Expression 1, p15

Average Error: 0.4 → 0.2

Time: 5.8s

Precision: binary64

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

Math

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

↓

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

FPCore

(FPCore (a b c d e) :precision binary64 (+ (+ (+ (+ e d) c) b) a))

↓

(FPCore (a b c d e) :precision binary64 (+ e (+ (+ c d) (+ b a))))

C

double code(double a, double b, double c, double d, double e) {
	return (((e + d) + c) + b) + a;
}

↓

double code(double a, double b, double c, double d, double e) {
	return e + ((c + d) + (b + a));
}

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	0.2

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

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+_binary64_626 0.3

   \[\leadsto \color{blue}{\left(\left(e + d\right) + c\right) + \left(b + a\right)}\]
4. Using strategy rm

5. Applied associate-+l+_binary64_626 0.3

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

6. Simplified 0.3

   \[\leadsto \left(e + \color{blue}{\left(c + d\right)}\right) + \left(b + a\right)\]
7. Using strategy rm

8. Applied associate-+l+_binary64_626 0.2

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

9. Final simplification 0.2

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

Reproduce

herbie shell --seed 2020231 
(FPCore (a b c d e)
  :name "Expression 1, p15"
  :precision binary64
  :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))