Expression, p14

Average Error: 0.0 → 0.0

Time: 7.7s

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

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

↓

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

C

double f(double a, double b, double c, double d) {
        double r88202 = a;
        double r88203 = b;
        double r88204 = c;
        double r88205 = r88203 + r88204;
        double r88206 = d;
        double r88207 = r88205 + r88206;
        double r88208 = r88202 * r88207;
        return r88208;
}

↓

double f(double a, double b, double c, double d) {
        double r88209 = a;
        double r88210 = b;
        double r88211 = c;
        double r88212 = r88210 + r88211;
        double r88213 = d;
        double r88214 = r88209 * r88213;
        double r88215 = fma(r88209, r88212, r88214);
        return r88215;
}

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. Using strategy rm

5. Applied fma-def 0.0

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

6. Final simplification 0.0

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

Reproduce

herbie shell --seed 2019305 +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)))