Average Error: 0.4 → 0.0
Time: 9.1s
Precision: 64
\[1 \le a \le 2 \le b \le 4 \le c \le 8 \le d \le 16 \le e \le 32\]
\[\left(\left(\left(e + d\right) + c\right) + b\right) + a\]
↓
\[\log \left(\left(e^{d} \cdot e^{c}\right) \cdot \left(\left(e^{a} \cdot e^{b}\right) \cdot e^{e}\right)\right)\]
\left(\left(\left(e + d\right) + c\right) + b\right) + a
↓
\log \left(\left(e^{d} \cdot e^{c}\right) \cdot \left(\left(e^{a} \cdot e^{b}\right) \cdot e^{e}\right)\right)
double f(double a, double b, double c, double d, double e) {
        double r65688 = e;
        double r65689 = d;
        double r65690 = r65688 + r65689;
        double r65691 = c;
        double r65692 = r65690 + r65691;
        double r65693 = b;
        double r65694 = r65692 + r65693;
        double r65695 = a;
        double r65696 = r65694 + r65695;
        return r65696;
}
↓
double f(double a, double b, double c, double d, double e) {
        double r65697 = d;
        double r65698 = exp(r65697);
        double r65699 = c;
        double r65700 = exp(r65699);
        double r65701 = r65698 * r65700;
        double r65702 = a;
        double r65703 = exp(r65702);
        double r65704 = b;
        double r65705 = exp(r65704);
        double r65706 = r65703 * r65705;
        double r65707 = e;
        double r65708 = exp(r65707);
        double r65709 = r65706 * r65708;
        double r65710 = r65701 * r65709;
        double r65711 = log(r65710);
        return r65711;
}

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. Simplified0.3

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

  4. Applied associate-+r+0.3

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

  6. Applied add-log-exp0.3

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

  7. Applied add-log-exp0.3

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

  8. Applied sum-log0.3

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

  9. Applied add-log-exp0.3

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

  10. Applied add-log-exp0.3

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

  11. Applied add-log-exp0.3

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

  12. Applied sum-log0.3

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

  13. Applied sum-log0.3

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

  14. Applied sum-log0.0

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

  15. Simplified0.3

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

  17. Applied add-log-exp0.3

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

  18. Applied add-log-exp0.3

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

  19. Applied sum-log0.3

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

  20. Applied add-log-exp0.3

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

  21. Applied sum-log0.2

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

  22. Applied add-log-exp0.2

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

  23. Applied add-log-exp0.2

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

  24. Applied sum-log0.2

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

  25. Applied sum-log0.0

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

  26. Applied rem-exp-log0.0

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

  27. Final simplification0.0

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

Reproduce

herbie shell --seed 2019196 
(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))