Average Error: 0.0 → 0.0
Time: 3.0s
Precision: 64
\[56789 \le a \le 98765 \land 0.0 \le b \le 1 \land 0.0 \le c \le 0.0016773000000000001 \land 0.0 \le d \le 0.0016773000000000001\]
\[a \cdot \left(\left(b + c\right) + d\right)\]
↓
\[\mathsf{fma}\left(d, a, \mathsf{fma}\left(a, b, a \cdot c\right)\right)\]
a \cdot \left(\left(b + c\right) + d\right)
↓
\mathsf{fma}\left(d, a, \mathsf{fma}\left(a, b, a \cdot c\right)\right)
double f(double a, double b, double c, double d) {
        double r75850 = a;
        double r75851 = b;
        double r75852 = c;
        double r75853 = r75851 + r75852;
        double r75854 = d;
        double r75855 = r75853 + r75854;
        double r75856 = r75850 * r75855;
        return r75856;
}
↓
double f(double a, double b, double c, double d) {
        double r75857 = d;
        double r75858 = a;
        double r75859 = b;
        double r75860 = c;
        double r75861 = r75858 * r75860;
        double r75862 = fma(r75858, r75859, r75861);
        double r75863 = fma(r75857, r75858, r75862);
        return r75863;
}

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. Taylor expanded around inf 0.0

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

  3. Simplified0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2020083 +o rules:numerics
(FPCore (a b c d)
  :name "Expression, p14"
  :precision binary64
  :pre (and (<= 56789 a 98765) (<= 0.0 b 1) (<= 0.0 c 0.0016773) (<= 0.0 d 0.0016773))

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

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