Average Error: 0.5 → 0.4
Time: 9.4s
Precision: 64
\[\left(\frac{\left(\left(d1 \cdot d2\right) - \left(d1 \cdot d3\right)\right)}{\left(d4 \cdot d1\right)}\right) - \left(d1 \cdot d1\right)\]
↓
\[d1 \cdot \left(d2 - \left(d3 - \left(d4 - d1\right)\right)\right)\]
\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(d3 - \left(d4 - d1\right)\right)\right)
double f(double d1, double d2, double d3, double d4) {
        double r1515471 = d1;
        double r1515472 = d2;
        double r1515473 = r1515471 * r1515472;
        double r1515474 = d3;
        double r1515475 = r1515471 * r1515474;
        double r1515476 = r1515473 - r1515475;
        double r1515477 = d4;
        double r1515478 = r1515477 * r1515471;
        double r1515479 = r1515476 + r1515478;
        double r1515480 = r1515471 * r1515471;
        double r1515481 = r1515479 - r1515480;
        return r1515481;
}
↓
double f(double d1, double d2, double d3, double d4) {
        double r1515482 = d1;
        double r1515483 = d2;
        double r1515484 = d3;
        double r1515485 = d4;
        double r1515486 = r1515485 - r1515482;
        double r1515487 = r1515484 - r1515486;
        double r1515488 = r1515483 - r1515487;
        double r1515489 = r1515482 * r1515488;
        return r1515489;
}

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 associate-+l-0.4

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

  5. Final simplification0.4

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

Reproduce

herbie shell --seed 2019104 
(FPCore (d1 d2 d3 d4)
  :name "FastMath dist4"
  (-.p16 (+.p16 (-.p16 (*.p16 d1 d2) (*.p16 d1 d3)) (*.p16 d4 d1)) (*.p16 d1 d1)))