Average Error: 0.4 → 0.0
Time: 30.4s
Precision: 64
\[\left(\left(\left(e + d\right) + c\right) + b\right) + a\]
↓
\[\log \left(e^{e} \cdot \left(e^{d} \cdot \left(e^{b} \cdot \left(e^{c} \cdot e^{a}\right)\right)\right)\right)\]
double f(double a, double b, double c, double d, double e) {
        double r11335510 = e;
        double r11335511 = d;
        double r11335512 = r11335510 + r11335511;
        double r11335513 = c;
        double r11335514 = r11335512 + r11335513;
        double r11335515 = b;
        double r11335516 = r11335514 + r11335515;
        double r11335517 = a;
        double r11335518 = r11335516 + r11335517;
        return r11335518;
}
↓
double f(double a, double b, double c, double d, double e) {
        double r11335519 = e;
        double r11335520 = exp(r11335519);
        double r11335521 = d;
        double r11335522 = exp(r11335521);
        double r11335523 = b;
        double r11335524 = exp(r11335523);
        double r11335525 = c;
        double r11335526 = exp(r11335525);
        double r11335527 = a;
        double r11335528 = exp(r11335527);
        double r11335529 = r11335526 * r11335528;
        double r11335530 = r11335524 * r11335529;
        double r11335531 = r11335522 * r11335530;
        double r11335532 = r11335520 * r11335531;
        double r11335533 = log(r11335532);
        return r11335533;
}


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

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Bits error versus e

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 add-log-exp0.4

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

  4. Applied add-log-exp0.4

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

  5. Applied add-log-exp0.4

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

  6. Applied add-log-exp0.4

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

  7. Applied sum-log0.4

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

  8. Applied sum-log0.3

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

  9. Applied sum-log0.2

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

  10. Simplified0.3

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

  12. Applied rem-log-exp0.3

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

  14. Applied add-log-exp0.3

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

  15. Applied add-log-exp0.3

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

  16. Applied sum-log0.3

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

  17. Simplified0.2

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

  19. Applied add-log-exp0.2

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

  20. Applied add-log-exp0.2

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

  21. Applied sum-log0.2

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

  22. Applied add-log-exp0.2

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

  23. Applied sum-log0.2

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

  24. Applied add-log-exp0.2

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

  25. Applied sum-log0.2

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

  26. Applied add-log-exp0.2

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

  27. Applied sum-log0.0

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

  28. Applied rem-exp-log0.0

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

  29. Final simplification0.0

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

Reproduce

herbie shell --seed 2019101 
(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))