Average Error: 0.5 → 0.4
Time: 22.1s
Precision: 64
\[\left(\frac{\left(\left(d1 \cdot d2\right) - \left(d1 \cdot d3\right)\right)}{\left(d4 \cdot d1\right)}\right) - \left(d1 \cdot d1\right)\]
↓
\[d1 \cdot \left(d2 + \left(d4 - \left(d3 + d1\right)\right)\right)\]
\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(d4 - \left(d3 + d1\right)\right)\right)
double f(double d1, double d2, double d3, double d4) {
        double r7546874 = d1;
        double r7546875 = d2;
        double r7546876 = r7546874 * r7546875;
        double r7546877 = d3;
        double r7546878 = r7546874 * r7546877;
        double r7546879 = r7546876 - r7546878;
        double r7546880 = d4;
        double r7546881 = r7546880 * r7546874;
        double r7546882 = r7546879 + r7546881;
        double r7546883 = r7546874 * r7546874;
        double r7546884 = r7546882 - r7546883;
        return r7546884;
}
↓
double f(double d1, double d2, double d3, double d4) {
        double r7546885 = d1;
        double r7546886 = d2;
        double r7546887 = d4;
        double r7546888 = d3;
        double r7546889 = r7546888 + r7546885;
        double r7546890 = r7546887 - r7546889;
        double r7546891 = r7546886 + r7546890;
        double r7546892 = r7546885 * r7546891;
        return r7546892;
}

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. Applied distribute-lft-in0.4

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

  7. Applied p16-distribute-lft-out0.4

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

  8. Final simplification0.4

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

Reproduce

herbie shell --seed 2019135 +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)))