Expression 1, p15

Average Error: 0.4 → 0.3

Time: 8.4s

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 r65187 = e;
        double r65188 = d;
        double r65189 = r65187 + r65188;
        double r65190 = c;
        double r65191 = r65189 + r65190;
        double r65192 = b;
        double r65193 = r65191 + r65192;
        double r65194 = a;
        double r65195 = r65193 + r65194;
        return r65195;
}

↓

double f(double a, double b, double c, double d, double e) {
        double r65196 = e;
        double r65197 = d;
        double r65198 = c;
        double r65199 = r65197 + r65198;
        double r65200 = r65196 + r65199;
        double r65201 = b;
        double r65202 = a;
        double r65203 = r65201 + r65202;
        double r65204 = r65200 + r65203;
        return r65204;
}

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 2019195 
(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))