Average Error: 0.4 → 0.0
Time: 12.0s
Precision: 64
\[1 \le a \le 2 \le b \le 4 \le c \le 8 \le d \le 16 \le e \le 32\]
\[\left(\left(\left(e + d\right) + c\right) + b\right) + a\]
↓
\[\log \left(\left(e^{e} \cdot e^{d}\right) \cdot \left(e^{a} \cdot \left(e^{c} \cdot e^{b}\right)\right)\right)\]
\left(\left(\left(e + d\right) + c\right) + b\right) + a
↓
\log \left(\left(e^{e} \cdot e^{d}\right) \cdot \left(e^{a} \cdot \left(e^{c} \cdot e^{b}\right)\right)\right)
double f(double a, double b, double c, double d, double e) {
        double r4138682 = e;
        double r4138683 = d;
        double r4138684 = r4138682 + r4138683;
        double r4138685 = c;
        double r4138686 = r4138684 + r4138685;
        double r4138687 = b;
        double r4138688 = r4138686 + r4138687;
        double r4138689 = a;
        double r4138690 = r4138688 + r4138689;
        return r4138690;
}
↓
double f(double a, double b, double c, double d, double e) {
        double r4138691 = e;
        double r4138692 = exp(r4138691);
        double r4138693 = d;
        double r4138694 = exp(r4138693);
        double r4138695 = r4138692 * r4138694;
        double r4138696 = a;
        double r4138697 = exp(r4138696);
        double r4138698 = c;
        double r4138699 = exp(r4138698);
        double r4138700 = b;
        double r4138701 = exp(r4138700);
        double r4138702 = r4138699 * r4138701;
        double r4138703 = r4138697 * r4138702;
        double r4138704 = r4138695 * r4138703;
        double r4138705 = log(r4138704);
        return r4138705;
}

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. Simplified0.3

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

  14. Applied add-log-exp0.3

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

  15. Applied add-log-exp0.3

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

  16. Applied add-log-exp0.3

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

  17. Applied sum-log0.3

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

  18. Applied sum-log0.3

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

  19. Applied add-log-exp0.3

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

  20. Applied add-log-exp0.3

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

  21. Applied sum-log0.3

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

  22. Applied sum-log0.0

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

  23. Applied rem-exp-log0.0

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

  24. Final simplification0.0

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

Reproduce

herbie shell --seed 2019163 +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))