Average Error: 0.4 → 0.0
Time: 11.4s
Precision: 64
\[1 \le a \le 2 \le b \le 4 \le c \le 8 \le d \le 16 \le e \le 32\]
\[\left(\left(\left(e + d\right) + c\right) + b\right) + a\]
↓
\[\log \left(\left(e^{e} \cdot e^{c}\right) \cdot \left(\left(e^{b} \cdot e^{a}\right) \cdot e^{d}\right)\right)\]
\left(\left(\left(e + d\right) + c\right) + b\right) + a
↓
\log \left(\left(e^{e} \cdot e^{c}\right) \cdot \left(\left(e^{b} \cdot e^{a}\right) \cdot e^{d}\right)\right)
double f(double a, double b, double c, double d, double e) {
        double r2317000 = e;
        double r2317001 = d;
        double r2317002 = r2317000 + r2317001;
        double r2317003 = c;
        double r2317004 = r2317002 + r2317003;
        double r2317005 = b;
        double r2317006 = r2317004 + r2317005;
        double r2317007 = a;
        double r2317008 = r2317006 + r2317007;
        return r2317008;
}
↓
double f(double a, double b, double c, double d, double e) {
        double r2317009 = e;
        double r2317010 = exp(r2317009);
        double r2317011 = c;
        double r2317012 = exp(r2317011);
        double r2317013 = r2317010 * r2317012;
        double r2317014 = b;
        double r2317015 = exp(r2317014);
        double r2317016 = a;
        double r2317017 = exp(r2317016);
        double r2317018 = r2317015 * r2317017;
        double r2317019 = d;
        double r2317020 = exp(r2317019);
        double r2317021 = r2317018 * r2317020;
        double r2317022 = r2317013 * r2317021;
        double r2317023 = log(r2317022);
        return r2317023;
}

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

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

  7. Applied add-log-exp0.4

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

  8. Applied sum-log0.4

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

  9. Applied sum-log0.3

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

  10. Applied sum-log0.2

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

  11. Applied sum-log0.0

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

  12. Simplified0.2

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

  14. Applied add-log-exp0.2

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

  15. Applied add-log-exp0.2

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

  16. Applied sum-log0.2

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

  17. Applied add-log-exp0.2

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

  18. Applied sum-log0.2

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

  19. Applied add-log-exp0.2

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

  20. Applied add-log-exp0.2

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

  21. Applied sum-log0.2

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

  22. Applied sum-log0.0

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

  23. Applied rem-exp-log0.0

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

  24. Final simplification0.0

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

Reproduce

herbie shell --seed 2019154 +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))