Expression 1, p15

Average Error: 0.4 → 0.3

Time: 8.6s

Precision: 64

\[1 \le a \le 2 \le b \le 4 \le c \le 8 \le d \le 16 \le e \le 32\]

Math

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

↓

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

C

double f(double a, double b, double c, double d, double e) {
        double r47305 = e;
        double r47306 = d;
        double r47307 = r47305 + r47306;
        double r47308 = c;
        double r47309 = r47307 + r47308;
        double r47310 = b;
        double r47311 = r47309 + r47310;
        double r47312 = a;
        double r47313 = r47311 + r47312;
        return r47313;
}

↓

double f(double a, double b, double c, double d, double e) {
        double r47314 = e;
        double r47315 = d;
        double r47316 = r47314 + r47315;
        double r47317 = c;
        double r47318 = b;
        double r47319 = a;
        double r47320 = r47318 + r47319;
        double r47321 = r47317 + r47320;
        double r47322 = r47316 + r47321;
        return r47322;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Bits error versus e

Target

Original	0.4
Target	0.2
Herbie	0.3

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

Derivation

1. Initial program 0.4

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

3. Applied associate-+l+ 0.3

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

5. Applied associate-+l+ 0.3

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

6. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019303 +o rules:numerics
(FPCore (a b c d e)
  :name "Expression 1, p15"
  :precision binary64
  :pre (<= 1 a 2 b 4 c 8 d 16 e 32)

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

  (+ (+ (+ (+ e d) c) b) a))