Expression, p14

Average Error: 0.0 → 0.0

Time: 15.2s

Precision: 64

\[56789 \le a \le 98765 \land 0 \le b \le 1 \land 0 \le c \le 0.0016773 \land 0 \le d \le 0.0016773\]

Math

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

↓

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

C

double f(double a, double b, double c, double d) {
        double r3655185 = a;
        double r3655186 = b;
        double r3655187 = c;
        double r3655188 = r3655186 + r3655187;
        double r3655189 = d;
        double r3655190 = r3655188 + r3655189;
        double r3655191 = r3655185 * r3655190;
        return r3655191;
}

↓

double f(double a, double b, double c, double d) {
        double r3655192 = c;
        double r3655193 = b;
        double r3655194 = r3655192 + r3655193;
        double r3655195 = a;
        double r3655196 = d;
        double r3655197 = r3655195 * r3655196;
        double r3655198 = fma(r3655194, r3655195, r3655197);
        return r3655198;
}

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-rgt-in 0.0

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

5. Applied fma-def 0.0

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

6. Final simplification 0.0

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

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (a b c d)
  :name "Expression, p14"
  :pre (and (<= 56789 a 98765) (<= 0 b 1) (<= 0 c 0.0016773) (<= 0 d 0.0016773))

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

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