Average Error: 0.4 → 0.0
Time: 22.7s
Precision: 64
\[1 \le a \le 2 \le b \le 4 \le c \le 8 \le d \le 16 \le e \le 32\]
\[\left(\left(\left(e + d\right) + c\right) + b\right) + a\]
↓
\[\log \left(\left(e^{c} \cdot e^{d}\right) \cdot \left(e^{a} \cdot \left(e^{e} \cdot e^{b}\right)\right)\right)\]
\left(\left(\left(e + d\right) + c\right) + b\right) + a
↓
\log \left(\left(e^{c} \cdot e^{d}\right) \cdot \left(e^{a} \cdot \left(e^{e} \cdot e^{b}\right)\right)\right)
double f(double a, double b, double c, double d, double e) {
        double r4993581 = e;
        double r4993582 = d;
        double r4993583 = r4993581 + r4993582;
        double r4993584 = c;
        double r4993585 = r4993583 + r4993584;
        double r4993586 = b;
        double r4993587 = r4993585 + r4993586;
        double r4993588 = a;
        double r4993589 = r4993587 + r4993588;
        return r4993589;
}
↓
double f(double a, double b, double c, double d, double e) {
        double r4993590 = c;
        double r4993591 = exp(r4993590);
        double r4993592 = d;
        double r4993593 = exp(r4993592);
        double r4993594 = r4993591 * r4993593;
        double r4993595 = a;
        double r4993596 = exp(r4993595);
        double r4993597 = e;
        double r4993598 = exp(r4993597);
        double r4993599 = b;
        double r4993600 = exp(r4993599);
        double r4993601 = r4993598 * r4993600;
        double r4993602 = r4993596 * r4993601;
        double r4993603 = r4993594 * r4993602;
        double r4993604 = log(r4993603);
        return r4993604;
}

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-log0.3

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

  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

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

  16. Applied add-log-exp0.3

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

  17. Applied add-log-exp0.3

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

  18. Applied add-log-exp0.3

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

  19. Applied sum-log0.3

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

  20. Applied sum-log0.2

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

  21. Applied add-log-exp0.2

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

  22. Applied add-log-exp0.2

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

  23. Applied sum-log0.2

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

  24. Applied sum-log0.0

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

  25. Applied rem-exp-log0.0

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

  26. Final simplification0.0

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

Reproduce

herbie shell --seed 2019143 +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))