Logistic function from Lakshay Garg

Average Error: 29.6 → 0.0

Time: 8.6s

Precision: 64

• unused variable y

Math

\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le -8.90063218864884623 \cdot 10^{-4}:\\ \;\;\;\;\frac{\frac{\frac{\frac{2}{\sqrt[3]{1 + e^{-2 \cdot x}} \cdot \sqrt[3]{1 + e^{-2 \cdot x}}}}{\sqrt[3]{1 + e^{-2 \cdot x}}}}{\sqrt[3]{1 + e^{-2 \cdot x}}} \cdot \frac{2}{\sqrt[3]{1 + e^{-2 \cdot x}} \cdot \sqrt[3]{1 + e^{-2 \cdot x}}} - 1 \cdot 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\\ \mathbf{elif}\;x \le 0.0012525288555290748:\\ \;\;\;\;1 \cdot x - \left(5.55112 \cdot 10^{-17} \cdot {x}^{4} + 0.33333333333333337 \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{\frac{\frac{\frac{2}{\sqrt[3]{1 + e^{-2 \cdot x}} \cdot \sqrt[3]{1 + e^{-2 \cdot x}}}}{\sqrt[3]{1 + e^{-2 \cdot x}}}}{\sqrt[3]{1 + e^{-2 \cdot x}}} \cdot \frac{2}{\sqrt[3]{1 + e^{-2 \cdot x}} \cdot \sqrt[3]{1 + e^{-2 \cdot x}}} - 1 \cdot 1}\right)}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\\ \end{array}\]

C

double f(double x, double __attribute__((unused)) y) {
        double r66969 = 2.0;
        double r66970 = 1.0;
        double r66971 = -2.0;
        double r66972 = x;
        double r66973 = r66971 * r66972;
        double r66974 = exp(r66973);
        double r66975 = r66970 + r66974;
        double r66976 = r66969 / r66975;
        double r66977 = r66976 - r66970;
        return r66977;
}

↓

double f(double x, double __attribute__((unused)) y) {
        double r66978 = x;
        double r66979 = -0.0008900632188648846;
        bool r66980 = r66978 <= r66979;
        double r66981 = 2.0;
        double r66982 = 1.0;
        double r66983 = -2.0;
        double r66984 = r66983 * r66978;
        double r66985 = exp(r66984);
        double r66986 = r66982 + r66985;
        double r66987 = cbrt(r66986);
        double r66988 = r66987 * r66987;
        double r66989 = r66981 / r66988;
        double r66990 = r66989 / r66987;
        double r66991 = r66990 / r66987;
        double r66992 = r66991 * r66989;
        double r66993 = r66982 * r66982;
        double r66994 = r66992 - r66993;
        double r66995 = r66981 / r66986;
        double r66996 = r66995 + r66982;
        double r66997 = r66994 / r66996;
        double r66998 = 0.0012525288555290748;
        bool r66999 = r66978 <= r66998;
        double r67000 = r66982 * r66978;
        double r67001 = 5.551115123125783e-17;
        double r67002 = 4.0;
        double r67003 = pow(r66978, r67002);
        double r67004 = r67001 * r67003;
        double r67005 = 0.33333333333333337;
        double r67006 = 3.0;
        double r67007 = pow(r66978, r67006);
        double r67008 = r67005 * r67007;
        double r67009 = r67004 + r67008;
        double r67010 = r67000 - r67009;
        double r67011 = exp(r66994);
        double r67012 = log(r67011);
        double r67013 = r67012 / r66996;
        double r67014 = r66999 ? r67010 : r67013;
        double r67015 = r66980 ? r66997 : r67014;
        return r67015;
}

Error

Bits error versus x

Bits error versus y

Derivation

1. Split input into 3 regimes

2. if x < -0.0008900632188648846

   1. Initial program 0.0

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
   2. Using strategy rm

   3. Applied flip-- 0.0

      \[\leadsto \color{blue}{\frac{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}} - 1 \cdot 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}}\]
   4. Using strategy rm

   5. Applied add-cube-cbrt 0.1

      \[\leadsto \frac{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{\color{blue}{\left(\sqrt[3]{1 + e^{-2 \cdot x}} \cdot \sqrt[3]{1 + e^{-2 \cdot x}}\right) \cdot \sqrt[3]{1 + e^{-2 \cdot x}}}} - 1 \cdot 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\]

   6. Applied add-sqr-sqrt 0.1

      \[\leadsto \frac{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{\color{blue}{\sqrt{2} \cdot \sqrt{2}}}{\left(\sqrt[3]{1 + e^{-2 \cdot x}} \cdot \sqrt[3]{1 + e^{-2 \cdot x}}\right) \cdot \sqrt[3]{1 + e^{-2 \cdot x}}} - 1 \cdot 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\]

   7. Applied times-frac 0.1

      \[\leadsto \frac{\frac{2}{1 + e^{-2 \cdot x}} \cdot \color{blue}{\left(\frac{\sqrt{2}}{\sqrt[3]{1 + e^{-2 \cdot x}} \cdot \sqrt[3]{1 + e^{-2 \cdot x}}} \cdot \frac{\sqrt{2}}{\sqrt[3]{1 + e^{-2 \cdot x}}}\right)} - 1 \cdot 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\]

   8. Applied add-cube-cbrt 0.1

      \[\leadsto \frac{\frac{2}{\color{blue}{\left(\sqrt[3]{1 + e^{-2 \cdot x}} \cdot \sqrt[3]{1 + e^{-2 \cdot x}}\right) \cdot \sqrt[3]{1 + e^{-2 \cdot x}}}} \cdot \left(\frac{\sqrt{2}}{\sqrt[3]{1 + e^{-2 \cdot x}} \cdot \sqrt[3]{1 + e^{-2 \cdot x}}} \cdot \frac{\sqrt{2}}{\sqrt[3]{1 + e^{-2 \cdot x}}}\right) - 1 \cdot 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\]

   9. Applied add-sqr-sqrt 0.1

      \[\leadsto \frac{\frac{\color{blue}{\sqrt{2} \cdot \sqrt{2}}}{\left(\sqrt[3]{1 + e^{-2 \cdot x}} \cdot \sqrt[3]{1 + e^{-2 \cdot x}}\right) \cdot \sqrt[3]{1 + e^{-2 \cdot x}}} \cdot \left(\frac{\sqrt{2}}{\sqrt[3]{1 + e^{-2 \cdot x}} \cdot \sqrt[3]{1 + e^{-2 \cdot x}}} \cdot \frac{\sqrt{2}}{\sqrt[3]{1 + e^{-2 \cdot x}}}\right) - 1 \cdot 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\]

   10. Applied times-frac 0.1

       \[\leadsto \frac{\color{blue}{\left(\frac{\sqrt{2}}{\sqrt[3]{1 + e^{-2 \cdot x}} \cdot \sqrt[3]{1 + e^{-2 \cdot x}}} \cdot \frac{\sqrt{2}}{\sqrt[3]{1 + e^{-2 \cdot x}}}\right)} \cdot \left(\frac{\sqrt{2}}{\sqrt[3]{1 + e^{-2 \cdot x}} \cdot \sqrt[3]{1 + e^{-2 \cdot x}}} \cdot \frac{\sqrt{2}}{\sqrt[3]{1 + e^{-2 \cdot x}}}\right) - 1 \cdot 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\]

   11. Applied swap-sqr 0.1

       \[\leadsto \frac{\color{blue}{\left(\frac{\sqrt{2}}{\sqrt[3]{1 + e^{-2 \cdot x}} \cdot \sqrt[3]{1 + e^{-2 \cdot x}}} \cdot \frac{\sqrt{2}}{\sqrt[3]{1 + e^{-2 \cdot x}} \cdot \sqrt[3]{1 + e^{-2 \cdot x}}}\right) \cdot \left(\frac{\sqrt{2}}{\sqrt[3]{1 + e^{-2 \cdot x}}} \cdot \frac{\sqrt{2}}{\sqrt[3]{1 + e^{-2 \cdot x}}}\right)} - 1 \cdot 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\]

   12. Simplified 0.1

       \[\leadsto \frac{\color{blue}{\frac{\frac{\frac{2}{\sqrt[3]{1 + e^{-2 \cdot x}} \cdot \sqrt[3]{1 + e^{-2 \cdot x}}}}{\sqrt[3]{1 + e^{-2 \cdot x}}}}{\sqrt[3]{1 + e^{-2 \cdot x}}}} \cdot \left(\frac{\sqrt{2}}{\sqrt[3]{1 + e^{-2 \cdot x}}} \cdot \frac{\sqrt{2}}{\sqrt[3]{1 + e^{-2 \cdot x}}}\right) - 1 \cdot 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\]

   13. Simplified 0.1

       \[\leadsto \frac{\frac{\frac{\frac{2}{\sqrt[3]{1 + e^{-2 \cdot x}} \cdot \sqrt[3]{1 + e^{-2 \cdot x}}}}{\sqrt[3]{1 + e^{-2 \cdot x}}}}{\sqrt[3]{1 + e^{-2 \cdot x}}} \cdot \color{blue}{\frac{2}{\sqrt[3]{1 + e^{-2 \cdot x}} \cdot \sqrt[3]{1 + e^{-2 \cdot x}}}} - 1 \cdot 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\]

   if -0.0008900632188648846 < x < 0.0012525288555290748

   1. Initial program 59.1

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1\]

   2. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{1 \cdot x - \left(5.55112 \cdot 10^{-17} \cdot {x}^{4} + 0.33333333333333337 \cdot {x}^{3}\right)}\]

   if 0.0012525288555290748 < x

   1. Initial program 0.1

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
   2. Using strategy rm

   3. Applied flip-- 0.1

      \[\leadsto \color{blue}{\frac{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}} - 1 \cdot 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}}\]
   4. Using strategy rm

   5. Applied add-cube-cbrt 0.1

      \[\leadsto \frac{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{\color{blue}{\left(\sqrt[3]{1 + e^{-2 \cdot x}} \cdot \sqrt[3]{1 + e^{-2 \cdot x}}\right) \cdot \sqrt[3]{1 + e^{-2 \cdot x}}}} - 1 \cdot 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\]

   6. Applied add-sqr-sqrt 1.1

      \[\leadsto \frac{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{\color{blue}{\sqrt{2} \cdot \sqrt{2}}}{\left(\sqrt[3]{1 + e^{-2 \cdot x}} \cdot \sqrt[3]{1 + e^{-2 \cdot x}}\right) \cdot \sqrt[3]{1 + e^{-2 \cdot x}}} - 1 \cdot 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\]

   7. Applied times-frac 1.1

      \[\leadsto \frac{\frac{2}{1 + e^{-2 \cdot x}} \cdot \color{blue}{\left(\frac{\sqrt{2}}{\sqrt[3]{1 + e^{-2 \cdot x}} \cdot \sqrt[3]{1 + e^{-2 \cdot x}}} \cdot \frac{\sqrt{2}}{\sqrt[3]{1 + e^{-2 \cdot x}}}\right)} - 1 \cdot 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\]

   8. Applied add-cube-cbrt 1.1

      \[\leadsto \frac{\frac{2}{\color{blue}{\left(\sqrt[3]{1 + e^{-2 \cdot x}} \cdot \sqrt[3]{1 + e^{-2 \cdot x}}\right) \cdot \sqrt[3]{1 + e^{-2 \cdot x}}}} \cdot \left(\frac{\sqrt{2}}{\sqrt[3]{1 + e^{-2 \cdot x}} \cdot \sqrt[3]{1 + e^{-2 \cdot x}}} \cdot \frac{\sqrt{2}}{\sqrt[3]{1 + e^{-2 \cdot x}}}\right) - 1 \cdot 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\]

   9. Applied add-sqr-sqrt 2.1

      \[\leadsto \frac{\frac{\color{blue}{\sqrt{2} \cdot \sqrt{2}}}{\left(\sqrt[3]{1 + e^{-2 \cdot x}} \cdot \sqrt[3]{1 + e^{-2 \cdot x}}\right) \cdot \sqrt[3]{1 + e^{-2 \cdot x}}} \cdot \left(\frac{\sqrt{2}}{\sqrt[3]{1 + e^{-2 \cdot x}} \cdot \sqrt[3]{1 + e^{-2 \cdot x}}} \cdot \frac{\sqrt{2}}{\sqrt[3]{1 + e^{-2 \cdot x}}}\right) - 1 \cdot 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\]

   10. Applied times-frac 2.0

       \[\leadsto \frac{\color{blue}{\left(\frac{\sqrt{2}}{\sqrt[3]{1 + e^{-2 \cdot x}} \cdot \sqrt[3]{1 + e^{-2 \cdot x}}} \cdot \frac{\sqrt{2}}{\sqrt[3]{1 + e^{-2 \cdot x}}}\right)} \cdot \left(\frac{\sqrt{2}}{\sqrt[3]{1 + e^{-2 \cdot x}} \cdot \sqrt[3]{1 + e^{-2 \cdot x}}} \cdot \frac{\sqrt{2}}{\sqrt[3]{1 + e^{-2 \cdot x}}}\right) - 1 \cdot 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\]

   11. Applied swap-sqr 2.1

       \[\leadsto \frac{\color{blue}{\left(\frac{\sqrt{2}}{\sqrt[3]{1 + e^{-2 \cdot x}} \cdot \sqrt[3]{1 + e^{-2 \cdot x}}} \cdot \frac{\sqrt{2}}{\sqrt[3]{1 + e^{-2 \cdot x}} \cdot \sqrt[3]{1 + e^{-2 \cdot x}}}\right) \cdot \left(\frac{\sqrt{2}}{\sqrt[3]{1 + e^{-2 \cdot x}}} \cdot \frac{\sqrt{2}}{\sqrt[3]{1 + e^{-2 \cdot x}}}\right)} - 1 \cdot 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\]

   12. Simplified 1.1

       \[\leadsto \frac{\color{blue}{\frac{\frac{\frac{2}{\sqrt[3]{1 + e^{-2 \cdot x}} \cdot \sqrt[3]{1 + e^{-2 \cdot x}}}}{\sqrt[3]{1 + e^{-2 \cdot x}}}}{\sqrt[3]{1 + e^{-2 \cdot x}}}} \cdot \left(\frac{\sqrt{2}}{\sqrt[3]{1 + e^{-2 \cdot x}}} \cdot \frac{\sqrt{2}}{\sqrt[3]{1 + e^{-2 \cdot x}}}\right) - 1 \cdot 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\]

   13. Simplified 0.1

       \[\leadsto \frac{\frac{\frac{\frac{2}{\sqrt[3]{1 + e^{-2 \cdot x}} \cdot \sqrt[3]{1 + e^{-2 \cdot x}}}}{\sqrt[3]{1 + e^{-2 \cdot x}}}}{\sqrt[3]{1 + e^{-2 \cdot x}}} \cdot \color{blue}{\frac{2}{\sqrt[3]{1 + e^{-2 \cdot x}} \cdot \sqrt[3]{1 + e^{-2 \cdot x}}}} - 1 \cdot 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\]
   14. Using strategy rm

   15. Applied add-log-exp 0.1

       \[\leadsto \frac{\frac{\frac{\frac{2}{\sqrt[3]{1 + e^{-2 \cdot x}} \cdot \sqrt[3]{1 + e^{-2 \cdot x}}}}{\sqrt[3]{1 + e^{-2 \cdot x}}}}{\sqrt[3]{1 + e^{-2 \cdot x}}} \cdot \frac{2}{\sqrt[3]{1 + e^{-2 \cdot x}} \cdot \sqrt[3]{1 + e^{-2 \cdot x}}} - \color{blue}{\log \left(e^{1 \cdot 1}\right)}}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\]

   16. Applied add-log-exp 0.1

       \[\leadsto \frac{\color{blue}{\log \left(e^{\frac{\frac{\frac{2}{\sqrt[3]{1 + e^{-2 \cdot x}} \cdot \sqrt[3]{1 + e^{-2 \cdot x}}}}{\sqrt[3]{1 + e^{-2 \cdot x}}}}{\sqrt[3]{1 + e^{-2 \cdot x}}} \cdot \frac{2}{\sqrt[3]{1 + e^{-2 \cdot x}} \cdot \sqrt[3]{1 + e^{-2 \cdot x}}}}\right)} - \log \left(e^{1 \cdot 1}\right)}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\]

   17. Applied diff-log 0.1

       \[\leadsto \frac{\color{blue}{\log \left(\frac{e^{\frac{\frac{\frac{2}{\sqrt[3]{1 + e^{-2 \cdot x}} \cdot \sqrt[3]{1 + e^{-2 \cdot x}}}}{\sqrt[3]{1 + e^{-2 \cdot x}}}}{\sqrt[3]{1 + e^{-2 \cdot x}}} \cdot \frac{2}{\sqrt[3]{1 + e^{-2 \cdot x}} \cdot \sqrt[3]{1 + e^{-2 \cdot x}}}}}{e^{1 \cdot 1}}\right)}}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\]

   18. Simplified 0.1

       \[\leadsto \frac{\log \color{blue}{\left(e^{\frac{\frac{\frac{2}{\sqrt[3]{1 + e^{-2 \cdot x}} \cdot \sqrt[3]{1 + e^{-2 \cdot x}}}}{\sqrt[3]{1 + e^{-2 \cdot x}}}}{\sqrt[3]{1 + e^{-2 \cdot x}}} \cdot \frac{2}{\sqrt[3]{1 + e^{-2 \cdot x}} \cdot \sqrt[3]{1 + e^{-2 \cdot x}}} - 1 \cdot 1}\right)}}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -8.90063218864884623 \cdot 10^{-4}:\\ \;\;\;\;\frac{\frac{\frac{\frac{2}{\sqrt[3]{1 + e^{-2 \cdot x}} \cdot \sqrt[3]{1 + e^{-2 \cdot x}}}}{\sqrt[3]{1 + e^{-2 \cdot x}}}}{\sqrt[3]{1 + e^{-2 \cdot x}}} \cdot \frac{2}{\sqrt[3]{1 + e^{-2 \cdot x}} \cdot \sqrt[3]{1 + e^{-2 \cdot x}}} - 1 \cdot 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\\ \mathbf{elif}\;x \le 0.0012525288555290748:\\ \;\;\;\;1 \cdot x - \left(5.55112 \cdot 10^{-17} \cdot {x}^{4} + 0.33333333333333337 \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{\frac{\frac{\frac{2}{\sqrt[3]{1 + e^{-2 \cdot x}} \cdot \sqrt[3]{1 + e^{-2 \cdot x}}}}{\sqrt[3]{1 + e^{-2 \cdot x}}}}{\sqrt[3]{1 + e^{-2 \cdot x}}} \cdot \frac{2}{\sqrt[3]{1 + e^{-2 \cdot x}} \cdot \sqrt[3]{1 + e^{-2 \cdot x}}} - 1 \cdot 1}\right)}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020034 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))