Average Error: 0.4 → 0.0
Time: 13.2s
Precision: 64
\[1 \le a \le 2 \le b \le 4 \le c \le 8 \le d \le 16 \le e \le 32\]
\[\left(\left(\left(e + d\right) + c\right) + b\right) + a\]
↓
\[\log \left(\left(e^{c} \cdot e^{e}\right) \cdot \left(\left(e^{b} \cdot e^{a}\right) \cdot e^{d}\right)\right)\]
\left(\left(\left(e + d\right) + c\right) + b\right) + a
↓
\log \left(\left(e^{c} \cdot e^{e}\right) \cdot \left(\left(e^{b} \cdot e^{a}\right) \cdot e^{d}\right)\right)
double f(double a, double b, double c, double d, double e) {
        double r5575564 = e;
        double r5575565 = d;
        double r5575566 = r5575564 + r5575565;
        double r5575567 = c;
        double r5575568 = r5575566 + r5575567;
        double r5575569 = b;
        double r5575570 = r5575568 + r5575569;
        double r5575571 = a;
        double r5575572 = r5575570 + r5575571;
        return r5575572;
}
↓
double f(double a, double b, double c, double d, double e) {
        double r5575573 = c;
        double r5575574 = exp(r5575573);
        double r5575575 = e;
        double r5575576 = exp(r5575575);
        double r5575577 = r5575574 * r5575576;
        double r5575578 = b;
        double r5575579 = exp(r5575578);
        double r5575580 = a;
        double r5575581 = exp(r5575580);
        double r5575582 = r5575579 * r5575581;
        double r5575583 = d;
        double r5575584 = exp(r5575583);
        double r5575585 = r5575582 * r5575584;
        double r5575586 = r5575577 * r5575585;
        double r5575587 = log(r5575586);
        return r5575587;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Bits error versus e

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original0.4
Target0.2
Herbie0.0
\[\left(d + \left(c + \left(a + b\right)\right)\right) + e\]

Derivation

  1. Initial program 0.4

    \[\left(\left(\left(e + d\right) + c\right) + b\right) + a\]
  2. Using strategy rm

  3. Applied associate-+l+0.3

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

  5. Applied associate-+l+0.3

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

  7. Applied add-log-exp0.3

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

  8. Applied add-log-exp0.3

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

  9. Applied sum-log0.3

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

  10. Applied add-log-exp0.3

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

  11. Applied add-log-exp0.3

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

  12. Applied sum-log0.3

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

  13. Applied add-log-exp0.3

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

  14. Applied sum-log0.3

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

  15. Applied sum-log0.0

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

  16. Simplified0.2

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

  18. Applied add-log-exp0.2

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

  19. Applied add-log-exp0.2

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

  20. Applied sum-log0.2

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

  21. Applied add-log-exp0.2

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

  22. Applied sum-log0.2

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

  23. Applied add-log-exp0.2

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

  24. Applied add-log-exp0.2

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

  25. Applied sum-log0.2

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

  26. Applied sum-log0.0

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

  27. Applied rem-exp-log0.0

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

  28. Final simplification0.0

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

Reproduce

herbie shell --seed 2019179 +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))