Average Error: 29.2 → 0.2
Time: 3.6s
Precision: binary64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
↓
\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.9011605904375583:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{fma}\left(x \cdot x, -0.5, x\right)\right)\\ \end{array} \]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;-2 \cdot x \leq -0.9011605904375583:\\
\;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\

\mathbf{else}:\\
\;\;\;\;\mathsf{expm1}\left(\mathsf{fma}\left(x \cdot x, -0.5, x\right)\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) -0.9011605904375583)
   (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0)
   (expm1 (fma (* x x) -0.5 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) <= -0.9011605904375583) {
		tmp = (2.0 / (1.0 + exp(-2.0 * x))) - 1.0;
	} else {
		tmp = expm1(fma((x * x), -0.5, x));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Split input into 2 regimes

  2. if (*.f64 -2 x) < -0.901160590437558251

    1. Initial program 0.0

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

    if -0.901160590437558251 < (*.f64 -2 x)

    1. Initial program 39.1

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

    2. Applied add-exp-log_binary6439.1

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

    3. Applied expm1-def_binary6439.1

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\log \left(\frac{2}{1 + e^{-2 \cdot x}}\right)\right)} \]

    4. Applied log-div_binary6439.1

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\log 2 - \log \left(1 + e^{-2 \cdot x}\right)}\right) \]

    5. Simplified39.0

      \[\leadsto \mathsf{expm1}\left(\log 2 - \color{blue}{\mathsf{log1p}\left(e^{x \cdot -2}\right)}\right) \]

    6. Taylor expanded in x around 0 0.3

      \[\leadsto \mathsf{expm1}\left(\color{blue}{x - 0.5 \cdot {x}^{2}}\right) \]

    7. Simplified0.3

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left(x \cdot x, -0.5, x\right)}\right) \]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.9011605904375583:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{fma}\left(x \cdot x, -0.5, x\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021280 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))