Expression, p6

Average Error: 3.7 → 2.8

Time: 2.5s

Precision: 64

\[-14 \le a \le -13 \land -3 \le b \le -2 \land 3 \le c \le 3.5 \land 12.5 \le d \le 13.5\]

Math

\[\left(a + \left(b + \left(c + d\right)\right)\right) \cdot 2\]

↓

\[\left(\left(a + \left(b + c\right)\right) + d\right) \cdot 2\]

C

double f(double a, double b, double c, double d) {
        double r81963 = a;
        double r81964 = b;
        double r81965 = c;
        double r81966 = d;
        double r81967 = r81965 + r81966;
        double r81968 = r81964 + r81967;
        double r81969 = r81963 + r81968;
        double r81970 = 2.0;
        double r81971 = r81969 * r81970;
        return r81971;
}

↓

double f(double a, double b, double c, double d) {
        double r81972 = a;
        double r81973 = b;
        double r81974 = c;
        double r81975 = r81973 + r81974;
        double r81976 = r81972 + r81975;
        double r81977 = d;
        double r81978 = r81976 + r81977;
        double r81979 = 2.0;
        double r81980 = r81978 * r81979;
        return r81980;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	3.7
Target	3.9
Herbie	2.8

\[\left(a + b\right) \cdot 2 + \left(c + d\right) \cdot 2\]

Derivation

1. Initial program 3.7

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

3. Applied associate-+r+ 2.8

   \[\leadsto \left(a + \color{blue}{\left(\left(b + c\right) + d\right)}\right) \cdot 2\]
4. Using strategy rm

5. Applied associate-+r+ 2.8

   \[\leadsto \color{blue}{\left(\left(a + \left(b + c\right)\right) + d\right)} \cdot 2\]

6. Final simplification 2.8

   \[\leadsto \left(\left(a + \left(b + c\right)\right) + d\right) \cdot 2\]

Reproduce

herbie shell --seed 2020062 +o rules:numerics
(FPCore (a b c d)
  :name "Expression, p6"
  :precision binary64
  :pre (and (<= -14 a -13) (<= -3 b -2) (<= 3 c 3.5) (<= 12.5 d 13.5))

  :herbie-target
  (+ (* (+ a b) 2) (* (+ c d) 2))

  (* (+ a (+ b (+ c d))) 2))