Expression 1, p15

Average Error: 0.4 → 0.3

Time: 14.6s

Precision: 64

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

Math

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

↓

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

C

double f(double a, double b, double c, double d, double e) {
        double r5480974 = e;
        double r5480975 = d;
        double r5480976 = r5480974 + r5480975;
        double r5480977 = c;
        double r5480978 = r5480976 + r5480977;
        double r5480979 = b;
        double r5480980 = r5480978 + r5480979;
        double r5480981 = a;
        double r5480982 = r5480980 + r5480981;
        return r5480982;
}

↓

double f(double a, double b, double c, double d, double e) {
        double r5480983 = b;
        double r5480984 = a;
        double r5480985 = r5480983 + r5480984;
        double r5480986 = c;
        double r5480987 = d;
        double r5480988 = r5480986 + r5480987;
        double r5480989 = e;
        double r5480990 = r5480988 + r5480989;
        double r5480991 = r5480985 + r5480990;
        return r5480991;
}

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

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

5. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019172 
(FPCore (a b c d e)
  :name "Expression 1, p15"
  :pre (<= 1.0 a 2.0 b 4.0 c 8.0 d 16.0 e 32.0)

  :herbie-target
  (+ (+ d (+ c (+ a b))) e)

  (+ (+ (+ (+ e d) c) b) a))