Average Error: 0.0 → 0.0
Time: 2.4s
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 r97801 = a;
        double r97802 = b;
        double r97803 = c;
        double r97804 = r97802 + r97803;
        double r97805 = d;
        double r97806 = r97804 + r97805;
        double r97807 = r97801 * r97806;
        return r97807;
}
↓
double f(double a, double b, double c, double d) {
        double r97808 = d;
        double r97809 = a;
        double r97810 = b;
        double r97811 = c;
        double r97812 = r97809 * r97811;
        double r97813 = fma(r97809, r97810, r97812);
        double r97814 = fma(r97808, r97809, r97813);
        return r97814;
}

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 *-un-lft-identity0.0

    \[\leadsto \color{blue}{\left(1 \cdot a\right)} \cdot \left(\left(b + c\right) + d\right)\]

  4. Applied associate-*l*0.0

    \[\leadsto \color{blue}{1 \cdot \left(a \cdot \left(\left(b + c\right) + d\right)\right)}\]

  5. Simplified0.0

    \[\leadsto 1 \cdot \color{blue}{\mathsf{fma}\left(d, a, \mathsf{fma}\left(a, b, a \cdot c\right)\right)}\]

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2020018 +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)))