Average Error: 0.5 → 0.4
Time: 28.4s
Precision: 64
\[\left(\left(\left(d1 \cdot d2\right) - \left(d1 \cdot d3\right)\right) + \left(d4 \cdot d1\right)\right) - \left(d1 \cdot d1\right)\]
↓
\[d1 \cdot \left(\left(d4 + d2\right) - \left(d3 + d1\right)\right)\]
\left(\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(\left(d4 + d2\right) - \left(d3 + d1\right)\right)
double f(double d1, double d2, double d3, double d4) {
        double r8044035 = d1;
        double r8044036 = d2;
        double r8044037 = r8044035 * r8044036;
        double r8044038 = d3;
        double r8044039 = r8044035 * r8044038;
        double r8044040 = r8044037 - r8044039;
        double r8044041 = d4;
        double r8044042 = r8044041 * r8044035;
        double r8044043 = r8044040 + r8044042;
        double r8044044 = r8044035 * r8044035;
        double r8044045 = r8044043 - r8044044;
        return r8044045;
}
↓
double f(double d1, double d2, double d3, double d4) {
        double r8044046 = d1;
        double r8044047 = d4;
        double r8044048 = d2;
        double r8044049 = r8044047 + r8044048;
        double r8044050 = d3;
        double r8044051 = r8044050 + r8044046;
        double r8044052 = r8044049 - r8044051;
        double r8044053 = r8044046 * r8044052;
        return r8044053;
}

Error

Bits error versus d1

Bits error versus d2

Bits error versus d3

Bits error versus d4

Derivation

  1. Initial program 0.5

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

  2. Simplified0.4

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

  4. Applied sub-neg0.4

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

  5. Applied associate-+l+0.4

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

  6. Simplified0.4

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

  8. Applied associate-+r-0.4

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 0 
(FPCore (d1 d2 d3 d4)
  :name "FastMath dist4"
  (-.p16 (+.p16 (-.p16 (*.p16 d1 d2) (*.p16 d1 d3)) (*.p16 d4 d1)) (*.p16 d1 d1)))