Average Error: 0.4 → 0.0
Time: 14.0s
Precision: 64
\[1 \le a \le 2 \le b \le 4 \le c \le 8 \le d \le 16 \le e \le 32\]
\[\left(\left(\left(e + d\right) + c\right) + b\right) + a\]
↓
\[\log \left(e^{e} \cdot \left(\left(e^{a} \cdot e^{b}\right) \cdot \left(e^{d} \cdot e^{c}\right)\right)\right)\]
\left(\left(\left(e + d\right) + c\right) + b\right) + a
↓
\log \left(e^{e} \cdot \left(\left(e^{a} \cdot e^{b}\right) \cdot \left(e^{d} \cdot e^{c}\right)\right)\right)
double f(double a, double b, double c, double d, double e) {
        double r4715416 = e;
        double r4715417 = d;
        double r4715418 = r4715416 + r4715417;
        double r4715419 = c;
        double r4715420 = r4715418 + r4715419;
        double r4715421 = b;
        double r4715422 = r4715420 + r4715421;
        double r4715423 = a;
        double r4715424 = r4715422 + r4715423;
        return r4715424;
}
↓
double f(double a, double b, double c, double d, double e) {
        double r4715425 = e;
        double r4715426 = exp(r4715425);
        double r4715427 = a;
        double r4715428 = exp(r4715427);
        double r4715429 = b;
        double r4715430 = exp(r4715429);
        double r4715431 = r4715428 * r4715430;
        double r4715432 = d;
        double r4715433 = exp(r4715432);
        double r4715434 = c;
        double r4715435 = exp(r4715434);
        double r4715436 = r4715433 * r4715435;
        double r4715437 = r4715431 * r4715436;
        double r4715438 = r4715426 * r4715437;
        double r4715439 = log(r4715438);
        return r4715439;
}

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

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

  5. Applied add-log-exp0.3

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

  6. Applied add-log-exp0.3

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

  7. Applied sum-log0.3

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

  8. Applied add-log-exp0.3

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

  9. Applied add-log-exp0.3

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

  10. Applied add-log-exp0.3

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

  11. Applied sum-log0.3

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

  12. Applied sum-log0.3

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

  13. Applied sum-log0.0

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

  14. Simplified0.3

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

  16. Applied add-log-exp0.3

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

  17. Applied add-log-exp0.3

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

  18. Applied add-log-exp0.3

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

  19. Applied sum-log0.3

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

  20. Applied add-log-exp0.3

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

  21. Applied add-log-exp0.3

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

  22. Applied sum-log0.3

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

  23. Applied sum-log0.2

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

  24. Applied sum-log0.0

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

  25. Applied rem-exp-log0.0

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

  26. Final simplification0.0

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

Reproduce

herbie shell --seed 2019170 +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))