Average Error: 0.4 → 0.0
Time: 14.3s
Precision: 64
\[1 \le a \le 2 \le b \le 4 \le c \le 8 \le d \le 16 \le e \le 32\]
\[\left(\left(\left(e + d\right) + c\right) + b\right) + a\]
↓
\[\log \left(e^{c} \cdot \left(\left(e^{e} \cdot e^{d}\right) \cdot \left(e^{b} \cdot e^{a}\right)\right)\right)\]
\left(\left(\left(e + d\right) + c\right) + b\right) + a
↓
\log \left(e^{c} \cdot \left(\left(e^{e} \cdot e^{d}\right) \cdot \left(e^{b} \cdot e^{a}\right)\right)\right)
double f(double a, double b, double c, double d, double e) {
        double r79870 = e;
        double r79871 = d;
        double r79872 = r79870 + r79871;
        double r79873 = c;
        double r79874 = r79872 + r79873;
        double r79875 = b;
        double r79876 = r79874 + r79875;
        double r79877 = a;
        double r79878 = r79876 + r79877;
        return r79878;
}
↓
double f(double a, double b, double c, double d, double e) {
        double r79879 = c;
        double r79880 = exp(r79879);
        double r79881 = e;
        double r79882 = exp(r79881);
        double r79883 = d;
        double r79884 = exp(r79883);
        double r79885 = r79882 * r79884;
        double r79886 = b;
        double r79887 = exp(r79886);
        double r79888 = a;
        double r79889 = exp(r79888);
        double r79890 = r79887 * r79889;
        double r79891 = r79885 * r79890;
        double r79892 = r79880 * r79891;
        double r79893 = log(r79892);
        return r79893;
}

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. Simplified0.3

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

  4. Applied add-log-exp0.3

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

  5. Applied add-log-exp0.3

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

  6. Applied sum-log0.3

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

  7. Applied add-log-exp0.3

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

  8. Applied add-log-exp0.3

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

  9. Applied sum-log0.3

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

  10. Applied add-log-exp0.3

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

  11. Applied sum-log0.3

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

  12. Applied sum-log0.0

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

  13. Simplified0.3

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

  15. Applied add-log-exp0.3

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

  16. Applied add-log-exp0.3

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

  17. Applied add-log-exp0.3

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

  18. Applied sum-log0.3

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

  19. Applied add-log-exp0.3

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

  20. Applied add-log-exp0.3

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

  21. Applied sum-log0.3

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

  22. Applied sum-log0.2

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

  23. Applied sum-log0.0

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

  24. Applied rem-exp-log0.0

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

  25. Final simplification0.0

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

Reproduce

herbie shell --seed 2019195 +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))