Average Error: 0.1 → 0.1
Time: 29.2s
Precision: 64
\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\]
↓
\[\mathsf{fma}\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}\right)\right), rand, 1\right) \cdot \left(a - \frac{1}{3}\right)\]
\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)
↓
\mathsf{fma}\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}\right)\right), rand, 1\right) \cdot \left(a - \frac{1}{3}\right)
double f(double a, double rand) {
        double r91081 = a;
        double r91082 = 1.0;
        double r91083 = 3.0;
        double r91084 = r91082 / r91083;
        double r91085 = r91081 - r91084;
        double r91086 = 9.0;
        double r91087 = r91086 * r91085;
        double r91088 = sqrt(r91087);
        double r91089 = r91082 / r91088;
        double r91090 = rand;
        double r91091 = r91089 * r91090;
        double r91092 = r91082 + r91091;
        double r91093 = r91085 * r91092;
        return r91093;
}
↓
double f(double a, double rand) {
        double r91094 = 1.0;
        double r91095 = 9.0;
        double r91096 = a;
        double r91097 = 3.0;
        double r91098 = r91094 / r91097;
        double r91099 = r91096 - r91098;
        double r91100 = r91095 * r91099;
        double r91101 = sqrt(r91100);
        double r91102 = r91094 / r91101;
        double r91103 = expm1(r91102);
        double r91104 = log1p(r91103);
        double r91105 = rand;
        double r91106 = fma(r91104, r91105, r91094);
        double r91107 = r91106 * r91099;
        return r91107;
}

Error

Bits error versus a

Bits error versus rand

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

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

  4. Applied log1p-expm1-u0.1

    \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}\right)\right)}, rand, 1\right) \cdot \left(a - \frac{1}{3}\right)\]

  5. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}\right)\right), rand, 1\right) \cdot \left(a - \frac{1}{3}\right)\]

Reproduce

herbie shell --seed 2019347 +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))))