        double r298425 = r298413 * r298424;
        double r298426 = sqrt(r298425);
        return r298426;
}

↓

double f(double p, double x) {
        double r298427 = 0.5;
        double r298428 = 1.0;
        double r298429 = x;
        double r298430 = 1.0;
        double r298431 = 4.0;
        double r298432 = p;
        double r298433 = r298431 * r298432;
        double r298434 = r298433 * r298432;
        double r298435 = r298429 * r298429;
        double r298436 = r298434 + r298435;
        double r298437 = sqrt(r298436);
        double r298438 = r298430 / r298437;
        double r298439 = r298429 * r298438;
        double r298440 = r298428 + r298439;
        double r298441 = r298427 * r298440;
        double r298442 = sqrt(r298441);
        return r298442;
}

Error

Bits error versus p

Bits error versus x

Target

Original	12.9
Target	12.9
Herbie	13.1

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

Derivation

1. Initial program 12.9

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

3. Applied div-inv 13.1

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

4. Final simplification 13.1

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

Reproduce

herbie shell --seed 2020047 
(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))))))))