Expression 1, p15

Average Error: 0.4 → 0.3

Time: 2.7s

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

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

↓

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

FPCore

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

↓

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

C

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

↓

double code(double a, double b, double c, double d, double e) {
	return ((double) (e + ((double) (((double) (d + ((double) (c + 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	0.3

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

Derivation

1. Initial program 0.4

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

2. Simplified 0.2

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

4. Applied associate-+r+ 0.2

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

6. Applied associate-+r+ 0.3

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

7. Simplified 0.3

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

8. Final simplification 0.3

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

Reproduce

herbie shell --seed 2020198 
(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))