Expression 1, p15

Average Error: 0.4 → 0.0

Time: 9.5s

Precision: 64

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

Math

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

↓

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

C

double f(double a, double b, double c, double d, double e) {
        double r2053597 = e;
        double r2053598 = d;
        double r2053599 = r2053597 + r2053598;
        double r2053600 = c;
        double r2053601 = r2053599 + r2053600;
        double r2053602 = b;
        double r2053603 = r2053601 + r2053602;
        double r2053604 = a;
        double r2053605 = r2053603 + r2053604;
        return r2053605;
}

↓

double f(double a, double b, double c, double d, double e) {
        double r2053606 = d;
        double r2053607 = exp(r2053606);
        double r2053608 = b;
        double r2053609 = exp(r2053608);
        double r2053610 = r2053607 * r2053609;
        double r2053611 = e;
        double r2053612 = exp(r2053611);
        double r2053613 = sqrt(r2053612);
        double r2053614 = c;
        double r2053615 = exp(r2053614);
        double r2053616 = sqrt(r2053615);
        double r2053617 = r2053613 * r2053616;
        double r2053618 = r2053617 * r2053617;
        double r2053619 = r2053610 * r2053618;
        double r2053620 = a;
        double r2053621 = exp(r2053620);
        double r2053622 = r2053619 * r2053621;
        double r2053623 = log(r2053622);
        return r2053623;
}

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 0.3

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

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

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

14. Applied add-log-exp 0.3

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

15. Applied add-log-exp 0.3

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

16. Applied sum-log 0.3

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

17. Applied add-log-exp 0.3

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

18. Applied add-log-exp 0.3

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

19. Applied sum-log 0.3

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

20. Applied sum-log 0.3

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

21. Applied add-log-exp 0.3

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

22. Applied sum-log 0.0

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

23. Applied rem-exp-log 0.0

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

25. Applied add-sqr-sqrt 0.0

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

26. Applied add-sqr-sqrt 0.0

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

27. Applied unswap-sqr 0.0

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

28. Final simplification 0.0

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

Reproduce

herbie shell --seed 2019156 +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))