Average Error: 0.5 → 0.4
Time: 20.2s
Precision: 64
\[\left(\frac{\left(\left(d1 \cdot d2\right) - \left(d1 \cdot d3\right)\right)}{\left(d4 \cdot d1\right)}\right) - \left(d1 \cdot d1\right)\]
↓
\[d1 \cdot \left(d2 + \left(\left(d4 - d3\right) - d1\right)\right)\]
\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 r5430756 = d1;
        double r5430757 = d2;
        double r5430758 = r5430756 * r5430757;
        double r5430759 = d3;
        double r5430760 = r5430756 * r5430759;
        double r5430761 = r5430758 - r5430760;
        double r5430762 = d4;
        double r5430763 = r5430762 * r5430756;
        double r5430764 = r5430761 + r5430763;
        double r5430765 = r5430756 * r5430756;
        double r5430766 = r5430764 - r5430765;
        return r5430766;
}
↓
double f(double d1, double d2, double d3, double d4) {
        double r5430767 = d1;
        double r5430768 = d2;
        double r5430769 = d4;
        double r5430770 = d3;
        double r5430771 = r5430769 - r5430770;
        double r5430772 = r5430771 - r5430767;
        double r5430773 = r5430768 + r5430772;
        double r5430774 = r5430767 * r5430773;
        return r5430774;
}

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 2019130 +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)))