Expression 1, p15

Average Error: 0.4 → 0.3

Time: 3.6s

Precision: 64

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

Math

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

↓

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

C

double f(double a, double b, double c, double d, double e) {
        double r98586 = e;
        double r98587 = d;
        double r98588 = r98586 + r98587;
        double r98589 = c;
        double r98590 = r98588 + r98589;
        double r98591 = b;
        double r98592 = r98590 + r98591;
        double r98593 = a;
        double r98594 = r98592 + r98593;
        return r98594;
}

↓

double f(double a, double b, double c, double d, double e) {
        double r98595 = d;
        double r98596 = e;
        double r98597 = c;
        double r98598 = r98596 + r98597;
        double r98599 = r98595 + r98598;
        double r98600 = b;
        double r98601 = a;
        double r98602 = r98600 + r98601;
        double r98603 = r98599 + r98602;
        return r98603;
}

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. Taylor expanded around 0 0.4

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

4. Applied associate-+l+ 0.3

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

5. Final simplification 0.3

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

Reproduce

herbie shell --seed 2020035 +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))