Octave 3.8, oct_fill_randg

Average Error: 0.1 → 1.7

Time: 5.3s

Precision: binary64

Math

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

↓

\[a \cdot \mathsf{fma}\left(\sqrt{\frac{1}{\left|\mathsf{fma}\left(a, 9, -3\right)\right|}}, rand, 1\right) \]

FPCore

(FPCore (a rand)
 :precision binary64
 (*
  (- a (/ 1.0 3.0))
  (+ 1.0 (* (/ 1.0 (sqrt (* 9.0 (- a (/ 1.0 3.0))))) rand))))

↓

(FPCore (a rand)
 :precision binary64
 (* a (fma (sqrt (/ 1.0 (fabs (fma a 9.0 -3.0)))) rand 1.0)))

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 * fma(sqrt((1.0 / fabs(fma(a, 9.0, -3.0)))), rand, 1.0);
}

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. Simplified 0.1

   \[\leadsto \color{blue}{\left(a - 0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{\mathsf{fma}\left(a, 9, -3\right)}}\right)} \]

3. Applied egg-rr 0.1

   \[\leadsto \left(a - 0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{\color{blue}{\left|\mathsf{fma}\left(a, 9, -3\right)\right|}}}\right) \]

4. Taylor expanded in a around inf 1.7

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

5. Simplified 1.7

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

6. Final simplification 1.7

   \[\leadsto a \cdot \mathsf{fma}\left(\sqrt{\frac{1}{\left|\mathsf{fma}\left(a, 9, -3\right)\right|}}, rand, 1\right) \]

Reproduce

herbie shell --seed 2022127 
(FPCore (a rand)
  :name "Octave 3.8, oct_fill_randg"
  :precision binary64
  (* (- a (/ 1.0 3.0)) (+ 1.0 (* (/ 1.0 (sqrt (* 9.0 (- a (/ 1.0 3.0))))) rand))))