Average Error: 0.2 → 0.2
Time: 22.9s
Precision: 64
\[\left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right) \cdot \left(\frac{\left(1\right)}{\left(\left(\frac{\left(1\right)}{\left(\sqrt{\left(\left(9\right) \cdot \left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right)\right)}\right)}\right) \cdot rand\right)}\right)\]
↓
\[1 \cdot \left(a - \frac{1.0}{3.0}\right) + \left(\frac{1}{\sqrt{a \cdot 9 + \left(-\frac{1.0}{3.0}\right) \cdot 9}} \cdot rand\right) \cdot \left(a - \frac{1.0}{3.0}\right)\]
\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)
↓
1 \cdot \left(a - \frac{1.0}{3.0}\right) + \left(\frac{1}{\sqrt{a \cdot 9 + \left(-\frac{1.0}{3.0}\right) \cdot 9}} \cdot rand\right) \cdot \left(a - \frac{1.0}{3.0}\right)
double f(double a, double rand) {
        double r2402104 = a;
        double r2402105 = 1.0;
        double r2402106 = /* ERROR: no posit support in C */;
        double r2402107 = 3.0;
        double r2402108 = /* ERROR: no posit support in C */;
        double r2402109 = r2402106 / r2402108;
        double r2402110 = r2402104 - r2402109;
        double r2402111 = 1.0;
        double r2402112 = /* ERROR: no posit support in C */;
        double r2402113 = 9.0;
        double r2402114 = /* ERROR: no posit support in C */;
        double r2402115 = r2402114 * r2402110;
        double r2402116 = sqrt(r2402115);
        double r2402117 = r2402112 / r2402116;
        double r2402118 = rand;
        double r2402119 = r2402117 * r2402118;
        double r2402120 = r2402112 + r2402119;
        double r2402121 = r2402110 * r2402120;
        return r2402121;
}
↓
double f(double a, double rand) {
        double r2402122 = 1.0;
        double r2402123 = a;
        double r2402124 = 1.0;
        double r2402125 = 3.0;
        double r2402126 = r2402124 / r2402125;
        double r2402127 = r2402123 - r2402126;
        double r2402128 = r2402122 * r2402127;
        double r2402129 = 9.0;
        double r2402130 = r2402123 * r2402129;
        double r2402131 = -r2402126;
        double r2402132 = r2402131 * r2402129;
        double r2402133 = r2402130 + r2402132;
        double r2402134 = sqrt(r2402133);
        double r2402135 = r2402122 / r2402134;
        double r2402136 = rand;
        double r2402137 = r2402135 * r2402136;
        double r2402138 = r2402137 * r2402127;
        double r2402139 = r2402128 + r2402138;
        return r2402139;
}

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-rgt-in0.2

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

  5. Applied sub-neg0.2

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

  6. Applied distribute-rgt-in0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2019121 
(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))))