Expression 1, p15

Average Error: 0.4 → 0.0

Time: 20.5s

Precision: 64

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

Math

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

↓

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

C

double f(double a, double b, double c, double d, double e) {
        double r4957001 = e;
        double r4957002 = d;
        double r4957003 = r4957001 + r4957002;
        double r4957004 = c;
        double r4957005 = r4957003 + r4957004;
        double r4957006 = b;
        double r4957007 = r4957005 + r4957006;
        double r4957008 = a;
        double r4957009 = r4957007 + r4957008;
        return r4957009;
}

↓

double f(double a, double b, double c, double d, double e) {
        double r4957010 = d;
        double r4957011 = exp(r4957010);
        double r4957012 = c;
        double r4957013 = exp(r4957012);
        double r4957014 = r4957011 * r4957013;
        double r4957015 = b;
        double r4957016 = exp(r4957015);
        double r4957017 = r4957014 * r4957016;
        double r4957018 = sqrt(r4957017);
        double r4957019 = r4957018 * r4957018;
        double r4957020 = e;
        double r4957021 = exp(r4957020);
        double r4957022 = a;
        double r4957023 = exp(r4957022);
        double r4957024 = r4957021 * r4957023;
        double r4957025 = r4957019 * r4957024;
        double r4957026 = log(r4957025);
        return r4957026;
}

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

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

14. Applied add-log-exp 0.3

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

15. Applied add-log-exp 0.3

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

16. Applied sum-log 0.3

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

17. Applied add-log-exp 0.3

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

18. Applied sum-log 0.2

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

19. Applied add-log-exp 0.2

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

20. Applied add-log-exp 0.2

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

21. Applied sum-log 0.2

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

22. Applied sum-log 0.0

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

23. Applied rem-exp-log 0.0

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

25. Applied add-sqr-sqrt 0.0

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

26. Final simplification 0.0

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

Reproduce

herbie shell --seed 2019152 +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))