Average Error: 28.5 → 0.0
Time: 6.1s
Precision: binary64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.06147715793666644:\\ \;\;\;\;\sqrt{\frac{\frac{4}{{\left(1 + e^{-2 \cdot x}\right)}^{2}} - 1}{1 + \frac{2}{1 + e^{-2 \cdot x}}}} \cdot \log \left(e^{\sqrt{\frac{\frac{4}{{\left(1 + e^{-2 \cdot x}\right)}^{2}} - 1}{1 + \frac{2}{1 + e^{-2 \cdot x}}}}}\right)\\ \mathbf{elif}\;-2 \cdot x \leq 0.028158795437731143:\\ \;\;\;\;\left(x + 0.13333333333333333 \cdot {x}^{5}\right) - \left(0.3333333333333333 \cdot {x}^{3} + 0.05396825396825397 \cdot {x}^{7}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{\frac{4}{\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)} + -1}{1 + \frac{2}{1 + e^{-2 \cdot x}}}}\right)\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;-2 \cdot x \leq -0.06147715793666644:\\
\;\;\;\;\sqrt{\frac{\frac{4}{{\left(1 + e^{-2 \cdot x}\right)}^{2}} - 1}{1 + \frac{2}{1 + e^{-2 \cdot x}}}} \cdot \log \left(e^{\sqrt{\frac{\frac{4}{{\left(1 + e^{-2 \cdot x}\right)}^{2}} - 1}{1 + \frac{2}{1 + e^{-2 \cdot x}}}}}\right)\\

\mathbf{elif}\;-2 \cdot x \leq 0.028158795437731143:\\
\;\;\;\;\left(x + 0.13333333333333333 \cdot {x}^{5}\right) - \left(0.3333333333333333 \cdot {x}^{3} + 0.05396825396825397 \cdot {x}^{7}\right)\\

\mathbf{else}:\\
\;\;\;\;\log \left(e^{\frac{\frac{4}{\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)} + -1}{1 + \frac{2}{1 + e^{-2 \cdot x}}}}\right)\\

\end{array}
(FPCore (x y) :precision binary64 (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))
↓
(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -0.06147715793666644)
   (*
    (sqrt
     (/
      (- (/ 4.0 (pow (+ 1.0 (exp (* -2.0 x))) 2.0)) 1.0)
      (+ 1.0 (/ 2.0 (+ 1.0 (exp (* -2.0 x)))))))
    (log
     (exp
      (sqrt
       (/
        (- (/ 4.0 (pow (+ 1.0 (exp (* -2.0 x))) 2.0)) 1.0)
        (+ 1.0 (/ 2.0 (+ 1.0 (exp (* -2.0 x))))))))))
   (if (<= (* -2.0 x) 0.028158795437731143)
     (-
      (+ x (* 0.13333333333333333 (pow x 5.0)))
      (+
       (* 0.3333333333333333 (pow x 3.0))
       (* 0.05396825396825397 (pow x 7.0))))
     (log
      (exp
       (/
        (+ (/ 4.0 (* (+ 1.0 (exp (* -2.0 x))) (+ 1.0 (exp (* -2.0 x))))) -1.0)
        (+ 1.0 (/ 2.0 (+ 1.0 (exp (* -2.0 x)))))))))))
double code(double x, double y) {
	return (2.0 / (1.0 + exp(-2.0 * x))) - 1.0;
}
↓
double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -0.06147715793666644) {
		tmp = sqrt(((4.0 / pow((1.0 + exp(-2.0 * x)), 2.0)) - 1.0) / (1.0 + (2.0 / (1.0 + exp(-2.0 * x))))) * log(exp(sqrt(((4.0 / pow((1.0 + exp(-2.0 * x)), 2.0)) - 1.0) / (1.0 + (2.0 / (1.0 + exp(-2.0 * x)))))));
	} else if ((-2.0 * x) <= 0.028158795437731143) {
		tmp = (x + (0.13333333333333333 * pow(x, 5.0))) - ((0.3333333333333333 * pow(x, 3.0)) + (0.05396825396825397 * pow(x, 7.0)));
	} else {
		tmp = log(exp(((4.0 / ((1.0 + exp(-2.0 * x)) * (1.0 + exp(-2.0 * x)))) + -1.0) / (1.0 + (2.0 / (1.0 + exp(-2.0 * x))))));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if (*.f64 -2 x) < -0.061477157936666443

    1. Initial program 0.0

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

    3. Applied flip--_binary640.0

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

    4. Simplified0.0

      \[\leadsto \frac{\color{blue}{\frac{4}{\left(1 + e^{x \cdot -2}\right) \cdot \left(1 + e^{x \cdot -2}\right)} + -1}}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\]

    5. Simplified0.0

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

    7. Applied add-log-exp_binary640.0

      \[\leadsto \color{blue}{\log \left(e^{\frac{\frac{4}{\left(1 + e^{x \cdot -2}\right) \cdot \left(1 + e^{x \cdot -2}\right)} + -1}{1 + \frac{2}{1 + e^{x \cdot -2}}}}\right)}\]

    8. Simplified0.0

      \[\leadsto \log \color{blue}{\left(e^{\frac{\frac{4}{{\left(1 + e^{x \cdot -2}\right)}^{2}} - 1}{1 + \frac{2}{1 + e^{x \cdot -2}}}}\right)}\]
    9. Using strategy rm

    10. Applied add-sqr-sqrt_binary640.0

      \[\leadsto \log \left(e^{\color{blue}{\sqrt{\frac{\frac{4}{{\left(1 + e^{x \cdot -2}\right)}^{2}} - 1}{1 + \frac{2}{1 + e^{x \cdot -2}}}} \cdot \sqrt{\frac{\frac{4}{{\left(1 + e^{x \cdot -2}\right)}^{2}} - 1}{1 + \frac{2}{1 + e^{x \cdot -2}}}}}}\right)\]

    11. Applied exp-prod_binary640.0

      \[\leadsto \log \color{blue}{\left({\left(e^{\sqrt{\frac{\frac{4}{{\left(1 + e^{x \cdot -2}\right)}^{2}} - 1}{1 + \frac{2}{1 + e^{x \cdot -2}}}}}\right)}^{\left(\sqrt{\frac{\frac{4}{{\left(1 + e^{x \cdot -2}\right)}^{2}} - 1}{1 + \frac{2}{1 + e^{x \cdot -2}}}}\right)}\right)}\]

    12. Applied log-pow_binary640.0

      \[\leadsto \color{blue}{\sqrt{\frac{\frac{4}{{\left(1 + e^{x \cdot -2}\right)}^{2}} - 1}{1 + \frac{2}{1 + e^{x \cdot -2}}}} \cdot \log \left(e^{\sqrt{\frac{\frac{4}{{\left(1 + e^{x \cdot -2}\right)}^{2}} - 1}{1 + \frac{2}{1 + e^{x \cdot -2}}}}}\right)}\]

    if -0.061477157936666443 < (*.f64 -2 x) < 0.0281587954377311

    1. Initial program 59.0

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

    2. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{\left(x + 0.13333333333333333 \cdot {x}^{5}\right) - \left(0.05396825396825397 \cdot {x}^{7} + 0.3333333333333333 \cdot {x}^{3}\right)}\]

    3. Simplified0.0

      \[\leadsto \color{blue}{\left(x + 0.13333333333333333 \cdot {x}^{5}\right) - \left(0.3333333333333333 \cdot {x}^{3} + 0.05396825396825397 \cdot {x}^{7}\right)}\]

    if 0.0281587954377311 < (*.f64 -2 x)

    1. Initial program 0.0

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

    3. Applied flip--_binary640.0

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

    4. Simplified0.0

      \[\leadsto \frac{\color{blue}{\frac{4}{\left(1 + e^{x \cdot -2}\right) \cdot \left(1 + e^{x \cdot -2}\right)} + -1}}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\]

    5. Simplified0.0

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

    7. Applied add-log-exp_binary640.0

      \[\leadsto \color{blue}{\log \left(e^{\frac{\frac{4}{\left(1 + e^{x \cdot -2}\right) \cdot \left(1 + e^{x \cdot -2}\right)} + -1}{1 + \frac{2}{1 + e^{x \cdot -2}}}}\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.06147715793666644:\\ \;\;\;\;\sqrt{\frac{\frac{4}{{\left(1 + e^{-2 \cdot x}\right)}^{2}} - 1}{1 + \frac{2}{1 + e^{-2 \cdot x}}}} \cdot \log \left(e^{\sqrt{\frac{\frac{4}{{\left(1 + e^{-2 \cdot x}\right)}^{2}} - 1}{1 + \frac{2}{1 + e^{-2 \cdot x}}}}}\right)\\ \mathbf{elif}\;-2 \cdot x \leq 0.028158795437731143:\\ \;\;\;\;\left(x + 0.13333333333333333 \cdot {x}^{5}\right) - \left(0.3333333333333333 \cdot {x}^{3} + 0.05396825396825397 \cdot {x}^{7}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{\frac{4}{\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)} + -1}{1 + \frac{2}{1 + e^{-2 \cdot x}}}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021175 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))