Expression 1, p15

Average Error: 0.4 → 0.0

Time: 29.3s

Precision: 64

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

Math

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

↓

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

C

double f(double a, double b, double c, double d, double e) {
        double r5251666 = e;
        double r5251667 = d;
        double r5251668 = r5251666 + r5251667;
        double r5251669 = c;
        double r5251670 = r5251668 + r5251669;
        double r5251671 = b;
        double r5251672 = r5251670 + r5251671;
        double r5251673 = a;
        double r5251674 = r5251672 + r5251673;
        return r5251674;
}

↓

double f(double a, double b, double c, double d, double e) {
        double r5251675 = d;
        double r5251676 = exp(r5251675);
        double r5251677 = c;
        double r5251678 = exp(r5251677);
        double r5251679 = r5251676 * r5251678;
        double r5251680 = b;
        double r5251681 = exp(r5251680);
        double r5251682 = sqrt(r5251681);
        double r5251683 = e;
        double r5251684 = exp(r5251683);
        double r5251685 = sqrt(r5251684);
        double r5251686 = r5251682 * r5251685;
        double r5251687 = r5251686 * r5251686;
        double r5251688 = r5251679 * r5251687;
        double r5251689 = a;
        double r5251690 = exp(r5251689);
        double r5251691 = r5251688 * r5251690;
        double r5251692 = log(r5251691);
        return r5251692;
}

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

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

5. Applied add-log-exp 0.3

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

6. Applied add-log-exp 0.3

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

7. Applied add-log-exp 0.3

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

8. Applied sum-log 0.3

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

9. Applied add-log-exp 0.3

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

10. Applied add-log-exp 0.3

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

11. Applied sum-log 0.3

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

12. Applied sum-log 0.2

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

13. Applied sum-log 0.0

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

14. Simplified 0.3

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

16. Applied add-log-exp 0.3

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

17. Applied add-log-exp 0.3

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

18. Applied sum-log 0.3

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

19. Applied add-log-exp 0.3

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

20. Applied add-log-exp 0.3

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

21. Applied sum-log 0.3

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

22. Applied sum-log 0.2

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

23. Applied add-log-exp 0.2

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

24. Applied sum-log 0.0

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

25. Applied rem-exp-log 0.0

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

27. Applied add-sqr-sqrt 0.0

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

28. Applied add-sqr-sqrt 0.0

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

29. Applied unswap-sqr 0.0

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

30. Final simplification 0.0

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

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (a b c d e)
  :name "Expression 1, p15"
  :pre (<= 1.0 a 2.0 b 4.0 c 8.0 d 16.0 e 32.0)

  :herbie-target
  (+ (+ d (+ c (+ a b))) e)

  (+ (+ (+ (+ e d) c) b) a))