Average Error: 0.0 → 0.0
Time: 2.2s
Precision: 64
\[56789 \le a \le 98765 \land 0.0 \le b \le 1 \land 0.0 \le c \le 0.0016773000000000001 \land 0.0 \le d \le 0.0016773000000000001\]
\[a \cdot \left(\left(b + c\right) + d\right)\]
↓
\[\mathsf{fma}\left(d, a, \mathsf{fma}\left(a, b, a \cdot c\right)\right)\]
a \cdot \left(\left(b + c\right) + d\right)
↓
\mathsf{fma}\left(d, a, \mathsf{fma}\left(a, b, a \cdot c\right)\right)
double f(double a, double b, double c, double d) {
        double r89678 = a;
        double r89679 = b;
        double r89680 = c;
        double r89681 = r89679 + r89680;
        double r89682 = d;
        double r89683 = r89681 + r89682;
        double r89684 = r89678 * r89683;
        return r89684;
}
↓
double f(double a, double b, double c, double d) {
        double r89685 = d;
        double r89686 = a;
        double r89687 = b;
        double r89688 = c;
        double r89689 = r89686 * r89688;
        double r89690 = fma(r89686, r89687, r89689);
        double r89691 = fma(r89685, r89686, r89690);
        return r89691;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original0.0
Target0.0
Herbie0.0
\[a \cdot b + a \cdot \left(c + d\right)\]

Derivation

  1. Initial program 0.0

    \[a \cdot \left(\left(b + c\right) + d\right)\]
  2. Using strategy rm

  3. Applied *-un-lft-identity0.0

    \[\leadsto \color{blue}{\left(1 \cdot a\right)} \cdot \left(\left(b + c\right) + d\right)\]

  4. Applied associate-*l*0.0

    \[\leadsto \color{blue}{1 \cdot \left(a \cdot \left(\left(b + c\right) + d\right)\right)}\]

  5. Simplified0.0

    \[\leadsto 1 \cdot \color{blue}{\mathsf{fma}\left(d, a, \mathsf{fma}\left(a, b, a \cdot c\right)\right)}\]

  6. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(d, a, \mathsf{fma}\left(a, b, a \cdot c\right)\right)\]

Reproduce

herbie shell --seed 2020081 +o rules:numerics
(FPCore (a b c d)
  :name "Expression, p14"
  :precision binary64
  :pre (and (<= 56789 a 98765) (<= 0.0 b 1) (<= 0.0 c 0.0016773) (<= 0.0 d 0.0016773))

  :herbie-target
  (+ (* a b) (* a (+ c d)))

  (* a (+ (+ b c) d)))