Given's Rotation SVD example

Average Error: 12.9 → 13.1

Time: 4.5s

Precision: 64

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

Math

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

↓

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

C

double f(double p, double x) {
        double r333065 = 0.5;
        double r333066 = 1.0;
        double r333067 = x;
        double r333068 = 4.0;
        double r333069 = p;
        double r333070 = r333068 * r333069;
        double r333071 = r333070 * r333069;
        double r333072 = r333067 * r333067;
        double r333073 = r333071 + r333072;
        double r333074 = sqrt(r333073);
        double r333075 = r333067 / r333074;
        double r333076 = r333066 + r333075;
        double r333077 = r333065 * r333076;
        double r333078 = sqrt(r333077);
        return r333078;
}

↓

double f(double p, double x) {
        double r333079 = 0.5;
        double r333080 = 1.0;
        double r333081 = x;
        double r333082 = 1.0;
        double r333083 = 4.0;
        double r333084 = p;
        double r333085 = r333083 * r333084;
        double r333086 = r333085 * r333084;
        double r333087 = r333081 * r333081;
        double r333088 = r333086 + r333087;
        double r333089 = sqrt(r333088);
        double r333090 = r333082 / r333089;
        double r333091 = r333081 * r333090;
        double r333092 = r333080 + r333091;
        double r333093 = r333079 * r333092;
        double r333094 = sqrt(r333093);
        return r333094;
}

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))))))))