Average Error: 0.0 → 0.0
Time: 7.7s
Precision: 64
\[56789 \le a \le 98765 \land 0 \le b \le 1 \land 0 \le c \le 0.0016773 \land 0 \le d \le 0.0016773\]
\[a \cdot \left(\left(b + c\right) + d\right)\]
↓
\[\mathsf{fma}\left(c + b, a, a \cdot d\right)\]
a \cdot \left(\left(b + c\right) + d\right)
↓
\mathsf{fma}\left(c + b, a, a \cdot d\right)
double f(double a, double b, double c, double d) {
        double r3075990 = a;
        double r3075991 = b;
        double r3075992 = c;
        double r3075993 = r3075991 + r3075992;
        double r3075994 = d;
        double r3075995 = r3075993 + r3075994;
        double r3075996 = r3075990 * r3075995;
        return r3075996;
}
↓
double f(double a, double b, double c, double d) {
        double r3075997 = c;
        double r3075998 = b;
        double r3075999 = r3075997 + r3075998;
        double r3076000 = a;
        double r3076001 = d;
        double r3076002 = r3076000 * r3076001;
        double r3076003 = fma(r3075999, r3076000, r3076002);
        return r3076003;
}

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-rgt-in0.0

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

  5. Applied fma-def0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019142 +o rules:numerics
(FPCore (a b c d)
  :name "Expression, p14"
  :pre (and (<= 56789 a 98765) (<= 0 b 1) (<= 0 c 0.0016773) (<= 0 d 0.0016773))

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

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