Given's Rotation SVD example

Average Error: 13.5 → 13.7

Time: 4.6s

Precision: 64

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

Math

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

↓

\[\sqrt{0.5 \cdot \left(\left(\sqrt[3]{1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}} \cdot \sqrt[3]{1 + \frac{x}{\left|\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}\right| \cdot \sqrt{\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}}}}\right) \cdot \sqrt[3]{1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right)}\]

C

double code(double p, double x) {
	return sqrt((0.5 * (1.0 + (x / sqrt((((4.0 * p) * p) + (x * x)))))));
}

↓

double code(double p, double x) {
	return sqrt((0.5 * ((cbrt((1.0 + (x / sqrt((((4.0 * p) * p) + (x * x)))))) * cbrt((1.0 + (x / (fabs(cbrt((((4.0 * p) * p) + (x * x)))) * sqrt(cbrt((((4.0 * p) * p) + (x * x))))))))) * cbrt((1.0 + (x / sqrt((((4.0 * p) * p) + (x * x)))))))));
}

Error

Bits error versus p

Bits error versus x

Target

Original	13.5
Target	13.5
Herbie	13.7

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

Derivation

1. Initial program 13.5

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

3. Applied add-cube-cbrt 13.7

   \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(\left(\sqrt[3]{1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}} \cdot \sqrt[3]{1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right) \cdot \sqrt[3]{1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right)}}\]
4. Using strategy rm

5. Applied add-cube-cbrt 13.7

   \[\leadsto \sqrt{0.5 \cdot \left(\left(\sqrt[3]{1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}} \cdot \sqrt[3]{1 + \frac{x}{\sqrt{\color{blue}{\left(\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x} \cdot \sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}\right) \cdot \sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}}}}}\right) \cdot \sqrt[3]{1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right)}\]

6. Applied sqrt-prod 13.7

   \[\leadsto \sqrt{0.5 \cdot \left(\left(\sqrt[3]{1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}} \cdot \sqrt[3]{1 + \frac{x}{\color{blue}{\sqrt{\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x} \cdot \sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \sqrt{\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}}}}}\right) \cdot \sqrt[3]{1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right)}\]

7. Simplified 13.7

   \[\leadsto \sqrt{0.5 \cdot \left(\left(\sqrt[3]{1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}} \cdot \sqrt[3]{1 + \frac{x}{\color{blue}{\left|\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}\right|} \cdot \sqrt{\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}}}}\right) \cdot \sqrt[3]{1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right)}\]

8. Final simplification 13.7

   \[\leadsto \sqrt{0.5 \cdot \left(\left(\sqrt[3]{1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}} \cdot \sqrt[3]{1 + \frac{x}{\left|\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}\right| \cdot \sqrt{\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}}}}\right) \cdot \sqrt[3]{1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right)}\]

Reproduce

herbie shell --seed 2020066 
(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))))))))