Average Error: 0.0 → 0.0
Time: 15.7s
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 r3334248 = a;
        double r3334249 = b;
        double r3334250 = c;
        double r3334251 = r3334249 + r3334250;
        double r3334252 = d;
        double r3334253 = r3334251 + r3334252;
        double r3334254 = r3334248 * r3334253;
        return r3334254;
}
↓
double f(double a, double b, double c, double d) {
        double r3334255 = c;
        double r3334256 = b;
        double r3334257 = r3334255 + r3334256;
        double r3334258 = a;
        double r3334259 = d;
        double r3334260 = r3334258 * r3334259;
        double r3334261 = fma(r3334257, r3334258, r3334260);
        return r3334261;
}

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 2019163 +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)))