Expression 1, p15

Average Error: 0.4 → 0.0

Time: 38.5s

Precision: 64

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

Math

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

↓

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

C

double f(double a, double b, double c, double d, double e) {
        double r16340582 = e;
        double r16340583 = d;
        double r16340584 = r16340582 + r16340583;
        double r16340585 = c;
        double r16340586 = r16340584 + r16340585;
        double r16340587 = b;
        double r16340588 = r16340586 + r16340587;
        double r16340589 = a;
        double r16340590 = r16340588 + r16340589;
        return r16340590;
}

↓

double f(double a, double b, double c, double d, double e) {
        double r16340591 = d;
        double r16340592 = exp(r16340591);
        double r16340593 = e;
        double r16340594 = exp(r16340593);
        double r16340595 = a;
        double r16340596 = exp(r16340595);
        double r16340597 = c;
        double r16340598 = exp(r16340597);
        double r16340599 = r16340596 * r16340598;
        double r16340600 = b;
        double r16340601 = exp(r16340600);
        double r16340602 = r16340599 * r16340601;
        double r16340603 = r16340594 * r16340602;
        double r16340604 = r16340592 * r16340603;
        double r16340605 = log(r16340604);
        return r16340605;
}

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. Using strategy rm

12. Applied associate-+l+ 0.2

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

14. Applied add-log-exp 0.2

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

15. Applied add-log-exp 0.2

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

16. Applied add-log-exp 0.2

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

17. Applied sum-log 0.2

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

18. Applied sum-log 0.2

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

19. Applied add-log-exp 0.2

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

20. Applied sum-log 0.2

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

21. Applied add-log-exp 0.2

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

22. Applied sum-log 0.0

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

23. Applied rem-exp-log 0.0

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

24. Final simplification 0.0

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

Reproduce

herbie shell --seed 2019124 +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))