Expression, p6

Average Error: 3.7 → 2.8

Time: 3.1s

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

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

↓

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

C

double f(double a, double b, double c, double d) {
        double r70672 = a;
        double r70673 = b;
        double r70674 = c;
        double r70675 = d;
        double r70676 = r70674 + r70675;
        double r70677 = r70673 + r70676;
        double r70678 = r70672 + r70677;
        double r70679 = 2.0;
        double r70680 = r70678 * r70679;
        return r70680;
}

↓

double f(double a, double b, double c, double d) {
        double r70681 = 2.0;
        double r70682 = a;
        double r70683 = b;
        double r70684 = c;
        double r70685 = r70683 + r70684;
        double r70686 = d;
        double r70687 = r70685 + r70686;
        double r70688 = r70682 + r70687;
        double r70689 = r70681 * r70688;
        return r70689;
}

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 pow1 2.8

   \[\leadsto \color{blue}{{\left(a + \left(\left(b + c\right) + d\right)\right)}^{1}} \cdot 2\]

6. Final simplification 2.8

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

Reproduce

herbie shell --seed 2020018 +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))