Average Error: 0.4 → 0.0
Time: 10.8s
Precision: 64
\[1 \le a \le 2 \le b \le 4 \le c \le 8 \le d \le 16 \le e \le 32\]
\[\left(\left(\left(e + d\right) + c\right) + b\right) + a\]
↓
\[\log \left(e^{e} \cdot \left(\left(e^{a} \cdot e^{b}\right) \cdot \left(e^{d} \cdot e^{c}\right)\right)\right)\]
\left(\left(\left(e + d\right) + c\right) + b\right) + a
↓
\log \left(e^{e} \cdot \left(\left(e^{a} \cdot e^{b}\right) \cdot \left(e^{d} \cdot e^{c}\right)\right)\right)
double f(double a, double b, double c, double d, double e) {
        double r1995667 = e;
        double r1995668 = d;
        double r1995669 = r1995667 + r1995668;
        double r1995670 = c;
        double r1995671 = r1995669 + r1995670;
        double r1995672 = b;
        double r1995673 = r1995671 + r1995672;
        double r1995674 = a;
        double r1995675 = r1995673 + r1995674;
        return r1995675;
}
↓
double f(double a, double b, double c, double d, double e) {
        double r1995676 = e;
        double r1995677 = exp(r1995676);
        double r1995678 = a;
        double r1995679 = exp(r1995678);
        double r1995680 = b;
        double r1995681 = exp(r1995680);
        double r1995682 = r1995679 * r1995681;
        double r1995683 = d;
        double r1995684 = exp(r1995683);
        double r1995685 = c;
        double r1995686 = exp(r1995685);
        double r1995687 = r1995684 * r1995686;
        double r1995688 = r1995682 * r1995687;
        double r1995689 = r1995677 * r1995688;
        double r1995690 = log(r1995689);
        return r1995690;
}

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+0.4

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

  5. Applied add-log-exp0.4

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

  6. Applied add-log-exp0.4

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

  7. Applied add-log-exp0.4

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

  8. Applied add-log-exp0.4

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

  9. Applied sum-log0.4

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

  10. Applied add-log-exp0.4

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

  11. Applied sum-log0.3

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

  12. Applied sum-log0.2

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

  13. Applied sum-log0.0

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

  14. Simplified0.2

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

  16. Applied add-log-exp0.2

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

  17. Applied add-log-exp0.2

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

  18. Applied sum-log0.2

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

  19. Applied add-log-exp0.2

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

  20. Applied add-log-exp0.2

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

  21. Applied sum-log0.2

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

  22. Applied sum-log0.2

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

  23. Applied add-log-exp0.2

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

  24. Applied sum-log0.0

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

  25. Applied rem-exp-log0.0

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

  26. Final simplification0.0

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

Reproduce

herbie shell --seed 2019153 
(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))