Average Error: 0.0 → 0.0
Time: 3.3s
Precision: 64
\[\left(d1 \cdot d2 + \left(d3 + 5\right) \cdot d1\right) + d1 \cdot 32\]
↓
\[\mathsf{fma}\left(37, d1, \mathsf{fma}\left(d1, d3, d1 \cdot d2\right)\right)\]
\left(d1 \cdot d2 + \left(d3 + 5\right) \cdot d1\right) + d1 \cdot 32
↓
\mathsf{fma}\left(37, d1, \mathsf{fma}\left(d1, d3, d1 \cdot d2\right)\right)
double f(double d1, double d2, double d3) {
        double r311011 = d1;
        double r311012 = d2;
        double r311013 = r311011 * r311012;
        double r311014 = d3;
        double r311015 = 5.0;
        double r311016 = r311014 + r311015;
        double r311017 = r311016 * r311011;
        double r311018 = r311013 + r311017;
        double r311019 = 32.0;
        double r311020 = r311011 * r311019;
        double r311021 = r311018 + r311020;
        return r311021;
}
↓
double f(double d1, double d2, double d3) {
        double r311022 = 37.0;
        double r311023 = d1;
        double r311024 = d3;
        double r311025 = d2;
        double r311026 = r311023 * r311025;
        double r311027 = fma(r311023, r311024, r311026);
        double r311028 = fma(r311022, r311023, r311027);
        return r311028;
}

Error

Bits error versus d1

Bits error versus d2

Bits error versus d3

Target

Original0.0
Target0.0
Herbie0.0
\[d1 \cdot \left(\left(37 + d3\right) + d2\right)\]

Derivation

  1. Initial program 0.0

    \[\left(d1 \cdot d2 + \left(d3 + 5\right) \cdot d1\right) + d1 \cdot 32\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(32, d1, \mathsf{fma}\left(d1, d2, \left(d3 + 5\right) \cdot d1\right)\right)}\]

  3. Taylor expanded around 0 0.0

    \[\leadsto \color{blue}{37 \cdot d1 + \left(d1 \cdot d3 + d1 \cdot d2\right)}\]

  4. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(37, d1, \mathsf{fma}\left(d1, d3, d1 \cdot d2\right)\right)}\]

  5. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(37, d1, \mathsf{fma}\left(d1, d3, d1 \cdot d2\right)\right)\]

Reproduce

herbie shell --seed 2020064 +o rules:numerics
(FPCore (d1 d2 d3)
  :name "FastMath dist3"
  :precision binary64

  :herbie-target
  (* d1 (+ (+ 37 d3) d2))

  (+ (+ (* d1 d2) (* (+ d3 5) d1)) (* d1 32)))