Average Error: 0.5 → 0.4
Time: 32.7s
Precision: 64
\[\left(\frac{\left(\left(d1 \cdot d2\right) - \left(d1 \cdot d3\right)\right)}{\left(d4 \cdot d1\right)}\right) - \left(d1 \cdot d1\right)\]
↓
\[d1 \cdot \left(\left(d4 + d2\right) - \left(d1 + d3\right)\right)\]
\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(\left(d4 + d2\right) - \left(d1 + d3\right)\right)
double f(double d1, double d2, double d3, double d4) {
        double r4704330 = d1;
        double r4704331 = d2;
        double r4704332 = r4704330 * r4704331;
        double r4704333 = d3;
        double r4704334 = r4704330 * r4704333;
        double r4704335 = r4704332 - r4704334;
        double r4704336 = d4;
        double r4704337 = r4704336 * r4704330;
        double r4704338 = r4704335 + r4704337;
        double r4704339 = r4704330 * r4704330;
        double r4704340 = r4704338 - r4704339;
        return r4704340;
}
↓
double f(double d1, double d2, double d3, double d4) {
        double r4704341 = d1;
        double r4704342 = d4;
        double r4704343 = d2;
        double r4704344 = r4704342 + r4704343;
        double r4704345 = d3;
        double r4704346 = r4704341 + r4704345;
        double r4704347 = r4704344 - r4704346;
        double r4704348 = r4704341 * r4704347;
        return r4704348;
}

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. Final simplification0.4

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

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)))