Expression 1, p15

Average Error: 0.4 → 0.0

Time: 16.2s

Precision: 64

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

Math

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

↓

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

C

double f(double a, double b, double c, double d, double e) {
        double r5660713 = e;
        double r5660714 = d;
        double r5660715 = r5660713 + r5660714;
        double r5660716 = c;
        double r5660717 = r5660715 + r5660716;
        double r5660718 = b;
        double r5660719 = r5660717 + r5660718;
        double r5660720 = a;
        double r5660721 = r5660719 + r5660720;
        return r5660721;
}

↓

double f(double a, double b, double c, double d, double e) {
        double r5660722 = a;
        double r5660723 = exp(r5660722);
        double r5660724 = d;
        double r5660725 = exp(r5660724);
        double r5660726 = e;
        double r5660727 = exp(r5660726);
        double r5660728 = b;
        double r5660729 = exp(r5660728);
        double r5660730 = c;
        double r5660731 = exp(r5660730);
        double r5660732 = r5660729 * r5660731;
        double r5660733 = r5660727 * r5660732;
        double r5660734 = r5660725 * r5660733;
        double r5660735 = r5660723 * r5660734;
        double r5660736 = log(r5660735);
        return r5660736;
}

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	0.0

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

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. Using strategy rm

5. Applied add-log-exp 0.3

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

6. Applied add-log-exp 0.3

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

7. Applied add-log-exp 0.3

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

8. Applied sum-log 0.3

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

9. Applied add-log-exp 0.3

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

10. Applied add-log-exp 0.3

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

11. Applied sum-log 0.3

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

12. Applied sum-log 0.2

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

13. Applied sum-log 0.0

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

14. Simplified 0.3

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

16. Applied add-log-exp 0.3

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

17. Applied add-log-exp 0.3

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

18. Applied add-log-exp 0.3

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

19. Applied add-log-exp 0.3

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

20. Applied add-log-exp 0.3

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

21. Applied sum-log 0.3

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

22. Applied sum-log 0.3

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

23. Applied sum-log 0.2

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

24. Applied sum-log 0.0

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

25. Applied rem-exp-log 0.0

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

26. Final simplification 0.0

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

Reproduce

herbie shell --seed 2019168 
(FPCore (a b c d e)
  :name "Expression 1, p15"
  :pre (<= 1 a 2 b 4 c 8 d 16 e 32)

  :herbie-target
  (+ (+ d (+ c (+ a b))) e)

  (+ (+ (+ (+ e d) c) b) a))