Expression, p6

Average Error: 3.7 → 2.8

Time: 2.9s

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 r83705 = a;
        double r83706 = b;
        double r83707 = c;
        double r83708 = d;
        double r83709 = r83707 + r83708;
        double r83710 = r83706 + r83709;
        double r83711 = r83705 + r83710;
        double r83712 = 2.0;
        double r83713 = r83711 * r83712;
        return r83713;
}

↓

double f(double a, double b, double c, double d) {
        double r83714 = a;
        double r83715 = b;
        double r83716 = c;
        double r83717 = r83715 + r83716;
        double r83718 = r83714 + r83717;
        double r83719 = d;
        double r83720 = r83718 + r83719;
        double r83721 = 2.0;
        double r83722 = r83720 * r83721;
        return r83722;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	3.7
Target	3.8
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 2020083 +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))