Expression 1, p15

Average Error: 0.4 → 0.3

Time: 3.3s

Precision: 64

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

Math

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

↓

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

C

double f(double a, double b, double c, double d, double e) {
        double r83132 = e;
        double r83133 = d;
        double r83134 = r83132 + r83133;
        double r83135 = c;
        double r83136 = r83134 + r83135;
        double r83137 = b;
        double r83138 = r83136 + r83137;
        double r83139 = a;
        double r83140 = r83138 + r83139;
        return r83140;
}

↓

double f(double a, double b, double c, double d, double e) {
        double r83141 = d;
        double r83142 = e;
        double r83143 = c;
        double r83144 = r83142 + r83143;
        double r83145 = r83141 + r83144;
        double r83146 = b;
        double r83147 = a;
        double r83148 = r83146 + r83147;
        double r83149 = r83145 + r83148;
        return r83149;
}

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. Taylor expanded around 0 0.3

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

5. Final simplification 0.3

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

Reproduce

herbie shell --seed 2020003 +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))