Average Error: 0.5 → 0.4
Time: 11.4s
Precision: 64
\[\left(\frac{\left(\left(d1 \cdot d2\right) - \left(d1 \cdot d3\right)\right)}{\left(d4 \cdot d1\right)}\right) - \left(d1 \cdot d1\right)\]
↓
\[d1 \cdot \left(d2 + \left(\left(-d3\right) + \left(d4 - d1\right)\right)\right)\]
\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(-d3\right) + \left(d4 - d1\right)\right)\right)
double f(double d1, double d2, double d3, double d4) {
        double r2295854 = d1;
        double r2295855 = d2;
        double r2295856 = r2295854 * r2295855;
        double r2295857 = d3;
        double r2295858 = r2295854 * r2295857;
        double r2295859 = r2295856 - r2295858;
        double r2295860 = d4;
        double r2295861 = r2295860 * r2295854;
        double r2295862 = r2295859 + r2295861;
        double r2295863 = r2295854 * r2295854;
        double r2295864 = r2295862 - r2295863;
        return r2295864;
}
↓
double f(double d1, double d2, double d3, double d4) {
        double r2295865 = d1;
        double r2295866 = d2;
        double r2295867 = d3;
        double r2295868 = -r2295867;
        double r2295869 = d4;
        double r2295870 = r2295869 - r2295865;
        double r2295871 = r2295868 + r2295870;
        double r2295872 = r2295866 + r2295871;
        double r2295873 = r2295865 * r2295872;
        return r2295873;
}

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

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

  4. Applied sub-neg0.4

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

  5. Applied associate-+l+0.4

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

  6. Final simplification0.4

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

Reproduce

herbie shell --seed 2019104 +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)))