Average Error: 0.1 → 0.1
Time: 6.7s
Precision: 64
\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\]
↓
\[\left(a - \frac{1}{3}\right) \cdot 1 + \frac{\left(\left(a - \frac{1}{3}\right) \cdot 1\right) \cdot \frac{rand}{\sqrt{a - \frac{1}{3}}}}{\sqrt{9}}\]
\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 - \frac{1}{3}\right) \cdot 1 + \frac{\left(\left(a - \frac{1}{3}\right) \cdot 1\right) \cdot \frac{rand}{\sqrt{a - \frac{1}{3}}}}{\sqrt{9}}
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 - (1.0 / 3.0)) * 1.0) + ((((a - (1.0 / 3.0)) * 1.0) * (rand / sqrt((a - (1.0 / 3.0))))) / 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. Using strategy rm

  3. Applied associate-*l/0.1

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

  5. Applied distribute-lft-in0.1

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

  7. Applied sqrt-prod0.1

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

  8. Applied times-frac0.1

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

  9. Applied associate-*r*0.1

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

  11. Applied associate-*r/0.1

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

  12. Applied associate-*l/0.1

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

  13. Final simplification0.1

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

Reproduce

herbie shell --seed 2020100 
(FPCore (a rand)
  :name "Octave 3.8, oct_fill_randg"
  :precision binary64
  (* (- a (/ 1 3)) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1 3))))) rand))))