Expression, p14

Average Error: 0.0 → 0.0

Time: 13.2s

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

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

↓

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

C

double f(double a, double b, double c, double d) {
        double r73900 = a;
        double r73901 = b;
        double r73902 = c;
        double r73903 = r73901 + r73902;
        double r73904 = d;
        double r73905 = r73903 + r73904;
        double r73906 = r73900 * r73905;
        return r73906;
}

↓

double f(double a, double b, double c, double d) {
        double r73907 = b;
        double r73908 = c;
        double r73909 = r73907 + r73908;
        double r73910 = a;
        double r73911 = r73909 * r73910;
        double r73912 = d;
        double r73913 = r73912 * r73910;
        double r73914 = r73911 + r73913;
        return r73914;
}

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. Simplified 0.0

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

6. Final simplification 0.0

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

Reproduce

herbie shell --seed 2019325 +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)))