Expression, p6

Average Error: 3.7 → 0.0

Time: 9.4s

Precision: 64

\[-14 \le a \le -13 \land -3 \le b \le -2 \land 3 \le c \le 3.5 \land 12.5 \le d \le 13.5\]

Math

\[\left(a + \left(b + \left(c + d\right)\right)\right) \cdot 2\]

↓

\[2 \cdot \left(c + \left(b + \left(a + d\right)\right)\right)\]

C

double f(double a, double b, double c, double d) {
        double r6627912 = a;
        double r6627913 = b;
        double r6627914 = c;
        double r6627915 = d;
        double r6627916 = r6627914 + r6627915;
        double r6627917 = r6627913 + r6627916;
        double r6627918 = r6627912 + r6627917;
        double r6627919 = 2.0;
        double r6627920 = r6627918 * r6627919;
        return r6627920;
}

↓

double f(double a, double b, double c, double d) {
        double r6627921 = 2.0;
        double r6627922 = c;
        double r6627923 = b;
        double r6627924 = a;
        double r6627925 = d;
        double r6627926 = r6627924 + r6627925;
        double r6627927 = r6627923 + r6627926;
        double r6627928 = r6627922 + r6627927;
        double r6627929 = r6627921 * r6627928;
        return r6627929;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	3.7
Target	3.8
Herbie	0.0

\[\left(a + b\right) \cdot 2 + \left(c + d\right) \cdot 2\]

Derivation

1. Initial program 3.7

   \[\left(a + \left(b + \left(c + d\right)\right)\right) \cdot 2\]
2. Using strategy rm

3. Applied associate-+r+ 2.8

   \[\leadsto \left(a + \color{blue}{\left(\left(b + c\right) + d\right)}\right) \cdot 2\]
4. Using strategy rm

5. Applied add-log-exp 2.8

   \[\leadsto \left(a + \left(\left(b + c\right) + \color{blue}{\log \left(e^{d}\right)}\right)\right) \cdot 2\]

6. Applied add-log-exp 2.8

   \[\leadsto \left(a + \left(\left(b + \color{blue}{\log \left(e^{c}\right)}\right) + \log \left(e^{d}\right)\right)\right) \cdot 2\]

7. Applied add-log-exp 2.8

   \[\leadsto \left(a + \left(\left(\color{blue}{\log \left(e^{b}\right)} + \log \left(e^{c}\right)\right) + \log \left(e^{d}\right)\right)\right) \cdot 2\]

8. Applied sum-log 2.8

   \[\leadsto \left(a + \left(\color{blue}{\log \left(e^{b} \cdot e^{c}\right)} + \log \left(e^{d}\right)\right)\right) \cdot 2\]

9. Applied sum-log 2.8

   \[\leadsto \left(a + \color{blue}{\log \left(\left(e^{b} \cdot e^{c}\right) \cdot e^{d}\right)}\right) \cdot 2\]

10. Applied add-log-exp 2.8

    \[\leadsto \left(\color{blue}{\log \left(e^{a}\right)} + \log \left(\left(e^{b} \cdot e^{c}\right) \cdot e^{d}\right)\right) \cdot 2\]

11. Applied sum-log 1.5

    \[\leadsto \color{blue}{\log \left(e^{a} \cdot \left(\left(e^{b} \cdot e^{c}\right) \cdot e^{d}\right)\right)} \cdot 2\]

12. Simplified 3.0

    \[\leadsto \log \color{blue}{\left(e^{\left(a + \left(d + b\right)\right) + c}\right)} \cdot 2\]
13. Using strategy rm

14. Applied associate-+r+ 0.9

    \[\leadsto \log \left(e^{\color{blue}{\left(\left(a + d\right) + b\right)} + c}\right) \cdot 2\]
15. Using strategy rm

16. Applied exp-sum 1.2

    \[\leadsto \log \color{blue}{\left(e^{\left(a + d\right) + b} \cdot e^{c}\right)} \cdot 2\]

17. Applied log-prod 0.0

    \[\leadsto \color{blue}{\left(\log \left(e^{\left(a + d\right) + b}\right) + \log \left(e^{c}\right)\right)} \cdot 2\]

18. Simplified 0.0

    \[\leadsto \left(\color{blue}{\left(\left(d + a\right) + b\right)} + \log \left(e^{c}\right)\right) \cdot 2\]

19. Simplified 0.0

    \[\leadsto \left(\left(\left(d + a\right) + b\right) + \color{blue}{c}\right) \cdot 2\]

20. Final simplification 0.0

    \[\leadsto 2 \cdot \left(c + \left(b + \left(a + d\right)\right)\right)\]

Reproduce

herbie shell --seed 2019172 
(FPCore (a b c d)
  :name "Expression, p6"
  :pre (and (<= -14.0 a -13.0) (<= -3.0 b -2.0) (<= 3.0 c 3.5) (<= 12.5 d 13.5))

  :herbie-target
  (+ (* (+ a b) 2.0) (* (+ c d) 2.0))

  (* (+ a (+ b (+ c d))) 2.0))