Average Error: 0.4 → 0.0
Time: 34.9s
Precision: 64
\[1 \le a \le 2 \le b \le 4 \le c \le 8 \le d \le 16 \le e \le 32\]
\[\left(\left(\left(e + d\right) + c\right) + b\right) + a\]
↓
\[\log \left(e^{d} \cdot \left(e^{e} \cdot \left(\left(e^{a} \cdot e^{c}\right) \cdot e^{b}\right)\right)\right)\]
\left(\left(\left(e + d\right) + c\right) + b\right) + a
↓
\log \left(e^{d} \cdot \left(e^{e} \cdot \left(\left(e^{a} \cdot e^{c}\right) \cdot e^{b}\right)\right)\right)
double f(double a, double b, double c, double d, double e) {
        double r11043927 = e;
        double r11043928 = d;
        double r11043929 = r11043927 + r11043928;
        double r11043930 = c;
        double r11043931 = r11043929 + r11043930;
        double r11043932 = b;
        double r11043933 = r11043931 + r11043932;
        double r11043934 = a;
        double r11043935 = r11043933 + r11043934;
        return r11043935;
}
↓
double f(double a, double b, double c, double d, double e) {
        double r11043936 = d;
        double r11043937 = exp(r11043936);
        double r11043938 = e;
        double r11043939 = exp(r11043938);
        double r11043940 = a;
        double r11043941 = exp(r11043940);
        double r11043942 = c;
        double r11043943 = exp(r11043942);
        double r11043944 = r11043941 * r11043943;
        double r11043945 = b;
        double r11043946 = exp(r11043945);
        double r11043947 = r11043944 * r11043946;
        double r11043948 = r11043939 * r11043947;
        double r11043949 = r11043937 * r11043948;
        double r11043950 = log(r11043949);
        return r11043950;
}

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

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

  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 associate-+l+0.3

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

  14. Applied add-log-exp0.3

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

  15. Applied add-log-exp0.3

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

  16. Applied add-log-exp0.3

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

  17. Applied sum-log0.3

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

  18. Applied sum-log0.3

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

  19. Applied add-log-exp0.3

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

  20. Applied sum-log0.2

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

  21. Applied add-log-exp0.2

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

  22. Applied sum-log0.0

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

  23. Applied rem-exp-log0.0

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

  24. Final simplification0.0

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

Reproduce

herbie shell --seed 2019104 
(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))