Average Error: 0.2 → 0.1
Time: 4.0s
Precision: binary64
\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\]
↓
\[1 \cdot \left(\left(a - \frac{1}{3}\right) + \sqrt{a - \frac{1}{3}} \cdot \frac{rand}{\sqrt{9}}\right)\]
\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) + \sqrt{a - \frac{1}{3}} \cdot \frac{rand}{\sqrt{9}}\right)
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) (1.0 * ((double) (((double) (a - ((double) (1.0 / 3.0)))) + ((double) (((double) sqrt(((double) (a - ((double) (1.0 / 3.0)))))) * ((double) (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 0.2

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

  2. Simplified0.1

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

  4. Applied distribute-lft-in0.1

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

  5. Simplified0.1

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

  7. Applied sqrt-prod0.1

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

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

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

  9. Applied times-frac0.1

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

  10. Applied associate-*r*0.1

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

  11. Simplified0.1

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

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

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

  14. Applied sqrt-prod0.1

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

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

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

  16. Applied times-frac0.1

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

  17. Applied associate-*r*0.1

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

  18. Simplified0.1

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

  19. Final simplification0.1

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

Reproduce

herbie shell --seed 2020184 
(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))))