Logistic function from Lakshay Garg

Average Error: 29.4 → 0.1

Time: 24.2s

Precision: 64

• unused variable y

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \le -0.006660654438723100610741933991221230826341:\\ \;\;\;\;\left(\sqrt[3]{\log \left(\sqrt{e^{\frac{2}{1 + e^{-2 \cdot x}} - 1}}\right)} \cdot \sqrt[3]{\log \left(\sqrt{e^{\frac{2}{1 + e^{-2 \cdot x}} - 1}}\right)}\right) \cdot \sqrt[3]{\log \left(\sqrt{e^{\frac{2}{1 + e^{-2 \cdot x}} - 1}}\right)} + \log \left(\sqrt{e^{\frac{2}{1 + e^{-2 \cdot x}} - 1}}\right)\\ \mathbf{elif}\;-2 \cdot x \le 5.083912517384328794752684776980800052115 \cdot 10^{-5}:\\ \;\;\;\;1 \cdot x - {x}^{3} \cdot \left(0.3333333333333333703407674875052180141211 + x \cdot 5.5511151231257827021181583404541015625 \cdot 10^{-17}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{e^{\left(\frac{\sqrt{2}}{\sqrt{1 + e^{-2 \cdot x}}} + \sqrt{1}\right) \cdot \left(\frac{\sqrt{2}}{\sqrt{1 + e^{-2 \cdot x}}} - \sqrt{1}\right)}}\right) + \log \left(\sqrt{e^{\frac{2}{1 + e^{-2 \cdot x}} - 1}}\right)\\ \end{array}\]

C

double f(double x, double __attribute__((unused)) y) {
        double r60944 = 2.0;
        double r60945 = 1.0;
        double r60946 = -2.0;
        double r60947 = x;
        double r60948 = r60946 * r60947;
        double r60949 = exp(r60948);
        double r60950 = r60945 + r60949;
        double r60951 = r60944 / r60950;
        double r60952 = r60951 - r60945;
        return r60952;
}

↓

double f(double x, double __attribute__((unused)) y) {
        double r60953 = -2.0;
        double r60954 = x;
        double r60955 = r60953 * r60954;
        double r60956 = -0.006660654438723101;
        bool r60957 = r60955 <= r60956;
        double r60958 = 2.0;
        double r60959 = 1.0;
        double r60960 = exp(r60955);
        double r60961 = r60959 + r60960;
        double r60962 = r60958 / r60961;
        double r60963 = r60962 - r60959;
        double r60964 = exp(r60963);
        double r60965 = sqrt(r60964);
        double r60966 = log(r60965);
        double r60967 = cbrt(r60966);
        double r60968 = r60967 * r60967;
        double r60969 = r60968 * r60967;
        double r60970 = r60969 + r60966;
        double r60971 = 5.083912517384329e-05;
        bool r60972 = r60955 <= r60971;
        double r60973 = r60959 * r60954;
        double r60974 = 3.0;
        double r60975 = pow(r60954, r60974);
        double r60976 = 0.33333333333333337;
        double r60977 = 5.551115123125783e-17;
        double r60978 = r60954 * r60977;
        double r60979 = r60976 + r60978;
        double r60980 = r60975 * r60979;
        double r60981 = r60973 - r60980;
        double r60982 = sqrt(r60958);
        double r60983 = sqrt(r60961);
        double r60984 = r60982 / r60983;
        double r60985 = sqrt(r60959);
        double r60986 = r60984 + r60985;
        double r60987 = r60984 - r60985;
        double r60988 = r60986 * r60987;
        double r60989 = exp(r60988);
        double r60990 = sqrt(r60989);
        double r60991 = log(r60990);
        double r60992 = r60991 + r60966;
        double r60993 = r60972 ? r60981 : r60992;
        double r60994 = r60957 ? r60970 : r60993;
        return r60994;
}

Error

Bits error versus x

Bits error versus y

Derivation

1. Split input into 3 regimes

2. if (* -2.0 x) < -0.006660654438723101

   1. Initial program 0.0

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

   3. Applied add-log-exp 0.0

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

   4. Applied add-log-exp 0.0

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

   5. Applied diff-log 0.0

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

   6. Simplified 0.0

      \[\leadsto \log \color{blue}{\left(e^{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right)}\]
   7. Using strategy rm

   8. Applied add-sqr-sqrt 0.0

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

   9. Applied log-prod 0.0

      \[\leadsto \color{blue}{\log \left(\sqrt{e^{\frac{2}{1 + e^{-2 \cdot x}} - 1}}\right) + \log \left(\sqrt{e^{\frac{2}{1 + e^{-2 \cdot x}} - 1}}\right)}\]
   10. Using strategy rm

   11. Applied add-sqr-sqrt 0.0

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

   12. Applied add-sqr-sqrt 0.0

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

   13. Applied add-sqr-sqrt 0.0

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

   14. Applied times-frac 0.0

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

   15. Applied difference-of-squares 0.0

       \[\leadsto \log \left(\sqrt{e^{\color{blue}{\left(\frac{\sqrt{2}}{\sqrt{1 + e^{-2 \cdot x}}} + \sqrt{1}\right) \cdot \left(\frac{\sqrt{2}}{\sqrt{1 + e^{-2 \cdot x}}} - \sqrt{1}\right)}}}\right) + \log \left(\sqrt{e^{\frac{2}{1 + e^{-2 \cdot x}} - 1}}\right)\]
   16. Using strategy rm

   17. Applied add-cube-cbrt 0.0

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

   18. Simplified 0.0

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

   19. Simplified 0.0

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

   if -0.006660654438723101 < (* -2.0 x) < 5.083912517384329e-05

   1. Initial program 59.2

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

   3. Applied add-log-exp 59.2

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

   4. Applied add-log-exp 59.2

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

   5. Applied diff-log 59.2

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

   6. Simplified 59.2

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

   7. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{1 \cdot x - \left(5.5511151231257827021181583404541015625 \cdot 10^{-17} \cdot {x}^{4} + 0.3333333333333333703407674875052180141211 \cdot {x}^{3}\right)}\]

   8. Simplified 0.0

      \[\leadsto \color{blue}{1 \cdot x - {x}^{3} \cdot \left(0.3333333333333333703407674875052180141211 + x \cdot 5.5511151231257827021181583404541015625 \cdot 10^{-17}\right)}\]

   if 5.083912517384329e-05 < (* -2.0 x)

   1. Initial program 0.1

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

   3. Applied add-log-exp 0.1

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

   4. Applied add-log-exp 0.1

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

   5. Applied diff-log 0.1

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

   6. Simplified 0.1

      \[\leadsto \log \color{blue}{\left(e^{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right)}\]
   7. Using strategy rm

   8. Applied add-sqr-sqrt 0.1

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

   9. Applied log-prod 0.1

      \[\leadsto \color{blue}{\log \left(\sqrt{e^{\frac{2}{1 + e^{-2 \cdot x}} - 1}}\right) + \log \left(\sqrt{e^{\frac{2}{1 + e^{-2 \cdot x}} - 1}}\right)}\]
   10. Using strategy rm

   11. Applied add-sqr-sqrt 0.1

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

   12. Applied add-sqr-sqrt 0.1

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

   13. Applied add-sqr-sqrt 0.1

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

   14. Applied times-frac 0.1

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

   15. Applied difference-of-squares 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \le -0.006660654438723100610741933991221230826341:\\ \;\;\;\;\left(\sqrt[3]{\log \left(\sqrt{e^{\frac{2}{1 + e^{-2 \cdot x}} - 1}}\right)} \cdot \sqrt[3]{\log \left(\sqrt{e^{\frac{2}{1 + e^{-2 \cdot x}} - 1}}\right)}\right) \cdot \sqrt[3]{\log \left(\sqrt{e^{\frac{2}{1 + e^{-2 \cdot x}} - 1}}\right)} + \log \left(\sqrt{e^{\frac{2}{1 + e^{-2 \cdot x}} - 1}}\right)\\ \mathbf{elif}\;-2 \cdot x \le 5.083912517384328794752684776980800052115 \cdot 10^{-5}:\\ \;\;\;\;1 \cdot x - {x}^{3} \cdot \left(0.3333333333333333703407674875052180141211 + x \cdot 5.5511151231257827021181583404541015625 \cdot 10^{-17}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{e^{\left(\frac{\sqrt{2}}{\sqrt{1 + e^{-2 \cdot x}}} + \sqrt{1}\right) \cdot \left(\frac{\sqrt{2}}{\sqrt{1 + e^{-2 \cdot x}}} - \sqrt{1}\right)}}\right) + \log \left(\sqrt{e^{\frac{2}{1 + e^{-2 \cdot x}} - 1}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))