FastMath repmul

Average Error: 0.1 → 0.2

Time: 6.6s

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 r1789318 = d1;
        double r1789319 = r1789318 * r1789318;
        double r1789320 = r1789319 * r1789318;
        double r1789321 = r1789320 * r1789318;
        return r1789321;
}

↓

double f(double d1) {
        double r1789322 = d1;
        double r1789323 = r1789322 * r1789322;
        double r1789324 = r1789323 * r1789323;
        return r1789324;
}

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 2019151 +o rules:numerics
(FPCore (d1)
  :name "FastMath repmul"
  (*.p16 (*.p16 (*.p16 d1 d1) d1) d1))