Expression, p14

Average Error: 0.0 → 0.0

Time: 6.9s

Precision: 64

\[56789 \le a \le 98765 \land 0.0 \le b \le 1 \land 0.0 \le c \le 0.001677300000000000058247850986958837893326 \land 0.0 \le d \le 0.001677300000000000058247850986958837893326\]

Math

\[a \cdot \left(\left(b + c\right) + d\right)\]

↓

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

C

double f(double a, double b, double c, double d) {
        double r49882 = a;
        double r49883 = b;
        double r49884 = c;
        double r49885 = r49883 + r49884;
        double r49886 = d;
        double r49887 = r49885 + r49886;
        double r49888 = r49882 * r49887;
        return r49888;
}

↓

double f(double a, double b, double c, double d) {
        double r49889 = b;
        double r49890 = c;
        double r49891 = r49889 + r49890;
        double r49892 = a;
        double r49893 = d;
        double r49894 = r49892 * r49893;
        double r49895 = fma(r49891, r49892, r49894);
        return r49895;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	0.0
Target	0.0
Herbie	0.0

\[a \cdot b + a \cdot \left(c + d\right)\]

Derivation

1. Initial program 0.0

   \[a \cdot \left(\left(b + c\right) + d\right)\]
2. Using strategy rm

3. Applied distribute-lft-in 0.0

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

4. Simplified 0.0

   \[\leadsto \color{blue}{\left(b + c\right) \cdot a} + a \cdot d\]
5. Using strategy rm

6. Applied fma-def 0.0

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

7. Final simplification 0.0

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

Reproduce

herbie shell --seed 2019209 +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 0.0016773000000000001) (<= 0.0 d 0.0016773000000000001))

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

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