Expression 1, p15

Average Error: 0.4 → 0.3

Time: 4.8s

Precision: 64

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

Math

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

↓

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

C

double f(double a, double b, double c, double d, double e) {
        double r116438 = e;
        double r116439 = d;
        double r116440 = r116438 + r116439;
        double r116441 = c;
        double r116442 = r116440 + r116441;
        double r116443 = b;
        double r116444 = r116442 + r116443;
        double r116445 = a;
        double r116446 = r116444 + r116445;
        return r116446;
}

↓

double f(double a, double b, double c, double d, double e) {
        double r116447 = e;
        double r116448 = d;
        double r116449 = r116447 + r116448;
        double r116450 = c;
        double r116451 = r116449 + r116450;
        double r116452 = b;
        double r116453 = a;
        double r116454 = r116452 + r116453;
        double r116455 = r116451 + r116454;
        return r116455;
}

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. Final simplification 0.3

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

Reproduce

herbie shell --seed 2020062 +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))