Expression 1, p15

Average Error: 0.4 → 0.4

Time: 6.1s

Precision: 64

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

Math

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

↓

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

C

double f(double a, double b, double c, double d, double e) {
        double r122741 = e;
        double r122742 = d;
        double r122743 = r122741 + r122742;
        double r122744 = c;
        double r122745 = r122743 + r122744;
        double r122746 = b;
        double r122747 = r122745 + r122746;
        double r122748 = a;
        double r122749 = r122747 + r122748;
        return r122749;
}

↓

double f(double a, double b, double c, double d, double e) {
        double r122750 = e;
        double r122751 = d;
        double r122752 = r122750 + r122751;
        double r122753 = c;
        double r122754 = b;
        double r122755 = r122753 + r122754;
        double r122756 = r122752 + r122755;
        double r122757 = a;
        double r122758 = r122756 + r122757;
        return r122758;
}

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	0.4

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

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+ 0.4

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

4. Final simplification 0.4

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

Reproduce

herbie shell --seed 2020056 +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))