FastMath repmul

Average Error: 0.1 → 0.2

Time: 3.4s

Precision: 64

Math

\[\left(\left(d1 \cdot d1\right) \cdot d1\right) \cdot d1\]

↓

\[\left(d1 \cdot d1\right) \cdot \left(d1 \cdot d1\right)\]

C

double f(double d1) {
        double r2355589 = d1;
        double r2355590 = r2355589 * r2355589;
        double r2355591 = r2355590 * r2355589;
        double r2355592 = r2355591 * r2355589;
        return r2355592;
}

↓

double f(double d1) {
        double r2355593 = d1;
        double r2355594 = r2355593 * r2355593;
        double r2355595 = r2355594 * r2355594;
        return r2355595;
}

Error

Bits error versus d1

Derivation

1. Initial program 0.1

   \[\left(\left(d1 \cdot d1\right) \cdot d1\right) \cdot d1\]

2. Simplified 0.2

   \[\leadsto \color{blue}{\left(d1 \cdot d1\right) \cdot \left(d1 \cdot d1\right)}\]

3. Final simplification 0.2

   \[\leadsto \left(d1 \cdot d1\right) \cdot \left(d1 \cdot d1\right)\]

Reproduce

herbie shell --seed 2019119 
(FPCore (d1)
  :name "FastMath repmul"
  (*.p16 (*.p16 (*.p16 d1 d1) d1) d1))