Average Error: 0.1 → 0.1
Time: 6.0s
Precision: binary64
\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\]
↓
\[\left(a - \frac{1}{3}\right) \cdot 1 + \left(\frac{1}{\sqrt{9}} \cdot \left(a - \frac{1}{3}\right)\right) \cdot \frac{rand}{\sqrt{a - \frac{1}{3}}}\]
\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(\frac{1}{\sqrt{9}} \cdot \left(a - \frac{1}{3}\right)\right) \cdot \frac{rand}{\sqrt{a - \frac{1}{3}}}
double code(double a, double rand) {
	return ((double) (((double) (a - ((double) (1.0 / 3.0)))) * ((double) (1.0 + ((double) (((double) (1.0 / ((double) sqrt(((double) (9.0 * ((double) (a - ((double) (1.0 / 3.0)))))))))) * rand))))));
}
↓
double code(double a, double rand) {
	return ((double) (((double) (((double) (a - ((double) (1.0 / 3.0)))) * 1.0)) + ((double) (((double) (((double) (1.0 / ((double) sqrt(9.0)))) * ((double) (a - ((double) (1.0 / 3.0)))))) * ((double) (rand / ((double) sqrt(((double) (a - ((double) (1.0 / 3.0))))))))))));
}

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 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 associate-*l/0.1

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

  8. Applied distribute-lft-in0.1

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

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

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

  11. Applied times-frac0.1

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

  12. Applied associate-*r*0.1

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

  13. Simplified0.1

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

  14. Final simplification0.1

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

Reproduce

herbie shell --seed 2020162 
(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))))