Average Error: 0.5 → 0.4
Time: 9.3s
Precision: 64
\[\left(\frac{\left(\left(d1 \cdot d2\right) - \left(d1 \cdot d3\right)\right)}{\left(d4 \cdot d1\right)}\right) - \left(d1 \cdot d1\right)\]
↓
\[d1 \cdot \left(d2 - \left(d3 - \left(d4 - d1\right)\right)\right)\]
\left(\frac{\left(\left(d1 \cdot d2\right) - \left(d1 \cdot d3\right)\right)}{\left(d4 \cdot d1\right)}\right) - \left(d1 \cdot d1\right)
↓
d1 \cdot \left(d2 - \left(d3 - \left(d4 - d1\right)\right)\right)
double f(double d1, double d2, double d3, double d4) {
        double r1307977 = d1;
        double r1307978 = d2;
        double r1307979 = r1307977 * r1307978;
        double r1307980 = d3;
        double r1307981 = r1307977 * r1307980;
        double r1307982 = r1307979 - r1307981;
        double r1307983 = d4;
        double r1307984 = r1307983 * r1307977;
        double r1307985 = r1307982 + r1307984;
        double r1307986 = r1307977 * r1307977;
        double r1307987 = r1307985 - r1307986;
        return r1307987;
}
↓
double f(double d1, double d2, double d3, double d4) {
        double r1307988 = d1;
        double r1307989 = d2;
        double r1307990 = d3;
        double r1307991 = d4;
        double r1307992 = r1307991 - r1307988;
        double r1307993 = r1307990 - r1307992;
        double r1307994 = r1307989 - r1307993;
        double r1307995 = r1307988 * r1307994;
        return r1307995;
}

Error

Bits error versus d1

Bits error versus d2

Bits error versus d3

Bits error versus d4

Derivation

  1. Initial program 0.5

    \[\left(\frac{\left(\left(d1 \cdot d2\right) - \left(d1 \cdot d3\right)\right)}{\left(d4 \cdot d1\right)}\right) - \left(d1 \cdot d1\right)\]

  2. Simplified0.4

    \[\leadsto \color{blue}{d1 \cdot \left(\frac{\left(d2 - d3\right)}{\left(d4 - d1\right)}\right)}\]
  3. Using strategy rm

  4. Applied associate-+l-0.4

    \[\leadsto d1 \cdot \color{blue}{\left(d2 - \left(d3 - \left(d4 - d1\right)\right)\right)}\]

  5. Final simplification0.4

    \[\leadsto d1 \cdot \left(d2 - \left(d3 - \left(d4 - d1\right)\right)\right)\]

Reproduce

herbie shell --seed 2019107 +o rules:numerics
(FPCore (d1 d2 d3 d4)
  :name "FastMath dist4"
  (-.p16 (+.p16 (-.p16 (*.p16 d1 d2) (*.p16 d1 d3)) (*.p16 d4 d1)) (*.p16 d1 d1)))