Expression 1, p15

Average Error: 0.4 → 0.3

Time: 8.5s

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 r77821 = e;
        double r77822 = d;
        double r77823 = r77821 + r77822;
        double r77824 = c;
        double r77825 = r77823 + r77824;
        double r77826 = b;
        double r77827 = r77825 + r77826;
        double r77828 = a;
        double r77829 = r77827 + r77828;
        return r77829;
}

↓

double f(double a, double b, double c, double d, double e) {
        double r77830 = e;
        double r77831 = d;
        double r77832 = c;
        double r77833 = r77831 + r77832;
        double r77834 = r77830 + r77833;
        double r77835 = b;
        double r77836 = a;
        double r77837 = r77835 + r77836;
        double r77838 = r77834 + r77837;
        return r77838;
}

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 2019294 
(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))