Average Error: 0.5 → 0.4
Time: 32.5s
Precision: 64
\[\left(\frac{\left(\left(d1 \cdot d2\right) - \left(d1 \cdot d3\right)\right)}{\left(d4 \cdot d1\right)}\right) - \left(d1 \cdot d1\right)\]
↓
\[d1 \cdot \left(\left(\frac{d4}{d2}\right) - \left(\frac{d1}{d3}\right)\right)\]
\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(\frac{d4}{d2}\right) - \left(\frac{d1}{d3}\right)\right)
double f(double d1, double d2, double d3, double d4) {
        double r3764853 = d1;
        double r3764854 = d2;
        double r3764855 = r3764853 * r3764854;
        double r3764856 = d3;
        double r3764857 = r3764853 * r3764856;
        double r3764858 = r3764855 - r3764857;
        double r3764859 = d4;
        double r3764860 = r3764859 * r3764853;
        double r3764861 = r3764858 + r3764860;
        double r3764862 = r3764853 * r3764853;
        double r3764863 = r3764861 - r3764862;
        return r3764863;
}
↓
double f(double d1, double d2, double d3, double d4) {
        double r3764864 = d1;
        double r3764865 = d4;
        double r3764866 = d2;
        double r3764867 = r3764865 + r3764866;
        double r3764868 = d3;
        double r3764869 = r3764864 + r3764868;
        double r3764870 = r3764867 - r3764869;
        double r3764871 = r3764864 * r3764870;
        return r3764871;
}

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

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

  8. Applied associate-+r-0.4

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 2019163 +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)))