Average Error: 0.5 → 0.4
Time: 24.8s
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 r963608 = d1;
        double r963609 = d2;
        double r963610 = r963608 * r963609;
        double r963611 = d3;
        double r963612 = r963608 * r963611;
        double r963613 = r963610 - r963612;
        double r963614 = d4;
        double r963615 = r963614 * r963608;
        double r963616 = r963613 + r963615;
        double r963617 = r963608 * r963608;
        double r963618 = r963616 - r963617;
        return r963618;
}
↓
double f(double d1, double d2, double d3, double d4) {
        double r963619 = d1;
        double r963620 = d4;
        double r963621 = d2;
        double r963622 = r963620 + r963621;
        double r963623 = d3;
        double r963624 = r963619 + r963623;
        double r963625 = r963622 - r963624;
        double r963626 = r963619 * r963625;
        return r963626;
}

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 2019151 +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)))