Average Error: 0.5 → 0.4
Time: 27.4s
Precision: 64
\[\left(\frac{\left(\left(d1 \cdot d2\right) - \left(d1 \cdot d3\right)\right)}{\left(d4 \cdot d1\right)}\right) - \left(d1 \cdot d1\right)\]
↓
\[d1 \cdot \left(\left(\left(d2 + d4\right) - d3\right) - d1\right)\]
\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(d2 + d4\right) - d3\right) - d1\right)
double f(double d1, double d2, double d3, double d4) {
        double r5565632 = d1;
        double r5565633 = d2;
        double r5565634 = r5565632 * r5565633;
        double r5565635 = d3;
        double r5565636 = r5565632 * r5565635;
        double r5565637 = r5565634 - r5565636;
        double r5565638 = d4;
        double r5565639 = r5565638 * r5565632;
        double r5565640 = r5565637 + r5565639;
        double r5565641 = r5565632 * r5565632;
        double r5565642 = r5565640 - r5565641;
        return r5565642;
}
↓
double f(double d1, double d2, double d3, double d4) {
        double r5565643 = d1;
        double r5565644 = d2;
        double r5565645 = d4;
        double r5565646 = r5565644 + r5565645;
        double r5565647 = d3;
        double r5565648 = r5565646 - r5565647;
        double r5565649 = r5565648 - r5565643;
        double r5565650 = r5565643 * r5565649;
        return r5565650;
}

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--r+0.4

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

  5. Final simplification0.4

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

Reproduce

herbie shell --seed 2019133 +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)))