Expression 1, p15

Average Error: 0.4 → 0.2

Time: 6.6s

Precision: binary64

• Falling back on racket egraph because egg-herbie package not installed

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

Math

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

↓

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

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) (d + ((double) (((double) (b + c)) + 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+ 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. Using strategy rm

7. Applied associate-+l+ 0.2

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

9. Applied associate-+r+ 0.2

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

10. Simplified 0.2

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

11. Final simplification 0.2

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

Reproduce

herbie shell --seed 2020182 
(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))