Average Error: 0.0 → 0.0
Time: 11.1s
Precision: 64
\[56789 \le a \le 98765 \land 0 \le b \le 1 \land 0 \le c \le 0.0016773 \land 0 \le d \le 0.0016773\]
\[a \cdot \left(\left(b + c\right) + d\right)\]
↓
\[\mathsf{fma}\left(a, \left(b + c\right), \left(d \cdot a\right)\right)\]
a \cdot \left(\left(b + c\right) + d\right)
↓
\mathsf{fma}\left(a, \left(b + c\right), \left(d \cdot a\right)\right)
double f(double a, double b, double c, double d) {
        double r2731325 = a;
        double r2731326 = b;
        double r2731327 = c;
        double r2731328 = r2731326 + r2731327;
        double r2731329 = d;
        double r2731330 = r2731328 + r2731329;
        double r2731331 = r2731325 * r2731330;
        return r2731331;
}
↓
double f(double a, double b, double c, double d) {
        double r2731332 = a;
        double r2731333 = b;
        double r2731334 = c;
        double r2731335 = r2731333 + r2731334;
        double r2731336 = d;
        double r2731337 = r2731336 * r2731332;
        double r2731338 = fma(r2731332, r2731335, r2731337);
        return r2731338;
}

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

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019129 +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)))