Average Error: 0.2 → 0.2
Time: 19.1s
Precision: 64
\[\left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right) \cdot \left(\frac{\left(1\right)}{\left(\left(\frac{\left(1\right)}{\left(\sqrt{\left(\left(9\right) \cdot \left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right)\right)}\right)}\right) \cdot rand\right)}\right)\]
↓
\[\left(a - \frac{1.0}{3.0}\right) \cdot 1 + \left(a - \frac{1.0}{3.0}\right) \cdot \left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]
\left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right) \cdot \left(\frac{\left(1\right)}{\left(\left(\frac{\left(1\right)}{\left(\sqrt{\left(\left(9\right) \cdot \left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right)\right)}\right)}\right) \cdot rand\right)}\right)
↓
\left(a - \frac{1.0}{3.0}\right) \cdot 1 + \left(a - \frac{1.0}{3.0}\right) \cdot \left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)
double f(double a, double rand) {
        double r1935110 = a;
        double r1935111 = 1.0;
        double r1935112 = /* ERROR: no posit support in C */;
        double r1935113 = 3.0;
        double r1935114 = /* ERROR: no posit support in C */;
        double r1935115 = r1935112 / r1935114;
        double r1935116 = r1935110 - r1935115;
        double r1935117 = 1.0;
        double r1935118 = /* ERROR: no posit support in C */;
        double r1935119 = 9.0;
        double r1935120 = /* ERROR: no posit support in C */;
        double r1935121 = r1935120 * r1935116;
        double r1935122 = sqrt(r1935121);
        double r1935123 = r1935118 / r1935122;
        double r1935124 = rand;
        double r1935125 = r1935123 * r1935124;
        double r1935126 = r1935118 + r1935125;
        double r1935127 = r1935116 * r1935126;
        return r1935127;
}
↓
double f(double a, double rand) {
        double r1935128 = a;
        double r1935129 = 1.0;
        double r1935130 = 3.0;
        double r1935131 = r1935129 / r1935130;
        double r1935132 = r1935128 - r1935131;
        double r1935133 = 1.0;
        double r1935134 = r1935132 * r1935133;
        double r1935135 = 9.0;
        double r1935136 = r1935135 * r1935132;
        double r1935137 = sqrt(r1935136);
        double r1935138 = r1935133 / r1935137;
        double r1935139 = rand;
        double r1935140 = r1935138 * r1935139;
        double r1935141 = r1935132 * r1935140;
        double r1935142 = r1935134 + r1935141;
        return r1935142;
}

Error

Bits error versus a

Bits error versus rand

Derivation

  1. Initial program 0.2

    \[\left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right) \cdot \left(\frac{\left(1\right)}{\left(\left(\frac{\left(1\right)}{\left(\sqrt{\left(\left(9\right) \cdot \left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right)\right)}\right)}\right) \cdot rand\right)}\right)\]
  2. Using strategy rm

  3. Applied distribute-lft-in0.2

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2019119 
(FPCore (a rand)
  :name "Octave 3.8, oct_fill_randg"
  (*.p16 (-.p16 a (/.p16 (real->posit16 1.0) (real->posit16 3.0))) (+.p16 (real->posit16 1) (*.p16 (/.p16 (real->posit16 1) (sqrt.p16 (*.p16 (real->posit16 9) (-.p16 a (/.p16 (real->posit16 1.0) (real->posit16 3.0)))))) rand))))