Expression 1, p15

Average Error: 0.4 → 0.3

Time: 8.9s

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 r87860 = e;
        double r87861 = d;
        double r87862 = r87860 + r87861;
        double r87863 = c;
        double r87864 = r87862 + r87863;
        double r87865 = b;
        double r87866 = r87864 + r87865;
        double r87867 = a;
        double r87868 = r87866 + r87867;
        return r87868;
}

↓

double f(double a, double b, double c, double d, double e) {
        double r87869 = e;
        double r87870 = d;
        double r87871 = c;
        double r87872 = r87870 + r87871;
        double r87873 = r87869 + r87872;
        double r87874 = b;
        double r87875 = a;
        double r87876 = r87874 + r87875;
        double r87877 = r87873 + r87876;
        return r87877;
}

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))