Average Error: 0.1 → 0.1
Time: 29.2s
Precision: 64
\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\]
↓
\[1 \cdot \left(a - \frac{1}{3}\right) + \frac{\left(a - \frac{1}{3}\right) \cdot 1}{\frac{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}{rand}}\]
\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(a - \frac{1}{3}\right) + \frac{\left(a - \frac{1}{3}\right) \cdot 1}{\frac{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}{rand}}
double f(double a, double rand) {
        double r72501 = a;
        double r72502 = 1.0;
        double r72503 = 3.0;
        double r72504 = r72502 / r72503;
        double r72505 = r72501 - r72504;
        double r72506 = 9.0;
        double r72507 = r72506 * r72505;
        double r72508 = sqrt(r72507);
        double r72509 = r72502 / r72508;
        double r72510 = rand;
        double r72511 = r72509 * r72510;
        double r72512 = r72502 + r72511;
        double r72513 = r72505 * r72512;
        return r72513;
}
↓
double f(double a, double rand) {
        double r72514 = 1.0;
        double r72515 = a;
        double r72516 = 3.0;
        double r72517 = r72514 / r72516;
        double r72518 = r72515 - r72517;
        double r72519 = r72514 * r72518;
        double r72520 = r72518 * r72514;
        double r72521 = 9.0;
        double r72522 = r72521 * r72518;
        double r72523 = sqrt(r72522);
        double r72524 = rand;
        double r72525 = r72523 / r72524;
        double r72526 = r72520 / r72525;
        double r72527 = r72519 + r72526;
        return r72527;
}

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 div-inv0.1

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

  4. Applied associate-*l*0.1

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

  5. Simplified0.1

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

  7. Applied distribute-lft-in0.1

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

  8. Simplified0.1

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

  10. Applied clear-num0.1

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

  12. Applied un-div-inv0.1

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

  13. Applied associate-*r/0.1

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

  14. Final simplification0.1

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

Reproduce

herbie shell --seed 2019326 
(FPCore (a rand)
  :name "Octave 3.8, oct_fill_randg"
  :precision binary64
  (* (- a (/ 1 3)) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1 3))))) rand))))