Average Error: 0.1 → 0.1
Time: 8.4s
Precision: 64
\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\]
↓
\[\left(a - \frac{1}{3}\right) \cdot 1 + \frac{a - \frac{1}{3}}{\frac{\sqrt{9} \cdot \sqrt{a - \frac{1}{3}}}{rand \cdot 1}}\]
\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{a - \frac{1}{3}}{\frac{\sqrt{9} \cdot \sqrt{a - \frac{1}{3}}}{rand \cdot 1}}
double f(double a, double rand) {
        double r151345 = a;
        double r151346 = 1.0;
        double r151347 = 3.0;
        double r151348 = r151346 / r151347;
        double r151349 = r151345 - r151348;
        double r151350 = 9.0;
        double r151351 = r151350 * r151349;
        double r151352 = sqrt(r151351);
        double r151353 = r151346 / r151352;
        double r151354 = rand;
        double r151355 = r151353 * r151354;
        double r151356 = r151346 + r151355;
        double r151357 = r151349 * r151356;
        return r151357;
}
↓
double f(double a, double rand) {
        double r151358 = a;
        double r151359 = 1.0;
        double r151360 = 3.0;
        double r151361 = r151359 / r151360;
        double r151362 = r151358 - r151361;
        double r151363 = r151362 * r151359;
        double r151364 = 9.0;
        double r151365 = sqrt(r151364);
        double r151366 = sqrt(r151362);
        double r151367 = r151365 * r151366;
        double r151368 = rand;
        double r151369 = r151368 * r151359;
        double r151370 = r151367 / r151369;
        double r151371 = r151362 / r151370;
        double r151372 = r151363 + r151371;
        return r151372;
}

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-prod0.2

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

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

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

  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. Using strategy rm

  7. Applied frac-times0.2

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

  8. Applied associate-*l/0.1

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

  9. Simplified0.1

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

  11. Applied clear-num0.2

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

  13. Applied distribute-lft-in0.1

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

  14. Simplified0.1

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

  15. Final simplification0.1

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

Reproduce

herbie shell --seed 2019354 
(FPCore (a rand)
  :name "Octave 3.8, oct_fill_randg"
  :precision binary64
  (* (- a (/ 1 3)) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1 3))))) rand))))