FastMath repmul

Average Error: 0.2 → 0.2

Time: 2.0m

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 r3786454 = d1;
        double r3786455 = r3786454 * r3786454;
        double r3786456 = r3786455 * r3786454;
        double r3786457 = r3786456 * r3786454;
        return r3786457;
}

↓

double f(double d1) {
        double r3786458 = d1;
        double r3786459 = r3786458 * r3786458;
        double r3786460 = r3786459 * r3786459;
        return r3786460;
}

Error

Bits error versus d1

Derivation

1. Initial program 0.2

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

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 2019168 +o rules:numerics
(FPCore (d1)
  :name "FastMath repmul"
  (*.p16 (*.p16 (*.p16 d1 d1) d1) d1))