Average Error: 0.5 → 0.4
Time: 1.5m
Precision: 64
\[\left(\frac{\left(\left(d1 \cdot d2\right) - \left(d1 \cdot d3\right)\right)}{\left(d4 \cdot d1\right)}\right) - \left(d1 \cdot d1\right)\]
↓
\[d1 \cdot \left(\left(d4 + d2\right) - \left(d3 + d1\right)\right)\]
\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(d4 + d2\right) - \left(d3 + d1\right)\right)
double f(double d1, double d2, double d3, double d4) {
        double r1555719 = d1;
        double r1555720 = d2;
        double r1555721 = r1555719 * r1555720;
        double r1555722 = d3;
        double r1555723 = r1555719 * r1555722;
        double r1555724 = r1555721 - r1555723;
        double r1555725 = d4;
        double r1555726 = r1555725 * r1555719;
        double r1555727 = r1555724 + r1555726;
        double r1555728 = r1555719 * r1555719;
        double r1555729 = r1555727 - r1555728;
        return r1555729;
}
↓
double f(double d1, double d2, double d3, double d4) {
        double r1555730 = d1;
        double r1555731 = d4;
        double r1555732 = d2;
        double r1555733 = r1555731 + r1555732;
        double r1555734 = d3;
        double r1555735 = r1555734 + r1555730;
        double r1555736 = r1555733 - r1555735;
        double r1555737 = r1555730 * r1555736;
        return r1555737;
}

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 sub-neg0.4

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

  5. Applied associate-+l+0.4

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

  6. Simplified0.4

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

  8. Applied associate-+r-0.4

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 2019154 +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)))