Given's Rotation SVD example

Average Error: 12.9 → 13.1

Time: 4.4s

Precision: 64

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

Math

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

↓

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

C

double f(double p, double x) {
        double r356502 = 0.5;
        double r356503 = 1.0;
        double r356504 = x;
        double r356505 = 4.0;
        double r356506 = p;
        double r356507 = r356505 * r356506;
        double r356508 = r356507 * r356506;
        double r356509 = r356504 * r356504;
        double r356510 = r356508 + r356509;
        double r356511 = sqrt(r356510);
        double r356512 = r356504 / r356511;
        double r356513 = r356503 + r356512;
        double r356514 = r356502 * r356513;
        double r356515 = sqrt(r356514);
        return r356515;
}

↓

double f(double p, double x) {
        double r356516 = 0.5;
        double r356517 = 1.0;
        double r356518 = x;
        double r356519 = 1.0;
        double r356520 = r356518 * r356518;
        double r356521 = 4.0;
        double r356522 = p;
        double r356523 = r356521 * r356522;
        double r356524 = r356523 * r356522;
        double r356525 = r356520 + r356524;
        double r356526 = sqrt(r356525);
        double r356527 = r356519 / r356526;
        double r356528 = r356518 * r356527;
        double r356529 = r356517 + r356528;
        double r356530 = r356516 * r356529;
        double r356531 = sqrt(r356530);
        return r356531;
}

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. Using strategy rm

5. Applied +-commutative 13.1

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

6. Final simplification 13.1

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

Reproduce

herbie shell --seed 2020060 +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))))))))