FastMath test3

Average Error: 0.1 → 0.0

Time: 8.1s

Precision: 64

Math

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

↓

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

C

double f(double d1, double d2, double d3) {
        double r14586912 = d1;
        double r14586913 = 3.0;
        double r14586914 = r14586912 * r14586913;
        double r14586915 = d2;
        double r14586916 = r14586912 * r14586915;
        double r14586917 = r14586914 + r14586916;
        double r14586918 = d3;
        double r14586919 = r14586912 * r14586918;
        double r14586920 = r14586917 + r14586919;
        return r14586920;
}

↓

double f(double d1, double d2, double d3) {
        double r14586921 = d1;
        double r14586922 = 3.0;
        double r14586923 = d2;
        double r14586924 = r14586921 * r14586923;
        double r14586925 = fma(r14586921, r14586922, r14586924);
        double r14586926 = d3;
        double r14586927 = r14586921 * r14586926;
        double r14586928 = r14586925 + r14586927;
        return r14586928;
}

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

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

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (d1 d2 d3)
  :name "FastMath test3"

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

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