Average Error: 0.1 → 0.1
Time: 27.7s
Precision: 64
\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\]
↓
\[\frac{rand}{\sqrt{\left(a - \frac{1}{3}\right) \cdot 9}} \cdot \left(1 \cdot \left(a - \frac{1}{3}\right)\right) + 1 \cdot \left(a - \frac{1}{3}\right)\]
\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)
↓
\frac{rand}{\sqrt{\left(a - \frac{1}{3}\right) \cdot 9}} \cdot \left(1 \cdot \left(a - \frac{1}{3}\right)\right) + 1 \cdot \left(a - \frac{1}{3}\right)
double f(double a, double rand) {
        double r3609965 = a;
        double r3609966 = 1.0;
        double r3609967 = 3.0;
        double r3609968 = r3609966 / r3609967;
        double r3609969 = r3609965 - r3609968;
        double r3609970 = 9.0;
        double r3609971 = r3609970 * r3609969;
        double r3609972 = sqrt(r3609971);
        double r3609973 = r3609966 / r3609972;
        double r3609974 = rand;
        double r3609975 = r3609973 * r3609974;
        double r3609976 = r3609966 + r3609975;
        double r3609977 = r3609969 * r3609976;
        return r3609977;
}
↓
double f(double a, double rand) {
        double r3609978 = rand;
        double r3609979 = a;
        double r3609980 = 1.0;
        double r3609981 = 3.0;
        double r3609982 = r3609980 / r3609981;
        double r3609983 = r3609979 - r3609982;
        double r3609984 = 9.0;
        double r3609985 = r3609983 * r3609984;
        double r3609986 = sqrt(r3609985);
        double r3609987 = r3609978 / r3609986;
        double r3609988 = r3609980 * r3609983;
        double r3609989 = r3609987 * r3609988;
        double r3609990 = r3609989 + r3609988;
        return r3609990;
}

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. Simplified0.1

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

  4. Applied fma-udef0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(FPCore (a rand)
  :name "Octave 3.8, oct_fill_randg"
  (* (- a (/ 1.0 3.0)) (+ 1.0 (* (/ 1.0 (sqrt (* 9.0 (- a (/ 1.0 3.0))))) rand))))