Average Error: 0.5 → 0.4
Time: 10.6s
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 r2225907 = d1;
        double r2225908 = d2;
        double r2225909 = r2225907 * r2225908;
        double r2225910 = d3;
        double r2225911 = r2225907 * r2225910;
        double r2225912 = r2225909 - r2225911;
        double r2225913 = d4;
        double r2225914 = r2225913 * r2225907;
        double r2225915 = r2225912 + r2225914;
        double r2225916 = r2225907 * r2225907;
        double r2225917 = r2225915 - r2225916;
        return r2225917;
}
↓
double f(double d1, double d2, double d3, double d4) {
        double r2225918 = d1;
        double r2225919 = d2;
        double r2225920 = d3;
        double r2225921 = -r2225920;
        double r2225922 = d4;
        double r2225923 = r2225922 - r2225918;
        double r2225924 = r2225921 + r2225923;
        double r2225925 = r2225919 + r2225924;
        double r2225926 = r2225918 * r2225925;
        return r2225926;
}

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 2019119 +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)))