Average Error: 0.5 → 0.4
Time: 21.2s
Precision: 64
\[\left(\frac{\left(\left(d1 \cdot d2\right) - \left(d1 \cdot d3\right)\right)}{\left(d4 \cdot d1\right)}\right) - \left(d1 \cdot d1\right)\]
↓
\[d1 \cdot \left(\left(\left(d4 - d1\right) - d3\right) + d2\right)\]
\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(\left(d4 - d1\right) - d3\right) + d2\right)
double f(double d1, double d2, double d3, double d4) {
        double r1627943 = d1;
        double r1627944 = d2;
        double r1627945 = r1627943 * r1627944;
        double r1627946 = d3;
        double r1627947 = r1627943 * r1627946;
        double r1627948 = r1627945 - r1627947;
        double r1627949 = d4;
        double r1627950 = r1627949 * r1627943;
        double r1627951 = r1627948 + r1627950;
        double r1627952 = r1627943 * r1627943;
        double r1627953 = r1627951 - r1627952;
        return r1627953;
}
↓
double f(double d1, double d2, double d3, double d4) {
        double r1627954 = d1;
        double r1627955 = d4;
        double r1627956 = r1627955 - r1627954;
        double r1627957 = d3;
        double r1627958 = r1627956 - r1627957;
        double r1627959 = d2;
        double r1627960 = r1627958 + r1627959;
        double r1627961 = r1627954 * r1627960;
        return r1627961;
}

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

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

  4. Applied associate--r+0.4

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

  5. Final simplification0.4

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

Reproduce

herbie shell --seed 2019152 +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)))