Average Error: 0.5 → 0.4
Time: 12.5s
Precision: 64
\[\left(\frac{\left(\left(d1 \cdot d2\right) - \left(d1 \cdot d3\right)\right)}{\left(d4 \cdot d1\right)}\right) - \left(d1 \cdot d1\right)\]
↓
\[d1 \cdot \left(\left(d2 - d3\right) + \left(d4 - d1\right)\right)\]
\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(\left(d2 - d3\right) + \left(d4 - d1\right)\right)
double f(double d1, double d2, double d3, double d4) {
        double r5145806 = d1;
        double r5145807 = d2;
        double r5145808 = r5145806 * r5145807;
        double r5145809 = d3;
        double r5145810 = r5145806 * r5145809;
        double r5145811 = r5145808 - r5145810;
        double r5145812 = d4;
        double r5145813 = r5145812 * r5145806;
        double r5145814 = r5145811 + r5145813;
        double r5145815 = r5145806 * r5145806;
        double r5145816 = r5145814 - r5145815;
        return r5145816;
}
↓
double f(double d1, double d2, double d3, double d4) {
        double r5145817 = d1;
        double r5145818 = d2;
        double r5145819 = d3;
        double r5145820 = r5145818 - r5145819;
        double r5145821 = d4;
        double r5145822 = r5145821 - r5145817;
        double r5145823 = r5145820 + r5145822;
        double r5145824 = r5145817 * r5145823;
        return r5145824;
}

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. Final simplification0.4

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

Reproduce

herbie shell --seed 2019125 +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)))