FastMath test3

Average Error: 0.1 → 0.0

Time: 12.3s

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 r8107287 = d1;
        double r8107288 = 3.0;
        double r8107289 = r8107287 * r8107288;
        double r8107290 = d2;
        double r8107291 = r8107287 * r8107290;
        double r8107292 = r8107289 + r8107291;
        double r8107293 = d3;
        double r8107294 = r8107287 * r8107293;
        double r8107295 = r8107292 + r8107294;
        return r8107295;
}

↓

double f(double d1, double d2, double d3) {
        double r8107296 = d3;
        double r8107297 = d1;
        double r8107298 = r8107296 * r8107297;
        double r8107299 = 3.0;
        double r8107300 = d2;
        double r8107301 = r8107297 * r8107300;
        double r8107302 = fma(r8107297, r8107299, r8107301);
        double r8107303 = r8107298 + r8107302;
        return r8107303;
}

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 2019164 +o rules:numerics
(FPCore (d1 d2 d3)
  :name "FastMath test3"

  :herbie-target
  (* d1 (+ (+ 3 d2) d3))

  (+ (+ (* d1 3) (* d1 d2)) (* d1 d3)))