Average Error: 0.1 → 0.1
Time: 8.6s
Precision: 64
\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\]
↓
\[\left(a - \frac{1}{3}\right) \cdot 1 + \left(a - \frac{1}{3}\right) \cdot \left(\frac{1}{\sqrt{1}} \cdot \frac{rand}{{\left(\left(a - \frac{1}{3}\right) \cdot 9\right)}^{\frac{1}{2}}}\right)\]
\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)
↓
\left(a - \frac{1}{3}\right) \cdot 1 + \left(a - \frac{1}{3}\right) \cdot \left(\frac{1}{\sqrt{1}} \cdot \frac{rand}{{\left(\left(a - \frac{1}{3}\right) \cdot 9\right)}^{\frac{1}{2}}}\right)
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(1.0)) * (rand / pow(((a - (1.0 / 3.0)) * 9.0), 0.5)))));
}

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 associate-/r*0.2

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

  6. Applied *-un-lft-identity0.2

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

  7. Applied sqrt-prod0.2

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

  8. Applied div-inv0.2

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

  9. Applied times-frac0.2

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

  10. Applied associate-*l*0.2

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

  11. Simplified0.1

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

  13. Applied distribute-lft-in0.1

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

  15. Applied pow1/20.1

    \[\leadsto \left(a - \frac{1}{3}\right) \cdot 1 + \left(a - \frac{1}{3}\right) \cdot \left(\frac{1}{\sqrt{1}} \cdot \frac{rand}{\sqrt{a - \frac{1}{3}} \cdot \color{blue}{{9}^{\frac{1}{2}}}}\right)\]

  16. Applied pow1/20.1

    \[\leadsto \left(a - \frac{1}{3}\right) \cdot 1 + \left(a - \frac{1}{3}\right) \cdot \left(\frac{1}{\sqrt{1}} \cdot \frac{rand}{\color{blue}{{\left(a - \frac{1}{3}\right)}^{\frac{1}{2}}} \cdot {9}^{\frac{1}{2}}}\right)\]

  17. Applied pow-prod-down0.1

    \[\leadsto \left(a - \frac{1}{3}\right) \cdot 1 + \left(a - \frac{1}{3}\right) \cdot \left(\frac{1}{\sqrt{1}} \cdot \frac{rand}{\color{blue}{{\left(\left(a - \frac{1}{3}\right) \cdot 9\right)}^{\frac{1}{2}}}}\right)\]

  18. Final simplification0.1

    \[\leadsto \left(a - \frac{1}{3}\right) \cdot 1 + \left(a - \frac{1}{3}\right) \cdot \left(\frac{1}{\sqrt{1}} \cdot \frac{rand}{{\left(\left(a - \frac{1}{3}\right) \cdot 9\right)}^{\frac{1}{2}}}\right)\]

Reproduce

herbie shell --seed 2020053 +o rules:numerics
(FPCore (a rand)
  :name "Octave 3.8, oct_fill_randg"
  :precision binary64
  (* (- a (/ 1 3)) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1 3))))) rand))))