FastMath test3

Average Error: 0.1 → 0.0

Time: 11.6s

Precision: 64

Math

\[\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3\]

↓

\[d3 \cdot d1 + \mathsf{fma}\left(d1, 3, d1 \cdot d2\right)\]

C

double f(double d1, double d2, double d3) {
        double r7900690 = d1;
        double r7900691 = 3.0;
        double r7900692 = r7900690 * r7900691;
        double r7900693 = d2;
        double r7900694 = r7900690 * r7900693;
        double r7900695 = r7900692 + r7900694;
        double r7900696 = d3;
        double r7900697 = r7900690 * r7900696;
        double r7900698 = r7900695 + r7900697;
        return r7900698;
}

↓

double f(double d1, double d2, double d3) {
        double r7900699 = d3;
        double r7900700 = d1;
        double r7900701 = r7900699 * r7900700;
        double r7900702 = 3.0;
        double r7900703 = d2;
        double r7900704 = r7900700 * r7900703;
        double r7900705 = fma(r7900700, r7900702, r7900704);
        double r7900706 = r7900701 + r7900705;
        return r7900706;
}

Error

Bits error versus d1

Bits error versus d2

Bits error versus d3

Target

Original	0.1
Target	0.1
Herbie	0.0

\[d1 \cdot \left(\left(3 + d2\right) + d3\right)\]

Derivation

1. Initial program 0.1

   \[\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3\]
2. Using strategy rm

3. Applied fma-def 0.0

   \[\leadsto \color{blue}{\mathsf{fma}\left(d1, 3, d1 \cdot d2\right)} + d1 \cdot d3\]

4. Final simplification 0.0

   \[\leadsto d3 \cdot d1 + \mathsf{fma}\left(d1, 3, d1 \cdot d2\right)\]

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(FPCore (d1 d2 d3)
  :name "FastMath test3"

  :herbie-target
  (* d1 (+ (+ 3.0 d2) d3))

  (+ (+ (* d1 3.0) (* d1 d2)) (* d1 d3)))