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 r7564965 = d1;
        double r7564966 = d2;
        double r7564967 = r7564965 * r7564966;
        double r7564968 = d3;
        double r7564969 = r7564965 * r7564968;
        double r7564970 = r7564967 - r7564969;
        double r7564971 = d4;
        double r7564972 = r7564971 * r7564965;
        double r7564973 = r7564970 + r7564972;
        double r7564974 = r7564965 * r7564965;
        double r7564975 = r7564973 - r7564974;
        return r7564975;
}
↓
double f(double d1, double d2, double d3, double d4) {
        double r7564976 = d1;
        double r7564977 = d2;
        double r7564978 = d4;
        double r7564979 = d3;
        double r7564980 = r7564978 - r7564979;
        double r7564981 = r7564980 - r7564976;
        double r7564982 = r7564977 + r7564981;
        double r7564983 = r7564976 * r7564982;
        return r7564983;
}

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 
(FPCore (d1 d2 d3 d4)
  :name "FastMath dist4"
  (-.p16 (+.p16 (-.p16 (*.p16 d1 d2) (*.p16 d1 d3)) (*.p16 d4 d1)) (*.p16 d1 d1)))