Expression 1, p15

Average Error: 0.4 → 0.3

Time: 3.0s

Precision: 64

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

Math

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

↓

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

C

double f(double a, double b, double c, double d, double e) {
        double r87962 = e;
        double r87963 = d;
        double r87964 = r87962 + r87963;
        double r87965 = c;
        double r87966 = r87964 + r87965;
        double r87967 = b;
        double r87968 = r87966 + r87967;
        double r87969 = a;
        double r87970 = r87968 + r87969;
        return r87970;
}

↓

double f(double a, double b, double c, double d, double e) {
        double r87971 = d;
        double r87972 = e;
        double r87973 = c;
        double r87974 = r87972 + r87973;
        double r87975 = r87971 + r87974;
        double r87976 = b;
        double r87977 = a;
        double r87978 = r87976 + r87977;
        double r87979 = r87975 + r87978;
        return r87979;
}

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

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

5. Final simplification 0.3

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

Reproduce

herbie shell --seed 2020027 +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))