FastMath repmul

Average Error: 0.1 → 0.2

Time: 3.3s

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 r3915331 = d1;
        double r3915332 = r3915331 * r3915331;
        double r3915333 = r3915332 * r3915331;
        double r3915334 = r3915333 * r3915331;
        return r3915334;
}

↓

double f(double d1) {
        double r3915335 = d1;
        double r3915336 = r3915335 * r3915335;
        double r3915337 = r3915336 * r3915336;
        return r3915337;
}

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 2019128 +o rules:numerics
(FPCore (d1)
  :name "FastMath repmul"
  (*.p16 (*.p16 (*.p16 d1 d1) d1) d1))