Expression 1, p15

Average Error: 0.4 → 0.0

Time: 15.1s

Precision: 64

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

Math

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

↓

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

C

double f(double a, double b, double c, double d, double e) {
        double r4394942 = e;
        double r4394943 = d;
        double r4394944 = r4394942 + r4394943;
        double r4394945 = c;
        double r4394946 = r4394944 + r4394945;
        double r4394947 = b;
        double r4394948 = r4394946 + r4394947;
        double r4394949 = a;
        double r4394950 = r4394948 + r4394949;
        return r4394950;
}

↓

double f(double a, double b, double c, double d, double e) {
        double r4394951 = e;
        double r4394952 = exp(r4394951);
        double r4394953 = d;
        double r4394954 = exp(r4394953);
        double r4394955 = r4394952 * r4394954;
        double r4394956 = a;
        double r4394957 = exp(r4394956);
        double r4394958 = c;
        double r4394959 = exp(r4394958);
        double r4394960 = b;
        double r4394961 = exp(r4394960);
        double r4394962 = r4394959 * r4394961;
        double r4394963 = r4394957 * r4394962;
        double r4394964 = r4394955 * r4394963;
        double r4394965 = log(r4394964);
        return r4394965;
}

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

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

7. Applied add-log-exp 0.4

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

8. Applied sum-log 0.4

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

9. Applied sum-log 0.3

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

10. Applied sum-log 0.2

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

11. Applied sum-log 0.0

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

12. Simplified 0.3

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

14. Applied add-log-exp 0.3

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

15. Applied add-log-exp 0.3

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

16. Applied add-log-exp 0.3

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

17. Applied sum-log 0.3

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

18. Applied sum-log 0.3

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

19. Applied add-log-exp 0.3

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

20. Applied add-log-exp 0.3

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

21. Applied sum-log 0.3

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

22. Applied sum-log 0.0

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

23. Applied rem-exp-log 0.0

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

24. Final simplification 0.0

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

Reproduce

herbie shell --seed 2019162 +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))