Expression 1, p15

Average Error: 0.4 → 0.0

Time: 25.0s

Precision: 64

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

Math

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

↓

\[\log \left(\left(\sqrt{e^{c} \cdot \left(\left(e^{e} \cdot e^{b}\right) \cdot e^{a}\right)} \cdot \sqrt{\sqrt[3]{\left(\left(e^{c} \cdot e^{c}\right) \cdot e^{c}\right) \cdot \left(\left(\left(e^{e} \cdot e^{b}\right) \cdot e^{a}\right) \cdot \left(\left(\left(e^{e} \cdot e^{b}\right) \cdot e^{a}\right) \cdot \left(\left(e^{e} \cdot e^{b}\right) \cdot e^{a}\right)\right)\right)}}\right) \cdot e^{d}\right)\]

C

double f(double a, double b, double c, double d, double e) {
        double r3068025 = e;
        double r3068026 = d;
        double r3068027 = r3068025 + r3068026;
        double r3068028 = c;
        double r3068029 = r3068027 + r3068028;
        double r3068030 = b;
        double r3068031 = r3068029 + r3068030;
        double r3068032 = a;
        double r3068033 = r3068031 + r3068032;
        return r3068033;
}

↓

double f(double a, double b, double c, double d, double e) {
        double r3068034 = c;
        double r3068035 = exp(r3068034);
        double r3068036 = e;
        double r3068037 = exp(r3068036);
        double r3068038 = b;
        double r3068039 = exp(r3068038);
        double r3068040 = r3068037 * r3068039;
        double r3068041 = a;
        double r3068042 = exp(r3068041);
        double r3068043 = r3068040 * r3068042;
        double r3068044 = r3068035 * r3068043;
        double r3068045 = sqrt(r3068044);
        double r3068046 = r3068035 * r3068035;
        double r3068047 = r3068046 * r3068035;
        double r3068048 = r3068043 * r3068043;
        double r3068049 = r3068043 * r3068048;
        double r3068050 = r3068047 * r3068049;
        double r3068051 = cbrt(r3068050);
        double r3068052 = sqrt(r3068051);
        double r3068053 = r3068045 * r3068052;
        double r3068054 = d;
        double r3068055 = exp(r3068054);
        double r3068056 = r3068053 * r3068055;
        double r3068057 = log(r3068056);
        return r3068057;
}

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

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

12. Applied add-log-exp 0.3

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

13. Applied add-log-exp 0.3

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

14. Applied add-log-exp 0.3

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

15. Applied sum-log 0.3

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

16. Applied sum-log 0.3

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

17. Applied add-log-exp 0.3

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

18. Applied sum-log 0.2

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

19. Applied add-log-exp 0.2

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

20. Applied sum-log 0.0

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

21. Applied rem-exp-log 0.0

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

23. Applied add-sqr-sqrt 0.0

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

25. Applied add-cbrt-cube 0.0

    \[\leadsto \log \left(e^{d} \cdot \left(\sqrt{e^{c} \cdot \left(\left(e^{b} \cdot e^{e}\right) \cdot e^{a}\right)} \cdot \sqrt{e^{c} \cdot \color{blue}{\sqrt[3]{\left(\left(\left(e^{b} \cdot e^{e}\right) \cdot e^{a}\right) \cdot \left(\left(e^{b} \cdot e^{e}\right) \cdot e^{a}\right)\right) \cdot \left(\left(e^{b} \cdot e^{e}\right) \cdot e^{a}\right)}}}\right)\right)\]

26. Applied add-cbrt-cube 0.0

    \[\leadsto \log \left(e^{d} \cdot \left(\sqrt{e^{c} \cdot \left(\left(e^{b} \cdot e^{e}\right) \cdot e^{a}\right)} \cdot \sqrt{\color{blue}{\sqrt[3]{\left(e^{c} \cdot e^{c}\right) \cdot e^{c}}} \cdot \sqrt[3]{\left(\left(\left(e^{b} \cdot e^{e}\right) \cdot e^{a}\right) \cdot \left(\left(e^{b} \cdot e^{e}\right) \cdot e^{a}\right)\right) \cdot \left(\left(e^{b} \cdot e^{e}\right) \cdot e^{a}\right)}}\right)\right)\]

27. Applied cbrt-unprod 0.0

    \[\leadsto \log \left(e^{d} \cdot \left(\sqrt{e^{c} \cdot \left(\left(e^{b} \cdot e^{e}\right) \cdot e^{a}\right)} \cdot \sqrt{\color{blue}{\sqrt[3]{\left(\left(e^{c} \cdot e^{c}\right) \cdot e^{c}\right) \cdot \left(\left(\left(\left(e^{b} \cdot e^{e}\right) \cdot e^{a}\right) \cdot \left(\left(e^{b} \cdot e^{e}\right) \cdot e^{a}\right)\right) \cdot \left(\left(e^{b} \cdot e^{e}\right) \cdot e^{a}\right)\right)}}}\right)\right)\]

28. Final simplification 0.0

    \[\leadsto \log \left(\left(\sqrt{e^{c} \cdot \left(\left(e^{e} \cdot e^{b}\right) \cdot e^{a}\right)} \cdot \sqrt{\sqrt[3]{\left(\left(e^{c} \cdot e^{c}\right) \cdot e^{c}\right) \cdot \left(\left(\left(e^{e} \cdot e^{b}\right) \cdot e^{a}\right) \cdot \left(\left(\left(e^{e} \cdot e^{b}\right) \cdot e^{a}\right) \cdot \left(\left(e^{e} \cdot e^{b}\right) \cdot e^{a}\right)\right)\right)}}\right) \cdot e^{d}\right)\]

Reproduce

herbie shell --seed 2019142 +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))