Average Error: 0.5 → 0.3
Time: 33.6s
Precision: 64
\[\left(\frac{\left(\left(d1 \cdot d2\right) - \left(d1 \cdot d3\right)\right)}{\left(d4 \cdot d1\right)}\right) - \left(d1 \cdot d1\right)\]
↓
\[d1 \cdot \left(\mathsf{qms}\left(\left(\mathsf{qms}\left(\left(\left(d2 + d4\right)\right), d1, 1.0\right)\right), d3, 1.0\right)\right)\]
\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(d2 + d4\right)\right), d1, 1.0\right)\right), d3, 1.0\right)\right)
double f(double d1, double d2, double d3, double d4) {
        double r4820363 = d1;
        double r4820364 = d2;
        double r4820365 = r4820363 * r4820364;
        double r4820366 = d3;
        double r4820367 = r4820363 * r4820366;
        double r4820368 = r4820365 - r4820367;
        double r4820369 = d4;
        double r4820370 = r4820369 * r4820363;
        double r4820371 = r4820368 + r4820370;
        double r4820372 = r4820363 * r4820363;
        double r4820373 = r4820371 - r4820372;
        return r4820373;
}
↓
double f(double d1, double d2, double d3, double d4) {
        double r4820374 = d1;
        double r4820375 = d2;
        double r4820376 = d4;
        double r4820377 = r4820375 + r4820376;
        double r4820378 = /*Error: no posit support in C */;
        double r4820379 = 1.0;
        double r4820380 = /*Error: no posit support in C */;
        double r4820381 = d3;
        double r4820382 = /*Error: no posit support in C */;
        double r4820383 = /*Error: no posit support in C */;
        double r4820384 = r4820374 * r4820383;
        return r4820384;
}

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

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

  4. Applied associate--r+0.4

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

  6. Applied introduce-quire0.4

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

  7. Applied insert-quire-sub0.4

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

  8. Applied insert-quire-sub0.3

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

  9. Final simplification0.3

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

Reproduce

herbie shell --seed 2019163 +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)))