Expression 1, p15

Average Error: 0.4 → 0.3

Time: 7.3s

Precision: 64

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

Math

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

↓

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

C

double f(double a, double b, double c, double d, double e) {
        double r61184 = e;
        double r61185 = d;
        double r61186 = r61184 + r61185;
        double r61187 = c;
        double r61188 = r61186 + r61187;
        double r61189 = b;
        double r61190 = r61188 + r61189;
        double r61191 = a;
        double r61192 = r61190 + r61191;
        return r61192;
}

↓

double f(double a, double b, double c, double d, double e) {
        double r61193 = e;
        double r61194 = d;
        double r61195 = r61193 + r61194;
        double r61196 = c;
        double r61197 = b;
        double r61198 = a;
        double r61199 = r61197 + r61198;
        double r61200 = r61196 + r61199;
        double r61201 = r61195 + r61200;
        return r61201;
}

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

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

6. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019304 +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))