Expression 1, p15

Average Error: 0.4 → 0.0

Time: 16.4s

Precision: 64

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

Math

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

↓

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

C

double f(double a, double b, double c, double d, double e) {
        double r4092469 = e;
        double r4092470 = d;
        double r4092471 = r4092469 + r4092470;
        double r4092472 = c;
        double r4092473 = r4092471 + r4092472;
        double r4092474 = b;
        double r4092475 = r4092473 + r4092474;
        double r4092476 = a;
        double r4092477 = r4092475 + r4092476;
        return r4092477;
}

↓

double f(double a, double b, double c, double d, double e) {
        double r4092478 = c;
        double r4092479 = exp(r4092478);
        double r4092480 = sqrt(r4092479);
        double r4092481 = e;
        double r4092482 = exp(r4092481);
        double r4092483 = r4092482 * r4092480;
        double r4092484 = r4092480 * r4092483;
        double r4092485 = a;
        double r4092486 = exp(r4092485);
        double r4092487 = b;
        double r4092488 = exp(r4092487);
        double r4092489 = r4092486 * r4092488;
        double r4092490 = r4092484 * r4092489;
        double r4092491 = d;
        double r4092492 = exp(r4092491);
        double r4092493 = r4092490 * r4092492;
        double r4092494 = log(r4092493);
        return r4092494;
}

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

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

14. Applied add-log-exp 0.3

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

15. Applied add-log-exp 0.3

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

16. Applied add-log-exp 0.3

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

17. Applied sum-log 0.3

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

18. Applied add-log-exp 0.3

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

19. Applied add-log-exp 0.3

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

20. Applied sum-log 0.3

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

21. Applied sum-log 0.2

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

22. Applied sum-log 0.0

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

23. Applied rem-exp-log 0.0

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

25. Applied add-sqr-sqrt 0.0

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

26. Applied associate-*l* 0.0

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

27. Final simplification 0.0

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

Reproduce

herbie shell --seed 2019152 
(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))