Expression 1, p15

Average Error: 0.4 → 0.0

Time: 39.5s

Precision: 64

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

Math

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

↓

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

C

double f(double a, double b, double c, double d, double e) {
        double r35998039 = e;
        double r35998040 = d;
        double r35998041 = r35998039 + r35998040;
        double r35998042 = c;
        double r35998043 = r35998041 + r35998042;
        double r35998044 = b;
        double r35998045 = r35998043 + r35998044;
        double r35998046 = a;
        double r35998047 = r35998045 + r35998046;
        return r35998047;
}

↓

double f(double a, double b, double c, double d, double e) {
        double r35998048 = e;
        double r35998049 = exp(r35998048);
        double r35998050 = a;
        double r35998051 = exp(r35998050);
        double r35998052 = b;
        double r35998053 = exp(r35998052);
        double r35998054 = d;
        double r35998055 = exp(r35998054);
        double r35998056 = c;
        double r35998057 = exp(r35998056);
        double r35998058 = r35998055 * r35998057;
        double r35998059 = r35998053 * r35998058;
        double r35998060 = r35998051 * r35998059;
        double r35998061 = r35998049 * r35998060;
        double r35998062 = log(r35998061);
        return r35998062;
}

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

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

13. Applied add-log-exp 0.3

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

14. Applied add-log-exp 0.3

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

15. Applied sum-log 0.3

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

16. Applied sum-log 0.1

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

17. Simplified 0.3

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

18. Taylor expanded around -inf 0.3

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

20. Applied add-log-exp 0.3

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

21. Applied add-log-exp 0.3

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

22. Applied sum-log 0.3

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

23. Applied add-log-exp 0.3

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

24. Applied sum-log 0.2

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

25. Applied add-log-exp 0.2

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

26. Applied sum-log 0.2

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

27. Applied add-log-exp 0.2

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

28. Applied sum-log 0.0

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

29. Applied rem-exp-log 0.0

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

30. Final simplification 0.0

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

Reproduce

herbie shell --seed 2019112 +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))