Average Error: 0.1 → 0.1
Time: 33.0s
Precision: 64
\[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]
↓
\[\frac{rand}{\sqrt{\left(a - \frac{1.0}{3.0}\right) \cdot 9}} \cdot \left(a - \frac{1.0}{3.0}\right) + \left(a - \frac{1.0}{3.0}\right)\]
\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)
↓
\frac{rand}{\sqrt{\left(a - \frac{1.0}{3.0}\right) \cdot 9}} \cdot \left(a - \frac{1.0}{3.0}\right) + \left(a - \frac{1.0}{3.0}\right)
double f(double a, double rand) {
        double r3100042 = a;
        double r3100043 = 1.0;
        double r3100044 = 3.0;
        double r3100045 = r3100043 / r3100044;
        double r3100046 = r3100042 - r3100045;
        double r3100047 = 1.0;
        double r3100048 = 9.0;
        double r3100049 = r3100048 * r3100046;
        double r3100050 = sqrt(r3100049);
        double r3100051 = r3100047 / r3100050;
        double r3100052 = rand;
        double r3100053 = r3100051 * r3100052;
        double r3100054 = r3100047 + r3100053;
        double r3100055 = r3100046 * r3100054;
        return r3100055;
}
↓
double f(double a, double rand) {
        double r3100056 = rand;
        double r3100057 = a;
        double r3100058 = 1.0;
        double r3100059 = 3.0;
        double r3100060 = r3100058 / r3100059;
        double r3100061 = r3100057 - r3100060;
        double r3100062 = 9.0;
        double r3100063 = r3100061 * r3100062;
        double r3100064 = sqrt(r3100063);
        double r3100065 = r3100056 / r3100064;
        double r3100066 = r3100065 * r3100061;
        double r3100067 = r3100066 + r3100061;
        return r3100067;
}

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

    \[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]

  2. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{rand}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}}, a - \frac{1.0}{3.0}, a - \frac{1.0}{3.0}\right)}\]
  3. Using strategy rm

  4. Applied fma-udef0.1

    \[\leadsto \color{blue}{\frac{rand}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot \left(a - \frac{1.0}{3.0}\right) + \left(a - \frac{1.0}{3.0}\right)}\]

  5. Final simplification0.1

    \[\leadsto \frac{rand}{\sqrt{\left(a - \frac{1.0}{3.0}\right) \cdot 9}} \cdot \left(a - \frac{1.0}{3.0}\right) + \left(a - \frac{1.0}{3.0}\right)\]

Reproduce

herbie shell --seed 2019135 +o rules:numerics
(FPCore (a rand)
  :name "Octave 3.8, oct_fill_randg"
  (* (- a (/ 1.0 3.0)) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand))))