Average Error: 28.9 → 0.1
Time: 6.0s
Precision: binary64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -1948.3925254814726:\\ \;\;\;\;\log \left(e^{\frac{2}{1 + e^{-2 \cdot x}}}\right) - 1\\ \mathbf{elif}\;-2 \cdot x \leq 0.001126977468769467:\\ \;\;\;\;\left(x - 0.3333333333333333 \cdot {x}^{3}\right) + 0.13333333333333333 \cdot {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(\sqrt{e^{\frac{2}{1 + e^{-2 \cdot x}}}}\right) \cdot \log \left(\sqrt{e^{\frac{2}{1 + e^{-2 \cdot x}}}}\right) - \left(-1 + \frac{2}{1 + e^{-2 \cdot x}} \cdot 0.5\right) \cdot \left(-1 + \frac{2}{1 + e^{-2 \cdot x}} \cdot 0.5\right)}{\log \left(\sqrt{e^{\frac{2}{1 + e^{-2 \cdot x}}}}\right) - \left(-1 + \frac{2}{1 + e^{-2 \cdot x}} \cdot 0.5\right)}\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;-2 \cdot x \leq -1948.3925254814726:\\
\;\;\;\;\log \left(e^{\frac{2}{1 + e^{-2 \cdot x}}}\right) - 1\\

\mathbf{elif}\;-2 \cdot x \leq 0.001126977468769467:\\
\;\;\;\;\left(x - 0.3333333333333333 \cdot {x}^{3}\right) + 0.13333333333333333 \cdot {x}^{5}\\

\mathbf{else}:\\
\;\;\;\;\frac{\log \left(\sqrt{e^{\frac{2}{1 + e^{-2 \cdot x}}}}\right) \cdot \log \left(\sqrt{e^{\frac{2}{1 + e^{-2 \cdot x}}}}\right) - \left(-1 + \frac{2}{1 + e^{-2 \cdot x}} \cdot 0.5\right) \cdot \left(-1 + \frac{2}{1 + e^{-2 \cdot x}} \cdot 0.5\right)}{\log \left(\sqrt{e^{\frac{2}{1 + e^{-2 \cdot x}}}}\right) - \left(-1 + \frac{2}{1 + e^{-2 \cdot x}} \cdot 0.5\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) -1948.3925254814726)
   (- (log (exp (/ 2.0 (+ 1.0 (exp (* -2.0 x)))))) 1.0)
   (if (<= (* -2.0 x) 0.001126977468769467)
     (+
      (- x (* 0.3333333333333333 (pow x 3.0)))
      (* 0.13333333333333333 (pow x 5.0)))
     (/
      (-
       (*
        (log (sqrt (exp (/ 2.0 (+ 1.0 (exp (* -2.0 x)))))))
        (log (sqrt (exp (/ 2.0 (+ 1.0 (exp (* -2.0 x))))))))
       (*
        (+ -1.0 (* (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 0.5))
        (+ -1.0 (* (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 0.5))))
      (-
       (log (sqrt (exp (/ 2.0 (+ 1.0 (exp (* -2.0 x)))))))
       (+ -1.0 (* (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 0.5)))))))
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) <= -1948.3925254814726) {
		tmp = log(exp(2.0 / (1.0 + exp(-2.0 * x)))) - 1.0;
	} else if ((-2.0 * x) <= 0.001126977468769467) {
		tmp = (x - (0.3333333333333333 * pow(x, 3.0))) + (0.13333333333333333 * pow(x, 5.0));
	} else {
		tmp = ((log(sqrt(exp(2.0 / (1.0 + exp(-2.0 * x))))) * log(sqrt(exp(2.0 / (1.0 + exp(-2.0 * x)))))) - ((-1.0 + ((2.0 / (1.0 + exp(-2.0 * x))) * 0.5)) * (-1.0 + ((2.0 / (1.0 + exp(-2.0 * x))) * 0.5)))) / (log(sqrt(exp(2.0 / (1.0 + exp(-2.0 * x))))) - (-1.0 + ((2.0 / (1.0 + exp(-2.0 * x))) * 0.5)));
	}
	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) < -1948.3925254814726

    1. Initial program 0

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

    3. Applied add-log-exp_binary64_7990

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

    4. Simplified0

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

    if -1948.3925254814726 < (*.f64 -2 x) < 0.00112697746876946696

    1. Initial program 58.7

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

    2. Taylor expanded around 0 0.3

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

    3. Simplified0.3

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

    if 0.00112697746876946696 < (*.f64 -2 x)

    1. Initial program 0.1

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

    3. Applied add-log-exp_binary64_7990.1

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

    4. Simplified0.1

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

    6. Applied add-sqr-sqrt_binary64_7820.1

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

    7. Applied log-prod_binary64_8460.1

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

    8. Applied associate--l+_binary64_6970.1

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

    9. Simplified0.1

      \[\leadsto \log \left(\sqrt{e^{\frac{2}{1 + e^{x \cdot -2}}}}\right) + \color{blue}{\left(-1 + 0.5 \cdot \frac{2}{e^{x \cdot -2} + 1}\right)}\]
    10. Using strategy rm

    11. Applied flip-+_binary64_7340.1

      \[\leadsto \color{blue}{\frac{\log \left(\sqrt{e^{\frac{2}{1 + e^{x \cdot -2}}}}\right) \cdot \log \left(\sqrt{e^{\frac{2}{1 + e^{x \cdot -2}}}}\right) - \left(-1 + 0.5 \cdot \frac{2}{e^{x \cdot -2} + 1}\right) \cdot \left(-1 + 0.5 \cdot \frac{2}{e^{x \cdot -2} + 1}\right)}{\log \left(\sqrt{e^{\frac{2}{1 + e^{x \cdot -2}}}}\right) - \left(-1 + 0.5 \cdot \frac{2}{e^{x \cdot -2} + 1}\right)}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -1948.3925254814726:\\ \;\;\;\;\log \left(e^{\frac{2}{1 + e^{-2 \cdot x}}}\right) - 1\\ \mathbf{elif}\;-2 \cdot x \leq 0.001126977468769467:\\ \;\;\;\;\left(x - 0.3333333333333333 \cdot {x}^{3}\right) + 0.13333333333333333 \cdot {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(\sqrt{e^{\frac{2}{1 + e^{-2 \cdot x}}}}\right) \cdot \log \left(\sqrt{e^{\frac{2}{1 + e^{-2 \cdot x}}}}\right) - \left(-1 + \frac{2}{1 + e^{-2 \cdot x}} \cdot 0.5\right) \cdot \left(-1 + \frac{2}{1 + e^{-2 \cdot x}} \cdot 0.5\right)}{\log \left(\sqrt{e^{\frac{2}{1 + e^{-2 \cdot x}}}}\right) - \left(-1 + \frac{2}{1 + e^{-2 \cdot x}} \cdot 0.5\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2021098 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))