Average Error: 0.4 → 0.0
Time: 27.3s
Precision: 64
\[1 \le a \le 2 \le b \le 4 \le c \le 8 \le d \le 16 \le e \le 32\]
\[\left(\left(\left(e + d\right) + c\right) + b\right) + a\]
↓
\[\log \left(\left(e^{a} \cdot e^{e}\right) \cdot \left(\left(e^{d} \cdot e^{c}\right) \cdot e^{b}\right)\right)\]
\left(\left(\left(e + d\right) + c\right) + b\right) + a
↓
\log \left(\left(e^{a} \cdot e^{e}\right) \cdot \left(\left(e^{d} \cdot e^{c}\right) \cdot e^{b}\right)\right)
double f(double a, double b, double c, double d, double e) {
        double r3977119 = e;
        double r3977120 = d;
        double r3977121 = r3977119 + r3977120;
        double r3977122 = c;
        double r3977123 = r3977121 + r3977122;
        double r3977124 = b;
        double r3977125 = r3977123 + r3977124;
        double r3977126 = a;
        double r3977127 = r3977125 + r3977126;
        return r3977127;
}
↓
double f(double a, double b, double c, double d, double e) {
        double r3977128 = a;
        double r3977129 = exp(r3977128);
        double r3977130 = e;
        double r3977131 = exp(r3977130);
        double r3977132 = r3977129 * r3977131;
        double r3977133 = d;
        double r3977134 = exp(r3977133);
        double r3977135 = c;
        double r3977136 = exp(r3977135);
        double r3977137 = r3977134 * r3977136;
        double r3977138 = b;
        double r3977139 = exp(r3977138);
        double r3977140 = r3977137 * r3977139;
        double r3977141 = r3977132 * r3977140;
        double r3977142 = log(r3977141);
        return r3977142;
}

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

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

  6. Applied sum-log0.4

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

  7. Applied sum-log0.3

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

  8. Simplified0.3

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

  10. Applied add-log-exp0.3

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

  11. Applied add-log-exp0.3

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

  12. Applied sum-log0.3

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

  13. Applied add-log-exp0.3

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

  14. Applied add-log-exp0.3

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

  15. Applied sum-log0.3

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

  16. Applied add-log-exp0.3

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

  17. Applied sum-log0.2

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

  18. Applied sum-log0.0

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

  19. Applied rem-exp-log0.0

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

  20. Final simplification0.0

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

Reproduce

herbie shell --seed 2019144 +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))