Average Error: 0.1 → 0.1
Time: 29.8s
Precision: 64
\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\]
↓
\[a \cdot \mathsf{fma}\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}, rand, 1\right) + \left(-\frac{1}{3} \cdot \mathsf{fma}\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}, rand, 1\right)\right)\]
\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)
↓
a \cdot \mathsf{fma}\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}, rand, 1\right) + \left(-\frac{1}{3} \cdot \mathsf{fma}\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}, rand, 1\right)\right)
double f(double a, double rand) {
        double r74131 = a;
        double r74132 = 1.0;
        double r74133 = 3.0;
        double r74134 = r74132 / r74133;
        double r74135 = r74131 - r74134;
        double r74136 = 9.0;
        double r74137 = r74136 * r74135;
        double r74138 = sqrt(r74137);
        double r74139 = r74132 / r74138;
        double r74140 = rand;
        double r74141 = r74139 * r74140;
        double r74142 = r74132 + r74141;
        double r74143 = r74135 * r74142;
        return r74143;
}
↓
double f(double a, double rand) {
        double r74144 = a;
        double r74145 = 1.0;
        double r74146 = 9.0;
        double r74147 = 3.0;
        double r74148 = r74145 / r74147;
        double r74149 = r74144 - r74148;
        double r74150 = r74146 * r74149;
        double r74151 = sqrt(r74150);
        double r74152 = r74145 / r74151;
        double r74153 = rand;
        double r74154 = fma(r74152, r74153, r74145);
        double r74155 = r74144 * r74154;
        double r74156 = r74148 * r74154;
        double r74157 = -r74156;
        double r74158 = r74155 + r74157;
        return r74158;
}

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 sub-neg0.1

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

  5. Applied distribute-lft-in0.1

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

  6. Simplified0.1

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

  7. Simplified0.1

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

  8. Final simplification0.1

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

Reproduce

herbie shell --seed 2019325 +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))))