Average Error: 0.5 → 0.4
Time: 31.8s
Precision: 64
\[\left(\frac{\left(\left(d1 \cdot d2\right) - \left(d1 \cdot d3\right)\right)}{\left(d4 \cdot d1\right)}\right) - \left(d1 \cdot d1\right)\]
↓
\[d1 \cdot \left(\frac{\left(d4 - \left(\frac{d1}{d3}\right)\right)}{d2}\right)\]
\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(\frac{\left(d4 - \left(\frac{d1}{d3}\right)\right)}{d2}\right)
double f(double d1, double d2, double d3, double d4) {
        double r3633658 = d1;
        double r3633659 = d2;
        double r3633660 = r3633658 * r3633659;
        double r3633661 = d3;
        double r3633662 = r3633658 * r3633661;
        double r3633663 = r3633660 - r3633662;
        double r3633664 = d4;
        double r3633665 = r3633664 * r3633658;
        double r3633666 = r3633663 + r3633665;
        double r3633667 = r3633658 * r3633658;
        double r3633668 = r3633666 - r3633667;
        return r3633668;
}
↓
double f(double d1, double d2, double d3, double d4) {
        double r3633669 = d1;
        double r3633670 = d4;
        double r3633671 = d3;
        double r3633672 = r3633669 + r3633671;
        double r3633673 = r3633670 - r3633672;
        double r3633674 = d2;
        double r3633675 = r3633673 + r3633674;
        double r3633676 = r3633669 * r3633675;
        return r3633676;
}

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. Final simplification0.4

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

Reproduce

herbie shell --seed 2019168 +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)))