Expression 1, p15

Average Error: 0.4 → 0.3

Time: 8.4s

Precision: 64

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

Math

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

↓

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

C

double f(double a, double b, double c, double d, double e) {
        double r84183 = e;
        double r84184 = d;
        double r84185 = r84183 + r84184;
        double r84186 = c;
        double r84187 = r84185 + r84186;
        double r84188 = b;
        double r84189 = r84187 + r84188;
        double r84190 = a;
        double r84191 = r84189 + r84190;
        return r84191;
}

↓

double f(double a, double b, double c, double d, double e) {
        double r84192 = e;
        double r84193 = d;
        double r84194 = c;
        double r84195 = r84193 + r84194;
        double r84196 = r84192 + r84195;
        double r84197 = b;
        double r84198 = a;
        double r84199 = r84197 + r84198;
        double r84200 = r84196 + r84199;
        return r84200;
}

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

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

6. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019325 +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))