Average Error: 0.5 → 0.4
Time: 34.5s
Precision: 64
\[\left(\frac{\left(\left(d1 \cdot d2\right) - \left(d1 \cdot d3\right)\right)}{\left(d4 \cdot d1\right)}\right) - \left(d1 \cdot d1\right)\]
↓
\[\left(\left(d4 + d2\right) - \left(d3 + d1\right)\right) \cdot d1\]
\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)
↓
\left(\left(d4 + d2\right) - \left(d3 + d1\right)\right) \cdot d1
double f(double d1, double d2, double d3, double d4) {
        double r3581875 = d1;
        double r3581876 = d2;
        double r3581877 = r3581875 * r3581876;
        double r3581878 = d3;
        double r3581879 = r3581875 * r3581878;
        double r3581880 = r3581877 - r3581879;
        double r3581881 = d4;
        double r3581882 = r3581881 * r3581875;
        double r3581883 = r3581880 + r3581882;
        double r3581884 = r3581875 * r3581875;
        double r3581885 = r3581883 - r3581884;
        return r3581885;
}
↓
double f(double d1, double d2, double d3, double d4) {
        double r3581886 = d4;
        double r3581887 = d2;
        double r3581888 = r3581886 + r3581887;
        double r3581889 = d3;
        double r3581890 = d1;
        double r3581891 = r3581889 + r3581890;
        double r3581892 = r3581888 - r3581891;
        double r3581893 = r3581892 * r3581890;
        return r3581893;
}

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. Using strategy rm

  10. Applied *-commutative0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2019162 +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)))