Average Error: 0.4 → 0.0
Time: 16.9s
Precision: 64
\[1 \le a \le 2 \le b \le 4 \le c \le 8 \le d \le 16 \le e \le 32\]
\[\left(\left(\left(e + d\right) + c\right) + b\right) + a\]
↓
\[\log \left(\left(e^{b} \cdot e^{c}\right) \cdot \left(e^{a} \cdot \left(e^{d} \cdot e^{e}\right)\right)\right)\]
\left(\left(\left(e + d\right) + c\right) + b\right) + a
↓
\log \left(\left(e^{b} \cdot e^{c}\right) \cdot \left(e^{a} \cdot \left(e^{d} \cdot e^{e}\right)\right)\right)
double f(double a, double b, double c, double d, double e) {
        double r4611689 = e;
        double r4611690 = d;
        double r4611691 = r4611689 + r4611690;
        double r4611692 = c;
        double r4611693 = r4611691 + r4611692;
        double r4611694 = b;
        double r4611695 = r4611693 + r4611694;
        double r4611696 = a;
        double r4611697 = r4611695 + r4611696;
        return r4611697;
}
↓
double f(double a, double b, double c, double d, double e) {
        double r4611698 = b;
        double r4611699 = exp(r4611698);
        double r4611700 = c;
        double r4611701 = exp(r4611700);
        double r4611702 = r4611699 * r4611701;
        double r4611703 = a;
        double r4611704 = exp(r4611703);
        double r4611705 = d;
        double r4611706 = exp(r4611705);
        double r4611707 = e;
        double r4611708 = exp(r4611707);
        double r4611709 = r4611706 * r4611708;
        double r4611710 = r4611704 * r4611709;
        double r4611711 = r4611702 * r4611710;
        double r4611712 = log(r4611711);
        return r4611712;
}

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 add-log-exp0.3

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

  6. Applied add-log-exp0.3

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

  7. Applied sum-log0.3

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

  8. Applied add-log-exp0.3

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

  9. Applied sum-log0.3

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

  10. Simplified0.3

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

  12. Applied add-log-exp0.3

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

  13. Applied add-log-exp0.3

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

  14. Applied sum-log0.3

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

  15. Applied add-log-exp0.3

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

  16. Applied add-log-exp0.3

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

  17. Applied add-log-exp0.3

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

  18. Applied sum-log0.3

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

  19. Applied sum-log0.2

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

  20. Applied sum-log0.0

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

  21. Applied rem-exp-log0.0

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

  22. Final simplification0.0

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

Reproduce

herbie shell --seed 2019134 
(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))