Average Error: 0.1 → 0.1
Time: 29.1s
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 r76137 = a;
        double r76138 = 1.0;
        double r76139 = 3.0;
        double r76140 = r76138 / r76139;
        double r76141 = r76137 - r76140;
        double r76142 = 9.0;
        double r76143 = r76142 * r76141;
        double r76144 = sqrt(r76143);
        double r76145 = r76138 / r76144;
        double r76146 = rand;
        double r76147 = r76145 * r76146;
        double r76148 = r76138 + r76147;
        double r76149 = r76141 * r76148;
        return r76149;
}
↓
double f(double a, double rand) {
        double r76150 = a;
        double r76151 = 1.0;
        double r76152 = 9.0;
        double r76153 = 3.0;
        double r76154 = r76151 / r76153;
        double r76155 = r76150 - r76154;
        double r76156 = r76152 * r76155;
        double r76157 = sqrt(r76156);
        double r76158 = r76151 / r76157;
        double r76159 = rand;
        double r76160 = fma(r76158, r76159, r76151);
        double r76161 = r76150 * r76160;
        double r76162 = r76154 * r76160;
        double r76163 = -r76162;
        double r76164 = r76161 + r76163;
        return r76164;
}

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))))