Average Error: 0.0 → 0.0
Time: 33.5s
Precision: 64
\[56789 \le a \le 98765 \land 0.0 \le b \le 1 \land 0.0 \le c \le 0.001677300000000000058247850986958837893326 \land 0.0 \le d \le 0.001677300000000000058247850986958837893326\]
\[a \cdot \left(\left(b + c\right) + d\right)\]
↓
\[\mathsf{fma}\left(a, b + c, d \cdot a\right)\]
a \cdot \left(\left(b + c\right) + d\right)
↓
\mathsf{fma}\left(a, b + c, d \cdot a\right)
double f(double a, double b, double c, double d) {
        double r4823755 = a;
        double r4823756 = b;
        double r4823757 = c;
        double r4823758 = r4823756 + r4823757;
        double r4823759 = d;
        double r4823760 = r4823758 + r4823759;
        double r4823761 = r4823755 * r4823760;
        return r4823761;
}
↓
double f(double a, double b, double c, double d) {
        double r4823762 = a;
        double r4823763 = b;
        double r4823764 = c;
        double r4823765 = r4823763 + r4823764;
        double r4823766 = d;
        double r4823767 = r4823766 * r4823762;
        double r4823768 = fma(r4823762, r4823765, r4823767);
        return r4823768;
}

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 distribute-lft-in0.0

    \[\leadsto \color{blue}{a \cdot \left(b + c\right) + a \cdot d}\]
  4. Using strategy rm

  5. Applied fma-def0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (a b c d)
  :name "Expression, p14"
  :pre (and (<= 56789.0 a 98765.0) (<= 0.0 b 1.0) (<= 0.0 c 0.0016773) (<= 0.0 d 0.0016773))

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

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