Expression 1, p15

Average Error: 0.4 → 0.0

Time: 33.8s

Precision: 64

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

Math

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

↓

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

C

double f(double a, double b, double c, double d, double e) {
        double r12923308 = e;
        double r12923309 = d;
        double r12923310 = r12923308 + r12923309;
        double r12923311 = c;
        double r12923312 = r12923310 + r12923311;
        double r12923313 = b;
        double r12923314 = r12923312 + r12923313;
        double r12923315 = a;
        double r12923316 = r12923314 + r12923315;
        return r12923316;
}

↓

double f(double a, double b, double c, double d, double e) {
        double r12923317 = c;
        double r12923318 = exp(r12923317);
        double r12923319 = e;
        double r12923320 = exp(r12923319);
        double r12923321 = r12923318 * r12923320;
        double r12923322 = d;
        double r12923323 = exp(r12923322);
        double r12923324 = b;
        double r12923325 = exp(r12923324);
        double r12923326 = r12923323 * r12923325;
        double r12923327 = a;
        double r12923328 = exp(r12923327);
        double r12923329 = r12923326 * r12923328;
        double r12923330 = r12923321 * r12923329;
        double r12923331 = log(r12923330);
        return r12923331;
}

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

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

4. Applied add-log-exp 0.4

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

5. Applied add-log-exp 0.4

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

6. Applied sum-log 0.4

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

7. Applied sum-log 0.3

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

8. Simplified 0.3

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

10. Applied add-log-exp 0.3

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

11. Applied sum-log 0.3

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

12. Simplified 0.2

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

14. Applied add-log-exp 0.2

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

15. Applied add-log-exp 0.2

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

16. Applied sum-log 0.2

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

17. Applied add-log-exp 0.2

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

18. Applied sum-log 0.2

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

19. Applied add-log-exp 0.2

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

20. Applied add-log-exp 0.2

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

21. Applied sum-log 0.2

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

22. Applied sum-log 0.0

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

23. Applied rem-exp-log 0.0

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

24. Simplified 0.2

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

26. Applied add-log-exp 0.2

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

27. Applied add-log-exp 0.2

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

28. Applied sum-log 0.2

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

29. Applied rem-exp-log 0.0

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

30. Final simplification 0.0

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

Reproduce

herbie shell --seed 2019121 +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))