Expression 1, p15

Average Error: 0.4 → 0.3

Time: 8.2s

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 r90507 = e;
        double r90508 = d;
        double r90509 = r90507 + r90508;
        double r90510 = c;
        double r90511 = r90509 + r90510;
        double r90512 = b;
        double r90513 = r90511 + r90512;
        double r90514 = a;
        double r90515 = r90513 + r90514;
        return r90515;
}

↓

double f(double a, double b, double c, double d, double e) {
        double r90516 = e;
        double r90517 = d;
        double r90518 = c;
        double r90519 = r90517 + r90518;
        double r90520 = r90516 + r90519;
        double r90521 = b;
        double r90522 = a;
        double r90523 = r90521 + r90522;
        double r90524 = r90520 + r90523;
        return r90524;
}

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))