Average Error: 0.1 → 0.2
Time: 7.7s
Precision: 64
\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\]
↓
\[\left(a - \frac{1}{3}\right) \cdot 1 + \frac{\left(\frac{1}{\sqrt{a - \frac{1}{3}}} \cdot rand\right) \cdot \left(a - \frac{1}{3}\right)}{\sqrt{9}}\]
\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{\left(\frac{1}{\sqrt{a - \frac{1}{3}}} \cdot rand\right) \cdot \left(a - \frac{1}{3}\right)}{\sqrt{9}}
double f(double a, double rand) {
        double r95043 = a;
        double r95044 = 1.0;
        double r95045 = 3.0;
        double r95046 = r95044 / r95045;
        double r95047 = r95043 - r95046;
        double r95048 = 9.0;
        double r95049 = r95048 * r95047;
        double r95050 = sqrt(r95049);
        double r95051 = r95044 / r95050;
        double r95052 = rand;
        double r95053 = r95051 * r95052;
        double r95054 = r95044 + r95053;
        double r95055 = r95047 * r95054;
        return r95055;
}
↓
double f(double a, double rand) {
        double r95056 = a;
        double r95057 = 1.0;
        double r95058 = 3.0;
        double r95059 = r95057 / r95058;
        double r95060 = r95056 - r95059;
        double r95061 = r95060 * r95057;
        double r95062 = sqrt(r95060);
        double r95063 = r95057 / r95062;
        double r95064 = rand;
        double r95065 = r95063 * r95064;
        double r95066 = r95065 * r95060;
        double r95067 = 9.0;
        double r95068 = sqrt(r95067);
        double r95069 = r95066 / r95068;
        double r95070 = r95061 + r95069;
        return r95070;
}

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 sqrt-prod0.1

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

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

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

  7. Applied times-frac0.2

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

  8. Applied associate-*l*0.2

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

  10. Applied associate-*l/0.1

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

  11. Applied associate-*r/0.2

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

  12. Simplified0.2

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

  13. Final simplification0.2

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

Reproduce

herbie shell --seed 2020046 +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))))