Expression 1, p15

Average Error: 0.4 → 0.0

Time: 29.7s

Precision: 64

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

Math

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

↓

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

C

double f(double a, double b, double c, double d, double e) {
        double r36476084 = e;
        double r36476085 = d;
        double r36476086 = r36476084 + r36476085;
        double r36476087 = c;
        double r36476088 = r36476086 + r36476087;
        double r36476089 = b;
        double r36476090 = r36476088 + r36476089;
        double r36476091 = a;
        double r36476092 = r36476090 + r36476091;
        return r36476092;
}

↓

double f(double a, double b, double c, double d, double e) {
        double r36476093 = e;
        double r36476094 = exp(r36476093);
        double r36476095 = a;
        double r36476096 = exp(r36476095);
        double r36476097 = b;
        double r36476098 = exp(r36476097);
        double r36476099 = d;
        double r36476100 = exp(r36476099);
        double r36476101 = c;
        double r36476102 = exp(r36476101);
        double r36476103 = r36476100 * r36476102;
        double r36476104 = r36476098 * r36476103;
        double r36476105 = r36476096 * r36476104;
        double r36476106 = r36476094 * r36476105;
        double r36476107 = log(r36476106);
        return r36476107;
}

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. Taylor expanded around -inf 0.3

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

13. Applied add-log-exp 0.3

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

14. Applied add-log-exp 0.3

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

15. Applied sum-log 0.3

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

16. Applied add-log-exp 0.3

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

17. Applied sum-log 0.2

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

18. Applied add-log-exp 0.2

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

19. Applied sum-log 0.2

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

20. Applied add-log-exp 0.2

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

21. Applied sum-log 0.0

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

22. Applied rem-exp-log 0.0

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

23. Final simplification 0.0

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

Reproduce

herbie shell --seed 2019119 
(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))