Expression 1, p15

Average Error: 0.4 → 0.2

Time: 7.7s

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 r58130 = e;
        double r58131 = d;
        double r58132 = r58130 + r58131;
        double r58133 = c;
        double r58134 = r58132 + r58133;
        double r58135 = b;
        double r58136 = r58134 + r58135;
        double r58137 = a;
        double r58138 = r58136 + r58137;
        return r58138;
}

↓

double f(double a, double b, double c, double d, double e) {
        double r58139 = d;
        double r58140 = exp(r58139);
        double r58141 = e;
        double r58142 = exp(r58141);
        double r58143 = c;
        double r58144 = exp(r58143);
        double r58145 = r58142 * r58144;
        double r58146 = r58140 * r58145;
        double r58147 = log(r58146);
        double r58148 = b;
        double r58149 = a;
        double r58150 = r58148 + r58149;
        double r58151 = r58147 + r58150;
        return r58151;
}

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))