Average Error: 0.1 → 0.1
Time: 5.2s
Precision: binary64
\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
↓
\[\left(-0.3333333333333333 + a\right) \cdot \left(1 + {\left(\left(-0.3333333333333333 + a\right) \cdot 9\right)}^{-0.5} \cdot rand\right) \]
\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)
↓
\left(-0.3333333333333333 + a\right) \cdot \left(1 + {\left(\left(-0.3333333333333333 + a\right) \cdot 9\right)}^{-0.5} \cdot rand\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
 (*
  (+ -0.3333333333333333 a)
  (+ 1.0 (* (pow (* (+ -0.3333333333333333 a) 9.0) -0.5) rand))))
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 (-0.3333333333333333 + a) * (1.0 + (pow(((-0.3333333333333333 + a) * 9.0), -0.5) * rand));
}

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. Applied pow1/2_binary640.1

    \[\leadsto \left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\color{blue}{{\left(9 \cdot \left(a - \frac{1}{3}\right)\right)}^{0.5}}} \cdot rand\right) \]

  3. Applied pow-flip_binary640.1

    \[\leadsto \left(a - \frac{1}{3}\right) \cdot \left(1 + \color{blue}{{\left(9 \cdot \left(a - \frac{1}{3}\right)\right)}^{\left(-0.5\right)}} \cdot rand\right) \]

  4. Simplified0.1

    \[\leadsto \left(a - \frac{1}{3}\right) \cdot \left(1 + {\left(9 \cdot \left(a - \frac{1}{3}\right)\right)}^{\color{blue}{-0.5}} \cdot rand\right) \]

  5. Applied *-un-lft-identity_binary640.1

    \[\leadsto \left(a - \frac{1}{3}\right) \cdot \color{blue}{\left(1 \cdot \left(1 + {\left(9 \cdot \left(a - \frac{1}{3}\right)\right)}^{-0.5} \cdot rand\right)\right)} \]

  6. Applied associate-*r*_binary640.1

    \[\leadsto \color{blue}{\left(\left(a - \frac{1}{3}\right) \cdot 1\right) \cdot \left(1 + {\left(9 \cdot \left(a - \frac{1}{3}\right)\right)}^{-0.5} \cdot rand\right)} \]

  7. Simplified0.1

    \[\leadsto \color{blue}{\left(-0.3333333333333333 + a\right)} \cdot \left(1 + {\left(9 \cdot \left(a - \frac{1}{3}\right)\right)}^{-0.5} \cdot rand\right) \]

  8. Final simplification0.1

    \[\leadsto \left(-0.3333333333333333 + a\right) \cdot \left(1 + {\left(\left(-0.3333333333333333 + a\right) \cdot 9\right)}^{-0.5} \cdot rand\right) \]

Reproduce

herbie shell --seed 2022004 
(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))))