Average Error: 0.5 → 0.3
Time: 18.7s
Precision: 64
\[\left(\frac{\left(\left(d1 \cdot d2\right) - \left(d1 \cdot d3\right)\right)}{\left(d4 \cdot d1\right)}\right) - \left(d1 \cdot d1\right)\]
↓
\[d1 \cdot \left(\mathsf{qms}\left(\left(\mathsf{qms}\left(\left(\left(d4 + d2\right)\right), d1, 1.0\right)\right), d3, 1.0\right)\right)\]
\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(\mathsf{qms}\left(\left(\mathsf{qms}\left(\left(\left(d4 + d2\right)\right), d1, 1.0\right)\right), d3, 1.0\right)\right)
double f(double d1, double d2, double d3, double d4) {
        double r1892622 = d1;
        double r1892623 = d2;
        double r1892624 = r1892622 * r1892623;
        double r1892625 = d3;
        double r1892626 = r1892622 * r1892625;
        double r1892627 = r1892624 - r1892626;
        double r1892628 = d4;
        double r1892629 = r1892628 * r1892622;
        double r1892630 = r1892627 + r1892629;
        double r1892631 = r1892622 * r1892622;
        double r1892632 = r1892630 - r1892631;
        return r1892632;
}
↓
double f(double d1, double d2, double d3, double d4) {
        double r1892633 = d1;
        double r1892634 = d4;
        double r1892635 = d2;
        double r1892636 = r1892634 + r1892635;
        double r1892637 = /*Error: no posit support in C */;
        double r1892638 = 1.0;
        double r1892639 = /*Error: no posit support in C */;
        double r1892640 = d3;
        double r1892641 = /*Error: no posit support in C */;
        double r1892642 = /*Error: no posit support in C */;
        double r1892643 = r1892633 * r1892642;
        return r1892643;
}

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

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

  8. Applied associate-+r-0.4

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

  10. Applied associate--r+0.4

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

  12. Applied introduce-quire0.4

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

  13. Applied insert-quire-sub0.4

    \[\leadsto d1 \cdot \left(\color{blue}{\left(\left(\mathsf{qms}\left(\left(\left(\frac{d4}{d2}\right)\right), d1, \left(1.0\right)\right)\right)\right)} - d3\right)\]

  14. Applied insert-quire-sub0.3

    \[\leadsto d1 \cdot \color{blue}{\left(\left(\mathsf{qms}\left(\left(\mathsf{qms}\left(\left(\left(\frac{d4}{d2}\right)\right), d1, \left(1.0\right)\right)\right), d3, \left(1.0\right)\right)\right)\right)}\]

  15. Final simplification0.3

    \[\leadsto d1 \cdot \left(\mathsf{qms}\left(\left(\mathsf{qms}\left(\left(\left(d4 + d2\right)\right), d1, 1.0\right)\right), d3, 1.0\right)\right)\]

Reproduce

herbie shell --seed 2019153 +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)))