Average Error: 0.1 → 0.1
Time: 4.7s
Precision: binary64
\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
↓
\[\left(a - 0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{a \cdot 9 - 3}}\right) \]
\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 - 0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{a \cdot 9 - 3}}\right)

(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 0.3333333333333333) (+ 1.0 (/ rand (sqrt (- (* a 9.0) 3.0))))))
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 - 0.3333333333333333) * (1.0 + (rand / sqrt(((a * 9.0) - 3.0))));
}

Error

Bits error versus a

Bits error versus rand

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Simplified0.1

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

  3. Taylor expanded in a around 0 0.1

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

  4. Final simplification0.1

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

Reproduce

herbie shell --seed 2022130 
(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))))