Average Error: 0.1 → 0.1
Time: 11.6s
Precision: 64
\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\]
↓
\[1 \cdot \left(a - \frac{1}{3}\right) + \frac{\frac{1 \cdot rand}{\sqrt{9}}}{\sqrt{a - \frac{1}{3}}} \cdot \left(a - \frac{1}{3}\right)\]
\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(a - \frac{1}{3}\right) + \frac{\frac{1 \cdot rand}{\sqrt{9}}}{\sqrt{a - \frac{1}{3}}} \cdot \left(a - \frac{1}{3}\right)
double f(double a, double rand) {
        double r85067 = a;
        double r85068 = 1.0;
        double r85069 = 3.0;
        double r85070 = r85068 / r85069;
        double r85071 = r85067 - r85070;
        double r85072 = 9.0;
        double r85073 = r85072 * r85071;
        double r85074 = sqrt(r85073);
        double r85075 = r85068 / r85074;
        double r85076 = rand;
        double r85077 = r85075 * r85076;
        double r85078 = r85068 + r85077;
        double r85079 = r85071 * r85078;
        return r85079;
}
↓
double f(double a, double rand) {
        double r85080 = 1.0;
        double r85081 = a;
        double r85082 = 3.0;
        double r85083 = r85080 / r85082;
        double r85084 = r85081 - r85083;
        double r85085 = r85080 * r85084;
        double r85086 = rand;
        double r85087 = r85080 * r85086;
        double r85088 = 9.0;
        double r85089 = sqrt(r85088);
        double r85090 = r85087 / r85089;
        double r85091 = sqrt(r85084);
        double r85092 = r85090 / r85091;
        double r85093 = r85092 * r85084;
        double r85094 = r85085 + r85093;
        return r85094;
}

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 associate-*l/0.1

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

  5. Applied distribute-lft-in0.1

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

  6. Simplified0.1

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

  7. Simplified0.1

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

  9. Applied sqrt-prod0.1

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

  10. Applied associate-/r*0.1

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

  11. Final simplification0.1

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

Reproduce

herbie shell --seed 2020045 +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))))