Average Error: 0.1 → 0.1
Time: 3.6s
Precision: binary64
\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\]
↓
\[1 \cdot \left(a - \frac{1}{3}\right) + \left(\left(a - \frac{1}{3}\right) \cdot \frac{1}{\sqrt{\left(a - \frac{1}{3}\right) \cdot 9}}\right) \cdot rand\]
\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)
↓
1 \cdot \left(a - \frac{1}{3}\right) + \left(\left(a - \frac{1}{3}\right) \cdot \frac{1}{\sqrt{\left(a - \frac{1}{3}\right) \cdot 9}}\right) \cdot rand
(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
 (+
  (* 1.0 (- a (/ 1.0 3.0)))
  (* (* (- a (/ 1.0 3.0)) (/ 1.0 (sqrt (* (- a (/ 1.0 3.0)) 9.0)))) rand)))
double code(double a, double rand) {
	return ((double) (((double) (a - (1.0 / 3.0))) * ((double) (1.0 + ((double) ((1.0 / ((double) sqrt(((double) (9.0 * ((double) (a - (1.0 / 3.0)))))))) * rand))))));
}
↓
double code(double a, double rand) {
	return ((double) (((double) (1.0 * ((double) (a - (1.0 / 3.0))))) + ((double) (((double) (((double) (a - (1.0 / 3.0))) * (1.0 / ((double) sqrt(((double) (((double) (a - (1.0 / 3.0))) * 9.0))))))) * 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. Using strategy rm

  3. Applied distribute-lft-in_binary640.1

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

  4. Simplified0.1

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

  6. Applied associate-*r*_binary640.1

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

  7. Simplified0.1

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

  8. Final simplification0.1

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

Reproduce

herbie shell --seed 2020205 
(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))))