Average Error: 0.5 → 0.4
Time: 10.0s
Precision: 64
\[\left(\frac{\left(\left(d1 \cdot d2\right) - \left(d1 \cdot d3\right)\right)}{\left(d4 \cdot d1\right)}\right) - \left(d1 \cdot d1\right)\]
↓
\[d1 \cdot \left(d2 + \left(\left(\left(-d3\right) + d4\right) - d1\right)\right)\]
\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(\left(\left(-d3\right) + d4\right) - d1\right)\right)
double f(double d1, double d2, double d3, double d4) {
        double r2070433 = d1;
        double r2070434 = d2;
        double r2070435 = r2070433 * r2070434;
        double r2070436 = d3;
        double r2070437 = r2070433 * r2070436;
        double r2070438 = r2070435 - r2070437;
        double r2070439 = d4;
        double r2070440 = r2070439 * r2070433;
        double r2070441 = r2070438 + r2070440;
        double r2070442 = r2070433 * r2070433;
        double r2070443 = r2070441 - r2070442;
        return r2070443;
}
↓
double f(double d1, double d2, double d3, double d4) {
        double r2070444 = d1;
        double r2070445 = d2;
        double r2070446 = d3;
        double r2070447 = -r2070446;
        double r2070448 = d4;
        double r2070449 = r2070447 + r2070448;
        double r2070450 = r2070449 - r2070444;
        double r2070451 = r2070445 + r2070450;
        double r2070452 = r2070444 * r2070451;
        return r2070452;
}

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 sub-neg0.4

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

  5. Applied associate-+l+0.4

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

  7. Applied associate-+r-0.4

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

  8. Final simplification0.4

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

Reproduce

herbie shell --seed 2019107 +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)))