Expression 1, p15

Average Error: 0.4 → 0.0

Time: 12.6s

Precision: 64

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

Math

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

↓

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

C

double f(double a, double b, double c, double d, double e) {
        double r1993144 = e;
        double r1993145 = d;
        double r1993146 = r1993144 + r1993145;
        double r1993147 = c;
        double r1993148 = r1993146 + r1993147;
        double r1993149 = b;
        double r1993150 = r1993148 + r1993149;
        double r1993151 = a;
        double r1993152 = r1993150 + r1993151;
        return r1993152;
}

↓

double f(double a, double b, double c, double d, double e) {
        double r1993153 = c;
        double r1993154 = exp(r1993153);
        double r1993155 = b;
        double r1993156 = exp(r1993155);
        double r1993157 = a;
        double r1993158 = exp(r1993157);
        double r1993159 = r1993156 * r1993158;
        double r1993160 = e;
        double r1993161 = exp(r1993160);
        double r1993162 = d;
        double r1993163 = exp(r1993162);
        double r1993164 = r1993161 * r1993163;
        double r1993165 = r1993159 * r1993164;
        double r1993166 = r1993154 * r1993165;
        double r1993167 = log(r1993166);
        return r1993167;
}

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 associate-+l+ 0.3

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

5. Applied add-log-exp 0.3

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

6. Applied add-log-exp 0.3

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

7. Applied add-log-exp 0.3

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

8. Applied add-log-exp 0.3

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

9. Applied sum-log 0.3

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

10. Applied sum-log 0.3

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

11. Applied sum-log 0.1

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

12. Simplified 0.3

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

14. Applied add-log-exp 0.3

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

15. Applied add-log-exp 0.3

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

16. Applied add-log-exp 0.3

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

17. Applied sum-log 0.3

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

18. Applied add-log-exp 0.3

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

19. Applied add-log-exp 0.3

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

20. Applied sum-log 0.3

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

21. Applied sum-log 0.2

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

22. Applied sum-log 0.0

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

23. Applied rem-exp-log 0.0

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

24. Final simplification 0.0

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

Reproduce

herbie shell --seed 2019151 
(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))