Average Error: 0.5 → 0.4
Time: 21.6s
Precision: 64
\[\left(\frac{\left(\left(d1 \cdot d2\right) - \left(d1 \cdot d3\right)\right)}{\left(d4 \cdot d1\right)}\right) - \left(d1 \cdot d1\right)\]
↓
\[d1 \cdot \left(d2 + \left(\left(d4 - d3\right) - d1\right)\right)\]
\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(\left(d4 - d3\right) - d1\right)\right)
double f(double d1, double d2, double d3, double d4) {
        double r6048972 = d1;
        double r6048973 = d2;
        double r6048974 = r6048972 * r6048973;
        double r6048975 = d3;
        double r6048976 = r6048972 * r6048975;
        double r6048977 = r6048974 - r6048976;
        double r6048978 = d4;
        double r6048979 = r6048978 * r6048972;
        double r6048980 = r6048977 + r6048979;
        double r6048981 = r6048972 * r6048972;
        double r6048982 = r6048980 - r6048981;
        return r6048982;
}
↓
double f(double d1, double d2, double d3, double d4) {
        double r6048983 = d1;
        double r6048984 = d2;
        double r6048985 = d4;
        double r6048986 = d3;
        double r6048987 = r6048985 - r6048986;
        double r6048988 = r6048987 - r6048983;
        double r6048989 = r6048984 + r6048988;
        double r6048990 = r6048983 * r6048989;
        return r6048990;
}

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

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

  4. Applied associate--l+0.4

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

  6. Applied associate--r+0.4

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

  7. Final simplification0.4

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

Reproduce

herbie shell --seed 2019134 +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)))