Average Error: 0.4 → 0.0
Time: 14.4s
Precision: 64
\[1 \le a \le 2 \le b \le 4 \le c \le 8 \le d \le 16 \le e \le 32\]
\[\left(\left(\left(e + d\right) + c\right) + b\right) + a\]
↓
\[\log \left(e^{c} \cdot \left(\left(e^{e} \cdot e^{d}\right) \cdot \left(e^{b} \cdot e^{a}\right)\right)\right)\]
\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 r3913287 = e;
        double r3913288 = d;
        double r3913289 = r3913287 + r3913288;
        double r3913290 = c;
        double r3913291 = r3913289 + r3913290;
        double r3913292 = b;
        double r3913293 = r3913291 + r3913292;
        double r3913294 = a;
        double r3913295 = r3913293 + r3913294;
        return r3913295;
}
↓
double f(double a, double b, double c, double d, double e) {
        double r3913296 = c;
        double r3913297 = exp(r3913296);
        double r3913298 = e;
        double r3913299 = exp(r3913298);
        double r3913300 = d;
        double r3913301 = exp(r3913300);
        double r3913302 = r3913299 * r3913301;
        double r3913303 = b;
        double r3913304 = exp(r3913303);
        double r3913305 = a;
        double r3913306 = exp(r3913305);
        double r3913307 = r3913304 * r3913306;
        double r3913308 = r3913302 * r3913307;
        double r3913309 = r3913297 * r3913308;
        double r3913310 = log(r3913309);
        return r3913310;
}

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

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

  5. Applied add-log-exp0.3

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

  6. Applied add-log-exp0.3

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

  7. Applied add-log-exp0.3

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

  8. Applied sum-log0.3

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

  9. Applied add-log-exp0.3

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

  10. Applied add-log-exp0.3

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

  11. Applied sum-log0.3

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

  12. Applied sum-log0.2

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

  13. Applied sum-log0.0

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

  14. Simplified0.3

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

  16. Applied add-log-exp0.3

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

  17. Applied add-log-exp0.3

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

  18. Applied add-log-exp0.3

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

  19. Applied sum-log0.3

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

  20. Applied add-log-exp0.3

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

  21. Applied add-log-exp0.3

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

  22. Applied sum-log0.3

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

  23. Applied sum-log0.2

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

  24. Applied sum-log0.0

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

  25. Applied rem-exp-log0.0

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

  26. 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 2019172 +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))