Average Error: 0.5 → 0.4
Time: 10.9s
Precision: 64
\[\left(\frac{\left(\left(d1 \cdot d2\right) - \left(d1 \cdot d3\right)\right)}{\left(d4 \cdot d1\right)}\right) - \left(d1 \cdot d1\right)\]
↓
\[d4 \cdot d1 + \left(d2 - \left(d3 + d1\right)\right) \cdot d1\]
\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)
↓
d4 \cdot d1 + \left(d2 - \left(d3 + d1\right)\right) \cdot d1
double f(double d1, double d2, double d3, double d4) {
        double r1238694 = d1;
        double r1238695 = d2;
        double r1238696 = r1238694 * r1238695;
        double r1238697 = d3;
        double r1238698 = r1238694 * r1238697;
        double r1238699 = r1238696 - r1238698;
        double r1238700 = d4;
        double r1238701 = r1238700 * r1238694;
        double r1238702 = r1238699 + r1238701;
        double r1238703 = r1238694 * r1238694;
        double r1238704 = r1238702 - r1238703;
        return r1238704;
}
↓
double f(double d1, double d2, double d3, double d4) {
        double r1238705 = d4;
        double r1238706 = d1;
        double r1238707 = r1238705 * r1238706;
        double r1238708 = d2;
        double r1238709 = d3;
        double r1238710 = r1238709 + r1238706;
        double r1238711 = r1238708 - r1238710;
        double r1238712 = r1238711 * r1238706;
        double r1238713 = r1238707 + r1238712;
        return r1238713;
}

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 associate-+l-0.4

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

  6. Applied sub-neg0.4

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

  7. Applied distribute-lft-in0.4

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

  8. Simplified0.4

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

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

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

  11. Simplified0.4

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

  13. Applied associate--l+0.4

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

  14. Applied distribute-rgt-in0.4

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

  15. Final simplification0.4

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

Reproduce

herbie shell --seed 2019120 
(FPCore (d1 d2 d3 d4)
  :name "FastMath dist4"
  (-.p16 (+.p16 (-.p16 (*.p16 d1 d2) (*.p16 d1 d3)) (*.p16 d4 d1)) (*.p16 d1 d1)))