Expression 1, p15

Average Error: 0.4 → 0.3

Time: 3.6s

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 r119131 = e;
        double r119132 = d;
        double r119133 = r119131 + r119132;
        double r119134 = c;
        double r119135 = r119133 + r119134;
        double r119136 = b;
        double r119137 = r119135 + r119136;
        double r119138 = a;
        double r119139 = r119137 + r119138;
        return r119139;
}

↓

double f(double a, double b, double c, double d, double e) {
        double r119140 = e;
        double r119141 = d;
        double r119142 = c;
        double r119143 = r119141 + r119142;
        double r119144 = r119140 + r119143;
        double r119145 = b;
        double r119146 = a;
        double r119147 = r119145 + r119146;
        double r119148 = r119144 + r119147;
        return r119148;
}

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 2019318 
(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))