Expression 1, p15

Average Error: 0.4 → 0.0

Time: 36.3s

Precision: 64

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

Math

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

↓

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

C

double f(double a, double b, double c, double d, double e) {
        double r21211953 = e;
        double r21211954 = d;
        double r21211955 = r21211953 + r21211954;
        double r21211956 = c;
        double r21211957 = r21211955 + r21211956;
        double r21211958 = b;
        double r21211959 = r21211957 + r21211958;
        double r21211960 = a;
        double r21211961 = r21211959 + r21211960;
        return r21211961;
}

↓

double f(double a, double b, double c, double d, double e) {
        double r21211962 = d;
        double r21211963 = exp(r21211962);
        double r21211964 = e;
        double r21211965 = exp(r21211964);
        double r21211966 = a;
        double r21211967 = exp(r21211966);
        double r21211968 = c;
        double r21211969 = exp(r21211968);
        double r21211970 = r21211967 * r21211969;
        double r21211971 = b;
        double r21211972 = exp(r21211971);
        double r21211973 = r21211970 * r21211972;
        double r21211974 = r21211965 * r21211973;
        double r21211975 = r21211963 * r21211974;
        double r21211976 = log(r21211975);
        return r21211976;
}

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 add-log-exp 0.4

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

4. Applied add-log-exp 0.4

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

5. Applied add-log-exp 0.4

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

6. Applied add-log-exp 0.4

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

7. Applied sum-log 0.4

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

8. Applied sum-log 0.3

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

9. Applied sum-log 0.2

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

10. Simplified 0.3

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

12. Applied add-log-exp 0.3

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

13. Applied add-log-exp 0.3

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

14. Applied add-log-exp 0.3

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

15. Applied sum-log 0.3

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

16. Applied sum-log 0.3

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

17. Applied add-log-exp 0.3

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

18. Applied add-log-exp 0.3

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

19. Applied sum-log 0.3

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

20. Applied sum-log 0.0

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

21. Applied rem-exp-log 0.0

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

22. Simplified 0.2

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

24. Applied exp-sum 0.0

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

25. Applied associate-*l* 0.0

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

26. Final simplification 0.0

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

Reproduce

herbie shell --seed 2019120 +o rules:numerics
(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))