Expression 1, p15

Average Error: 0.4 → 0.3

Time: 5.7s

Precision: 64

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

Math

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

↓

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

C

double f(double a, double b, double c, double d, double e) {
        double r133884 = e;
        double r133885 = d;
        double r133886 = r133884 + r133885;
        double r133887 = c;
        double r133888 = r133886 + r133887;
        double r133889 = b;
        double r133890 = r133888 + r133889;
        double r133891 = a;
        double r133892 = r133890 + r133891;
        return r133892;
}

↓

double f(double a, double b, double c, double d, double e) {
        double r133893 = e;
        double r133894 = d;
        double r133895 = r133893 + r133894;
        double r133896 = c;
        double r133897 = b;
        double r133898 = r133896 + r133897;
        double r133899 = r133895 + r133898;
        double r133900 = a;
        double r133901 = r133899 + r133900;
        return r133901;
}

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

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

4. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019353 +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))