Expression 1, p15

Average Error: 0.4 → 0.0

Time: 40.4s

Precision: 64

\[1.0 \le a \le 2.0 \le b \le 4.0 \le c \le 8.0 \le d \le 16.0 \le e \le 32.0\]

Math

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

↓

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

C

double f(double a, double b, double c, double d, double e) {
        double r5191291 = e;
        double r5191292 = d;
        double r5191293 = r5191291 + r5191292;
        double r5191294 = c;
        double r5191295 = r5191293 + r5191294;
        double r5191296 = b;
        double r5191297 = r5191295 + r5191296;
        double r5191298 = a;
        double r5191299 = r5191297 + r5191298;
        return r5191299;
}

↓

double f(double a, double b, double c, double d, double e) {
        double r5191300 = e;
        double r5191301 = exp(r5191300);
        double r5191302 = d;
        double r5191303 = exp(r5191302);
        double r5191304 = r5191301 * r5191303;
        double r5191305 = c;
        double r5191306 = exp(r5191305);
        double r5191307 = b;
        double r5191308 = exp(r5191307);
        double r5191309 = r5191306 * r5191308;
        double r5191310 = a;
        double r5191311 = exp(r5191310);
        double r5191312 = r5191309 * r5191311;
        double r5191313 = r5191304 * r5191312;
        double r5191314 = sqrt(r5191313);
        double r5191315 = r5191314 * r5191314;
        double r5191316 = log(r5191315);
        return r5191316;
}

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

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

14. Applied add-log-exp 0.3

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

15. Applied add-log-exp 0.3

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

16. Applied sum-log 0.3

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

17. Applied add-log-exp 0.3

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

18. Applied sum-log 0.3

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

19. Applied add-log-exp 0.3

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

20. Applied add-log-exp 0.3

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

21. Applied sum-log 0.3

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

22. Applied sum-log 0.0

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

23. Applied rem-exp-log 0.0

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

25. Applied add-sqr-sqrt 0.0

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

26. Final simplification 0.0

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

Reproduce

herbie shell --seed 2019165 +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))