Expression 1, p15

Average Error: 0.4 → 0.3

Time: 5.8s

Precision: 64

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

Math

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

↓

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

C

double f(double a, double b, double c, double d, double e) {
        double r120565 = e;
        double r120566 = d;
        double r120567 = r120565 + r120566;
        double r120568 = c;
        double r120569 = r120567 + r120568;
        double r120570 = b;
        double r120571 = r120569 + r120570;
        double r120572 = a;
        double r120573 = r120571 + r120572;
        return r120573;
}

↓

double f(double a, double b, double c, double d, double e) {
        double r120574 = e;
        double r120575 = d;
        double r120576 = r120574 + r120575;
        double r120577 = c;
        double r120578 = b;
        double r120579 = r120577 + r120578;
        double r120580 = r120576 + r120579;
        double r120581 = a;
        double r120582 = r120580 + r120581;
        return r120582;
}

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

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

4. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019353 +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))