Expression 1, p15

Average Error: 0.4 → 0.0

Time: 40.9s

Precision: 64

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

Math

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

↓

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

C

double f(double a, double b, double c, double d, double e) {
        double r25614265 = e;
        double r25614266 = d;
        double r25614267 = r25614265 + r25614266;
        double r25614268 = c;
        double r25614269 = r25614267 + r25614268;
        double r25614270 = b;
        double r25614271 = r25614269 + r25614270;
        double r25614272 = a;
        double r25614273 = r25614271 + r25614272;
        return r25614273;
}

↓

double f(double a, double b, double c, double d, double e) {
        double r25614274 = e;
        double r25614275 = exp(r25614274);
        double r25614276 = d;
        double r25614277 = exp(r25614276);
        double r25614278 = b;
        double r25614279 = exp(r25614278);
        double r25614280 = c;
        double r25614281 = exp(r25614280);
        double r25614282 = a;
        double r25614283 = exp(r25614282);
        double r25614284 = r25614281 * r25614283;
        double r25614285 = r25614279 * r25614284;
        double r25614286 = r25614277 * r25614285;
        double r25614287 = r25614275 * r25614286;
        double r25614288 = log(r25614287);
        double r25614289 = exp(r25614288);
        double r25614290 = log(r25614289);
        return r25614290;
}

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

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

14. Applied sum-log 0.3

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

15. Simplified 0.2

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

17. Applied add-log-exp 0.2

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

18. Applied add-log-exp 0.2

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

19. Applied sum-log 0.2

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

20. Applied add-log-exp 0.2

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

21. Applied sum-log 0.2

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

22. Applied add-log-exp 0.2

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

23. Applied sum-log 0.2

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

24. Applied add-log-exp 0.2

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

25. Applied sum-log 0.0

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

26. Applied rem-exp-log 0.0

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

27. Final simplification 0.0

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

Reproduce

herbie shell --seed 2019104 +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))