Octave 3.8, oct_fill_randg

Average Error: 0.1 → 0.1

Time: 7.6s

Precision: 64

Math

\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\]

↓

\[\left(a - \frac{1}{3}\right) \cdot \left(1 + 1 \cdot \frac{\frac{rand}{\sqrt{a - \frac{1}{3}}}}{\sqrt{9}}\right)\]

C

double code(double a, double rand) {
	return ((a - (1.0 / 3.0)) * (1.0 + ((1.0 / sqrt((9.0 * (a - (1.0 / 3.0))))) * rand)));
}

↓

double code(double a, double rand) {
	return ((a - (1.0 / 3.0)) * (1.0 + (1.0 * ((rand / sqrt((a - (1.0 / 3.0)))) / sqrt(9.0)))));
}

Error

Bits error versus a

Bits error versus rand

Derivation

1. Initial program 0.1

   \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\]
2. Using strategy rm

3. Applied sqrt-prod 0.2

   \[\leadsto \left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\color{blue}{\sqrt{9} \cdot \sqrt{a - \frac{1}{3}}}} \cdot rand\right)\]
4. Using strategy rm

5. Applied div-inv 0.2

   \[\leadsto \left(a - \frac{1}{3}\right) \cdot \left(1 + \color{blue}{\left(1 \cdot \frac{1}{\sqrt{9} \cdot \sqrt{a - \frac{1}{3}}}\right)} \cdot rand\right)\]

6. Applied associate-*l* 0.2

   \[\leadsto \left(a - \frac{1}{3}\right) \cdot \left(1 + \color{blue}{1 \cdot \left(\frac{1}{\sqrt{9} \cdot \sqrt{a - \frac{1}{3}}} \cdot rand\right)}\right)\]

7. Simplified 0.1

   \[\leadsto \left(a - \frac{1}{3}\right) \cdot \left(1 + 1 \cdot \color{blue}{\frac{\frac{rand}{\sqrt{a - \frac{1}{3}}}}{\sqrt{9}}}\right)\]

8. Final simplification 0.1

   \[\leadsto \left(a - \frac{1}{3}\right) \cdot \left(1 + 1 \cdot \frac{\frac{rand}{\sqrt{a - \frac{1}{3}}}}{\sqrt{9}}\right)\]

Reproduce

herbie shell --seed 2020102 
(FPCore (a rand)
  :name "Octave 3.8, oct_fill_randg"
  :precision binary64
  (* (- a (/ 1 3)) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1 3))))) rand))))