Average Error: 0.4 → 0.0
Time: 19.0s
Precision: 64
\[1 \le a \le 2 \le b \le 4 \le c \le 8 \le d \le 16 \le e \le 32\]
\[\left(\left(\left(e + d\right) + c\right) + b\right) + a\]
↓
\[\log \left(\left(e^{a} \cdot e^{c}\right) \cdot \left(\left(e^{d} \cdot e^{e}\right) \cdot e^{b}\right)\right)\]
\left(\left(\left(e + d\right) + c\right) + b\right) + a
↓
\log \left(\left(e^{a} \cdot e^{c}\right) \cdot \left(\left(e^{d} \cdot e^{e}\right) \cdot e^{b}\right)\right)
double f(double a, double b, double c, double d, double e) {
        double r4087020 = e;
        double r4087021 = d;
        double r4087022 = r4087020 + r4087021;
        double r4087023 = c;
        double r4087024 = r4087022 + r4087023;
        double r4087025 = b;
        double r4087026 = r4087024 + r4087025;
        double r4087027 = a;
        double r4087028 = r4087026 + r4087027;
        return r4087028;
}
↓
double f(double a, double b, double c, double d, double e) {
        double r4087029 = a;
        double r4087030 = exp(r4087029);
        double r4087031 = c;
        double r4087032 = exp(r4087031);
        double r4087033 = r4087030 * r4087032;
        double r4087034 = d;
        double r4087035 = exp(r4087034);
        double r4087036 = e;
        double r4087037 = exp(r4087036);
        double r4087038 = r4087035 * r4087037;
        double r4087039 = b;
        double r4087040 = exp(r4087039);
        double r4087041 = r4087038 * r4087040;
        double r4087042 = r4087033 * r4087041;
        double r4087043 = log(r4087042);
        return r4087043;
}

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

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

  5. Applied sum-log0.4

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

  6. Simplified0.3

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

  8. Applied add-log-exp0.3

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

  9. Applied add-log-exp0.3

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

  10. Applied sum-log0.3

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

  11. Applied add-log-exp0.3

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

  12. Applied add-log-exp0.3

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

  13. Applied sum-log0.3

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

  14. Applied add-log-exp0.3

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

  15. Applied sum-log0.2

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

  16. Applied sum-log0.0

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

  17. Applied rem-exp-log0.0

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

  18. Final simplification0.0

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

Reproduce

herbie shell --seed 2019151 +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))