Average Error: 0.0 → 0.0
Time: 15.1s
Precision: 64
\[56789 \le a \le 98765 \land 0.0 \le b \le 1 \land 0.0 \le c \le 0.001677300000000000058247850986958837893326 \land 0.0 \le d \le 0.001677300000000000058247850986958837893326\]
\[a \cdot \left(\left(b + c\right) + d\right)\]
↓
\[\mathsf{fma}\left(c + b, a, a \cdot d\right)\]
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 r5044909 = a;
        double r5044910 = b;
        double r5044911 = c;
        double r5044912 = r5044910 + r5044911;
        double r5044913 = d;
        double r5044914 = r5044912 + r5044913;
        double r5044915 = r5044909 * r5044914;
        return r5044915;
}
↓
double f(double a, double b, double c, double d) {
        double r5044916 = c;
        double r5044917 = b;
        double r5044918 = r5044916 + r5044917;
        double r5044919 = a;
        double r5044920 = d;
        double r5044921 = r5044919 * r5044920;
        double r5044922 = fma(r5044918, r5044919, r5044921);
        return r5044922;
}

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 2019170 +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)))