Expression 1, p15

Average Error: 0.4 → 0.0

Time: 19.7s

Precision: 64

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

Math

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

↓

\[\log \left(\left(\left(e^{b} \cdot e^{c}\right) \cdot e^{a}\right) \cdot \left(\left(\left(\sqrt[3]{e^{d}} \cdot \sqrt[3]{e^{d}}\right) \cdot e^{e}\right) \cdot \sqrt[3]{e^{d}}\right)\right)\]

C

double f(double a, double b, double c, double d, double e) {
        double r4535388 = e;
        double r4535389 = d;
        double r4535390 = r4535388 + r4535389;
        double r4535391 = c;
        double r4535392 = r4535390 + r4535391;
        double r4535393 = b;
        double r4535394 = r4535392 + r4535393;
        double r4535395 = a;
        double r4535396 = r4535394 + r4535395;
        return r4535396;
}

↓

double f(double a, double b, double c, double d, double e) {
        double r4535397 = b;
        double r4535398 = exp(r4535397);
        double r4535399 = c;
        double r4535400 = exp(r4535399);
        double r4535401 = r4535398 * r4535400;
        double r4535402 = a;
        double r4535403 = exp(r4535402);
        double r4535404 = r4535401 * r4535403;
        double r4535405 = d;
        double r4535406 = exp(r4535405);
        double r4535407 = cbrt(r4535406);
        double r4535408 = r4535407 * r4535407;
        double r4535409 = e;
        double r4535410 = exp(r4535409);
        double r4535411 = r4535408 * r4535410;
        double r4535412 = r4535411 * r4535407;
        double r4535413 = r4535404 * r4535412;
        double r4535414 = log(r4535413);
        return r4535414;
}

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

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

12. Applied add-log-exp 0.3

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

13. Applied add-log-exp 0.3

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

14. Applied add-log-exp 0.3

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

15. Applied sum-log 0.3

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

16. Applied sum-log 0.2

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

17. Applied add-log-exp 0.2

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

18. Applied add-log-exp 0.2

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

19. Applied sum-log 0.2

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

20. Applied sum-log 0.0

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

21. Applied rem-exp-log 0.0

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

23. Applied add-cube-cbrt 0.0

    \[\leadsto \log \left(\left(e^{e} \cdot \color{blue}{\left(\left(\sqrt[3]{e^{d}} \cdot \sqrt[3]{e^{d}}\right) \cdot \sqrt[3]{e^{d}}\right)}\right) \cdot \left(\left(e^{c} \cdot e^{b}\right) \cdot e^{a}\right)\right)\]

24. Applied associate-*r* 0.0

    \[\leadsto \log \left(\color{blue}{\left(\left(e^{e} \cdot \left(\sqrt[3]{e^{d}} \cdot \sqrt[3]{e^{d}}\right)\right) \cdot \sqrt[3]{e^{d}}\right)} \cdot \left(\left(e^{c} \cdot e^{b}\right) \cdot e^{a}\right)\right)\]

25. Final simplification 0.0

    \[\leadsto \log \left(\left(\left(e^{b} \cdot e^{c}\right) \cdot e^{a}\right) \cdot \left(\left(\left(\sqrt[3]{e^{d}} \cdot \sqrt[3]{e^{d}}\right) \cdot e^{e}\right) \cdot \sqrt[3]{e^{d}}\right)\right)\]

Reproduce

herbie shell --seed 2019142 
(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))