Average Error: 0.0 → 0.0
Time: 12.7s
Precision: 64
\[56789 \le a \le 98765 \land 0 \le b \le 1 \land 0 \le c \le 0.0016773 \land 0 \le d \le 0.0016773\]
\[a \cdot \left(\left(b + c\right) + d\right)\]
↓
\[\mathsf{fma}\left(a, b + c, d \cdot a\right)\]
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 r3491400 = a;
        double r3491401 = b;
        double r3491402 = c;
        double r3491403 = r3491401 + r3491402;
        double r3491404 = d;
        double r3491405 = r3491403 + r3491404;
        double r3491406 = r3491400 * r3491405;
        return r3491406;
}
↓
double f(double a, double b, double c, double d) {
        double r3491407 = a;
        double r3491408 = b;
        double r3491409 = c;
        double r3491410 = r3491408 + r3491409;
        double r3491411 = d;
        double r3491412 = r3491411 * r3491407;
        double r3491413 = fma(r3491407, r3491410, r3491412);
        return r3491413;
}

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 2019164 +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)))