Average Error: 0.4 → 0.0
Time: 16.5s
Precision: 64
\[1 \le a \le 2 \le b \le 4 \le c \le 8 \le d \le 16 \le e \le 32\]
\[\left(\left(\left(e + d\right) + c\right) + b\right) + a\]
↓
\[\log \left(e^{a} \cdot \left(\left(\sqrt{e^{d} \cdot e^{b}} \cdot \sqrt{e^{d} \cdot e^{b}}\right) \cdot \left(e^{c} \cdot e^{e}\right)\right)\right)\]
\left(\left(\left(e + d\right) + c\right) + b\right) + a
↓
\log \left(e^{a} \cdot \left(\left(\sqrt{e^{d} \cdot e^{b}} \cdot \sqrt{e^{d} \cdot e^{b}}\right) \cdot \left(e^{c} \cdot e^{e}\right)\right)\right)
double f(double a, double b, double c, double d, double e) {
        double r2305009 = e;
        double r2305010 = d;
        double r2305011 = r2305009 + r2305010;
        double r2305012 = c;
        double r2305013 = r2305011 + r2305012;
        double r2305014 = b;
        double r2305015 = r2305013 + r2305014;
        double r2305016 = a;
        double r2305017 = r2305015 + r2305016;
        return r2305017;
}
↓
double f(double a, double b, double c, double d, double e) {
        double r2305018 = a;
        double r2305019 = exp(r2305018);
        double r2305020 = d;
        double r2305021 = exp(r2305020);
        double r2305022 = b;
        double r2305023 = exp(r2305022);
        double r2305024 = r2305021 * r2305023;
        double r2305025 = sqrt(r2305024);
        double r2305026 = r2305025 * r2305025;
        double r2305027 = c;
        double r2305028 = exp(r2305027);
        double r2305029 = e;
        double r2305030 = exp(r2305029);
        double r2305031 = r2305028 * r2305030;
        double r2305032 = r2305026 * r2305031;
        double r2305033 = r2305019 * r2305032;
        double r2305034 = log(r2305033);
        return r2305034;
}

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

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

  12. Applied add-log-exp0.3

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

  13. Applied add-log-exp0.3

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

  14. Applied sum-log0.3

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

  15. Applied add-log-exp0.3

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

  16. Applied add-log-exp0.3

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

  17. Applied sum-log0.3

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

  18. Applied sum-log0.2

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

  19. Applied add-log-exp0.2

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

  20. Applied sum-log0.0

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

  21. Applied rem-exp-log0.0

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

  23. Applied add-sqr-sqrt0.0

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

  24. Final simplification0.0

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

Reproduce

herbie shell --seed 2019151 +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))