Average Error: 0.0 → 0.0
Time: 2.7s
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 r228606 = d1;
        double r228607 = d2;
        double r228608 = r228606 * r228607;
        double r228609 = d3;
        double r228610 = 5.0;
        double r228611 = r228609 + r228610;
        double r228612 = r228611 * r228606;
        double r228613 = r228608 + r228612;
        double r228614 = 32.0;
        double r228615 = r228606 * r228614;
        double r228616 = r228613 + r228615;
        return r228616;
}
↓
double f(double d1, double d2, double d3) {
        double r228617 = 37.0;
        double r228618 = d1;
        double r228619 = d3;
        double r228620 = d2;
        double r228621 = r228618 * r228620;
        double r228622 = fma(r228618, r228619, r228621);
        double r228623 = fma(r228617, r228618, r228622);
        return r228623;
}

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 2020060 +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)))