Average Error: 0.1 → 0.0
Time: 27.1s
Precision: 64
\[\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3\]
↓
\[\mathsf{fma}\left(d1, 3 + d2, d3 \cdot d1\right)\]
\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3
↓
\mathsf{fma}\left(d1, 3 + d2, d3 \cdot d1\right)
double f(double d1, double d2, double d3) {
        double r8581632 = d1;
        double r8581633 = 3.0;
        double r8581634 = r8581632 * r8581633;
        double r8581635 = d2;
        double r8581636 = r8581632 * r8581635;
        double r8581637 = r8581634 + r8581636;
        double r8581638 = d3;
        double r8581639 = r8581632 * r8581638;
        double r8581640 = r8581637 + r8581639;
        return r8581640;
}
↓
double f(double d1, double d2, double d3) {
        double r8581641 = d1;
        double r8581642 = 3.0;
        double r8581643 = d2;
        double r8581644 = r8581642 + r8581643;
        double r8581645 = d3;
        double r8581646 = r8581645 * r8581641;
        double r8581647 = fma(r8581641, r8581644, r8581646);
        return r8581647;
}

Error

Bits error versus d1

Bits error versus d2

Bits error versus d3

Target

Original0.1
Target0.1
Herbie0.0
\[d1 \cdot \left(\left(3 + d2\right) + d3\right)\]

Derivation

  1. Initial program 0.1

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

  3. Applied distribute-lft-out0.1

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

  4. Applied fma-def0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (d1 d2 d3)
  :name "FastMath test3"

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

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