Expression 1, p15

Average Error: 0.4 → 0.2

Time: 19.1s

Precision: 64

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

Math

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

↓

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

C

double f(double a, double b, double c, double d, double e) {
        double r4526107 = e;
        double r4526108 = d;
        double r4526109 = r4526107 + r4526108;
        double r4526110 = c;
        double r4526111 = r4526109 + r4526110;
        double r4526112 = b;
        double r4526113 = r4526111 + r4526112;
        double r4526114 = a;
        double r4526115 = r4526113 + r4526114;
        return r4526115;
}

↓

double f(double a, double b, double c, double d, double e) {
        double r4526116 = e;
        double r4526117 = b;
        double r4526118 = a;
        double r4526119 = c;
        double r4526120 = r4526118 + r4526119;
        double r4526121 = r4526117 + r4526120;
        double r4526122 = d;
        double r4526123 = r4526121 + r4526122;
        double r4526124 = r4526116 + r4526123;
        return r4526124;
}

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	0.2

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

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

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

4. Applied add-log-exp 0.4

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

5. Applied sum-log 0.4

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

6. Simplified 0.3

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

8. Applied add-log-exp 0.3

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

9. Applied add-log-exp 0.3

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

10. Applied sum-log 0.3

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

11. Applied add-log-exp 0.3

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

12. Applied sum-log 0.3

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

13. Applied add-log-exp 0.3

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

14. Applied sum-log 0.2

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

15. Applied rem-exp-log 0.2

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

16. Taylor expanded around -inf 0.2

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

17. Simplified 0.2

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

19. Applied associate-+l+ 0.2

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

20. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019130 +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))