Average Error: 0.1 → 0.2
Time: 6.7s
Precision: 64
\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\]
↓
\[\left(a - \frac{1}{3}\right) \cdot 1 + \frac{\left(a - \frac{1}{3}\right) \cdot \left(\frac{1}{\sqrt{a - \frac{1}{3}}} \cdot rand\right)}{\sqrt{9}}\]
\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(a - \frac{1}{3}\right) \cdot \left(\frac{1}{\sqrt{a - \frac{1}{3}}} \cdot rand\right)}{\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 / sqrt((a - (1.0 / 3.0)))) * rand)) / 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 sqrt-prod0.1

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

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

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

  5. Applied times-frac0.2

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

  6. Applied associate-*l*0.2

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

  8. Applied distribute-lft-in0.2

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

  9. Simplified0.2

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

  10. Final simplification0.2

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

Reproduce

herbie shell --seed 2020091 
(FPCore (a rand)
  :name "Octave 3.8, oct_fill_randg"
  :precision binary64
  (* (- a (/ 1 3)) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1 3))))) rand))))