Average Error: 0.4 → 0.3
Time: 37.0s
Precision: 64
\[1 \le a \le 2 \le b \le 4 \le c \le 8 \le d \le 16 \le e \le 32\]
\[\left(\left(\left(e + d\right) + c\right) + b\right) + a\]
↓
\[\log \left(e^{\log \left(e^{\left(c + a\right) + \left(e + \left(d + b\right)\right)}\right)}\right)\]
\left(\left(\left(e + d\right) + c\right) + b\right) + a
↓
\log \left(e^{\log \left(e^{\left(c + a\right) + \left(e + \left(d + b\right)\right)}\right)}\right)
double f(double a, double b, double c, double d, double e) {
        double r14636331 = e;
        double r14636332 = d;
        double r14636333 = r14636331 + r14636332;
        double r14636334 = c;
        double r14636335 = r14636333 + r14636334;
        double r14636336 = b;
        double r14636337 = r14636335 + r14636336;
        double r14636338 = a;
        double r14636339 = r14636337 + r14636338;
        return r14636339;
}
↓
double f(double a, double b, double c, double d, double e) {
        double r14636340 = c;
        double r14636341 = a;
        double r14636342 = r14636340 + r14636341;
        double r14636343 = e;
        double r14636344 = d;
        double r14636345 = b;
        double r14636346 = r14636344 + r14636345;
        double r14636347 = r14636343 + r14636346;
        double r14636348 = r14636342 + r14636347;
        double r14636349 = exp(r14636348);
        double r14636350 = log(r14636349);
        double r14636351 = exp(r14636350);
        double r14636352 = log(r14636351);
        return r14636352;
}

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
Herbie0.3
\[\left(d + \left(c + \left(a + b\right)\right)\right) + e\]

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

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

  13. Applied add-log-exp0.3

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

  14. Applied add-log-exp0.3

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

  15. Applied sum-log0.3

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

  16. Applied sum-log0.1

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

  17. Simplified0.3

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

  18. Final simplification0.3

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

Reproduce

herbie shell --seed 2019121 
(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))