Expression 1, p15

Average Error: 0.4 → 0.2

Time: 8.4s

Precision: 64

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

Math

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

↓

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

C

double f(double a, double b, double c, double d, double e) {
        double r85194 = e;
        double r85195 = d;
        double r85196 = r85194 + r85195;
        double r85197 = c;
        double r85198 = r85196 + r85197;
        double r85199 = b;
        double r85200 = r85198 + r85199;
        double r85201 = a;
        double r85202 = r85200 + r85201;
        return r85202;
}

↓

double f(double a, double b, double c, double d, double e) {
        double r85203 = d;
        double r85204 = exp(r85203);
        double r85205 = e;
        double r85206 = exp(r85205);
        double r85207 = c;
        double r85208 = exp(r85207);
        double r85209 = r85206 * r85208;
        double r85210 = r85204 * r85209;
        double r85211 = log(r85210);
        double r85212 = b;
        double r85213 = a;
        double r85214 = r85212 + r85213;
        double r85215 = r85211 + r85214;
        return r85215;
}

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 associate-+l+ 0.3

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

5. Applied add-log-exp 0.3

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

6. Applied add-log-exp 0.3

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

7. Applied add-log-exp 0.3

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

8. Applied sum-log 0.3

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

9. Applied sum-log 0.2

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

10. Simplified 0.3

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

12. Applied add-log-exp 0.3

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

13. Applied add-log-exp 0.3

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

14. Applied sum-log 0.3

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

15. Applied add-log-exp 0.3

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

16. Applied sum-log 0.2

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

17. Applied rem-exp-log 0.2

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

18. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019326 
(FPCore (a b c d e)
  :name "Expression 1, p15"
  :precision binary64
  :pre (<= 1 a 2 b 4 c 8 d 16 e 32)

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

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