Average Error: 0.4 → 0.0
Time: 1.3m
Precision: 64
\[1 \le a \le 2 \le b \le 4 \le c \le 8 \le d \le 16 \le e \le 32\]
\[\left(\left(\left(e + d\right) + c\right) + b\right) + a\]
↓
\[\log \left(\left(e^{e} \cdot e^{d}\right) \cdot \left(e^{b} \cdot \left(e^{c} \cdot e^{a}\right)\right)\right)\]
\left(\left(\left(e + d\right) + c\right) + b\right) + a
↓
\log \left(\left(e^{e} \cdot e^{d}\right) \cdot \left(e^{b} \cdot \left(e^{c} \cdot e^{a}\right)\right)\right)
double f(double a, double b, double c, double d, double e) {
        double r45490055 = e;
        double r45490056 = d;
        double r45490057 = r45490055 + r45490056;
        double r45490058 = c;
        double r45490059 = r45490057 + r45490058;
        double r45490060 = b;
        double r45490061 = r45490059 + r45490060;
        double r45490062 = a;
        double r45490063 = r45490061 + r45490062;
        return r45490063;
}
↓
double f(double a, double b, double c, double d, double e) {
        double r45490064 = e;
        double r45490065 = exp(r45490064);
        double r45490066 = d;
        double r45490067 = exp(r45490066);
        double r45490068 = r45490065 * r45490067;
        double r45490069 = b;
        double r45490070 = exp(r45490069);
        double r45490071 = c;
        double r45490072 = exp(r45490071);
        double r45490073 = a;
        double r45490074 = exp(r45490073);
        double r45490075 = r45490072 * r45490074;
        double r45490076 = r45490070 * r45490075;
        double r45490077 = r45490068 * r45490076;
        double r45490078 = log(r45490077);
        return r45490078;
}

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 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 add-log-exp0.3

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

  13. Applied add-log-exp0.3

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

  14. Applied add-log-exp0.3

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

  15. Applied sum-log0.3

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

  16. Applied sum-log0.3

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

  17. Applied add-log-exp0.3

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

  18. Applied add-log-exp0.3

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

  19. Applied sum-log0.3

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

  20. Applied sum-log0.0

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

  21. Applied rem-exp-log0.0

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

  22. Final simplification0.0

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

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(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))