Expression 1, p15

Average Error: 0.4 → 0.3

Time: 19.0s

Precision: 64

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

Math

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

↓

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

C

double f(double a, double b, double c, double d, double e) {
        double r4120794 = e;
        double r4120795 = d;
        double r4120796 = r4120794 + r4120795;
        double r4120797 = c;
        double r4120798 = r4120796 + r4120797;
        double r4120799 = b;
        double r4120800 = r4120798 + r4120799;
        double r4120801 = a;
        double r4120802 = r4120800 + r4120801;
        return r4120802;
}

↓

double f(double a, double b, double c, double d, double e) {
        double r4120803 = c;
        double r4120804 = a;
        double r4120805 = d;
        double r4120806 = b;
        double r4120807 = r4120805 + r4120806;
        double r4120808 = r4120804 + r4120807;
        double r4120809 = r4120803 + r4120808;
        double r4120810 = e;
        double r4120811 = exp(r4120810);
        double r4120812 = log(r4120811);
        double r4120813 = r4120809 + r4120812;
        return r4120813;
}

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	0.3

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

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

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

4. Applied add-log-exp 0.4

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

5. Applied add-log-exp 0.4

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

6. Applied add-log-exp 0.4

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

7. Applied sum-log 0.4

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

8. Applied sum-log 0.3

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

9. Applied sum-log 0.2

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

10. Simplified 0.3

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

12. Applied exp-sum 0.3

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

13. Applied log-prod 0.3

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

14. Applied associate-+l+ 0.3

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

15. Simplified 0.3

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

17. Applied associate-+l+ 0.3

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

18. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019141 +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))