Average Error: 0.5 → 0.2
Time: 21.5s
Precision: 64
\[\frac{\left(\frac{\left(d1 \cdot \left(3\right)\right)}{\left(d1 \cdot d2\right)}\right)}{\left(d1 \cdot d3\right)}\]
↓
\[\left(\mathsf{qma}\left(\left(\mathsf{qma}\left(\left(3\right), d3, 1.0\right)\right), d2, 1.0\right)\right) \cdot d1\]
\frac{\left(\frac{\left(d1 \cdot \left(3\right)\right)}{\left(d1 \cdot d2\right)}\right)}{\left(d1 \cdot d3\right)}
↓
\left(\mathsf{qma}\left(\left(\mathsf{qma}\left(\left(3\right), d3, 1.0\right)\right), d2, 1.0\right)\right) \cdot d1
double f(double d1, double d2, double d3) {
        double r1078035 = d1;
        double r1078036 = 3.0;
        double r1078037 = /* ERROR: no posit support in C */;
        double r1078038 = r1078035 * r1078037;
        double r1078039 = d2;
        double r1078040 = r1078035 * r1078039;
        double r1078041 = r1078038 + r1078040;
        double r1078042 = d3;
        double r1078043 = r1078035 * r1078042;
        double r1078044 = r1078041 + r1078043;
        return r1078044;
}
↓
double f(double d1, double d2, double d3) {
        double r1078045 = 3.0;
        double r1078046 = /*Error: no posit support in C */;
        double r1078047 = d3;
        double r1078048 = 1.0;
        double r1078049 = /*Error: no posit support in C */;
        double r1078050 = d2;
        double r1078051 = /*Error: no posit support in C */;
        double r1078052 = /*Error: no posit support in C */;
        double r1078053 = d1;
        double r1078054 = r1078052 * r1078053;
        return r1078054;
}

Error

Bits error versus d1

Bits error versus d2

Bits error versus d3

Derivation

  1. Initial program 0.5

    \[\frac{\left(\frac{\left(d1 \cdot \left(3\right)\right)}{\left(d1 \cdot d2\right)}\right)}{\left(d1 \cdot d3\right)}\]

  2. Simplified0.3

    \[\leadsto \color{blue}{\left(\frac{\left(\frac{\left(3\right)}{d3}\right)}{d2}\right) \cdot d1}\]
  3. Using strategy rm

  4. Applied introduce-quire0.3

    \[\leadsto \left(\frac{\left(\frac{\color{blue}{\left(\left(\left(3\right)\right)\right)}}{d3}\right)}{d2}\right) \cdot d1\]

  5. Applied insert-quire-add0.3

    \[\leadsto \left(\frac{\color{blue}{\left(\left(\mathsf{qma}\left(\left(\left(3\right)\right), d3, \left(1.0\right)\right)\right)\right)}}{d2}\right) \cdot d1\]

  6. Applied insert-quire-add0.2

    \[\leadsto \color{blue}{\left(\left(\mathsf{qma}\left(\left(\mathsf{qma}\left(\left(\left(3\right)\right), d3, \left(1.0\right)\right)\right), d2, \left(1.0\right)\right)\right)\right)} \cdot d1\]

  7. Final simplification0.2

    \[\leadsto \left(\mathsf{qma}\left(\left(\mathsf{qma}\left(\left(3\right), d3, 1.0\right)\right), d2, 1.0\right)\right) \cdot d1\]

Reproduce

herbie shell --seed 2019153 +o rules:numerics
(FPCore (d1 d2 d3)
  :name "FastMath test3"
  (+.p16 (+.p16 (*.p16 d1 (real->posit16 3)) (*.p16 d1 d2)) (*.p16 d1 d3)))