Average Error: 0.1 → 0.1
Time: 7.9s
Precision: 64
\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\]
↓
\[\left(a - \frac{1}{3}\right) \cdot 1 + \left(\frac{\sqrt{a - \frac{1}{3}}}{\sqrt{1}} \cdot \left(\frac{\sqrt{a - \frac{1}{3}}}{\sqrt{9}} \cdot \frac{1}{\sqrt{a - \frac{1}{3}}}\right)\right) \cdot rand\]
\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{\sqrt{a - \frac{1}{3}}}{\sqrt{1}} \cdot \left(\frac{\sqrt{a - \frac{1}{3}}}{\sqrt{9}} \cdot \frac{1}{\sqrt{a - \frac{1}{3}}}\right)\right) \cdot rand
double f(double a, double rand) {
        double r88537 = a;
        double r88538 = 1.0;
        double r88539 = 3.0;
        double r88540 = r88538 / r88539;
        double r88541 = r88537 - r88540;
        double r88542 = 9.0;
        double r88543 = r88542 * r88541;
        double r88544 = sqrt(r88543);
        double r88545 = r88538 / r88544;
        double r88546 = rand;
        double r88547 = r88545 * r88546;
        double r88548 = r88538 + r88547;
        double r88549 = r88541 * r88548;
        return r88549;
}
↓
double f(double a, double rand) {
        double r88550 = a;
        double r88551 = 1.0;
        double r88552 = 3.0;
        double r88553 = r88551 / r88552;
        double r88554 = r88550 - r88553;
        double r88555 = r88554 * r88551;
        double r88556 = sqrt(r88554);
        double r88557 = 1.0;
        double r88558 = sqrt(r88557);
        double r88559 = r88556 / r88558;
        double r88560 = 9.0;
        double r88561 = sqrt(r88560);
        double r88562 = r88556 / r88561;
        double r88563 = r88551 / r88556;
        double r88564 = r88562 * r88563;
        double r88565 = r88559 * r88564;
        double r88566 = rand;
        double r88567 = r88565 * r88566;
        double r88568 = r88555 + r88567;
        return r88568;
}

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 distribute-lft-in0.1

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

  5. Applied associate-*r*0.1

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

  7. Applied sqrt-prod0.1

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

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

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

  9. Applied times-frac0.2

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

  10. Applied associate-*r*0.2

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

  11. Simplified0.1

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

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

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

  14. Applied sqrt-prod0.1

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

  15. Applied add-sqr-sqrt0.2

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

  16. Applied times-frac0.1

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

  17. Applied associate-*l*0.1

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

  18. Final simplification0.1

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

Reproduce

herbie shell --seed 2020039 +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))))