Average Error: 0.1 → 0.2
Time: 6.2s
Precision: binary64
\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\]
↓
\[1 \cdot \left(\left(a - \frac{1}{3}\right) + \frac{\sqrt{a - \frac{1}{3}} \cdot rand}{\sqrt{9}}\right)\]
\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(\left(a - \frac{1}{3}\right) + \frac{\sqrt{a - \frac{1}{3}} \cdot rand}{\sqrt{9}}\right)
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) (1.0 * ((double) (((double) (a - (1.0 / 3.0))) + (((double) (((double) sqrt(((double) (a - (1.0 / 3.0))))) * rand)) / ((double) sqrt(9.0)))))));
}

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

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

  4. Applied distribute-lft-in0.1

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

  5. Simplified0.1

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

  7. Applied sqrt-prod0.1

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

  8. Applied *-un-lft-identity0.1

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

  9. Applied times-frac0.1

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

  10. Applied associate-*r*0.1

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

  11. Simplified0.1

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

  13. Applied associate-*r/0.2

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

  14. Simplified0.2

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

  15. Final simplification0.2

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

Reproduce

herbie shell --seed 2020196 
(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))))