\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

↓

\[\frac{\frac{\mathsf{fma}\left(x, 2, 2\right)}{e^{x}}}{2} \]

\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}

↓

\frac{\frac{\mathsf{fma}\left(x, 2, 2\right)}{e^{x}}}{2}

(FPCore (x eps)
 :precision binary64
 (/
  (-
   (* (+ 1.0 (/ 1.0 eps)) (exp (- (* (- 1.0 eps) x))))
   (* (- (/ 1.0 eps) 1.0) (exp (- (* (+ 1.0 eps) x)))))
  2.0))

↓

(FPCore (x eps) :precision binary64 (/ (/ (fma x 2.0 2.0) (exp x)) 2.0))

double code(double x, double eps) {
	return (((1.0 + (1.0 / eps)) * exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * exp(-((1.0 + eps) * x)))) / 2.0;
}

↓

double code(double x, double eps) {
	return (fma(x, 2.0, 2.0) / exp(x)) / 2.0;
}

Derivation

Initial program 29.7
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
Taylor expanded in eps around 0 0.5
\[\leadsto \frac{\color{blue}{2 \cdot \left(x \cdot e^{-x}\right) + 2 \cdot e^{-x}}}{2} \]
Taylor expanded in x around -inf 0.5
\[\leadsto \frac{\color{blue}{2 \cdot e^{-1 \cdot x} + 2 \cdot \left(e^{-1 \cdot x} \cdot x\right)}}{2} \]
Simplified0.5
\[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(x, 2, 2\right)}{e^{x}}}}{2} \]
Final simplification0.5
\[\leadsto \frac{\frac{\mathsf{fma}\left(x, 2, 2\right)}{e^{x}}}{2} \]

Reproduce

herbie shell --seed 2022068 
(FPCore (x eps)
  :name "NMSE Section 6.1 mentioned, A"
  :precision binary64
  (/ (- (* (+ 1.0 (/ 1.0 eps)) (exp (- (* (- 1.0 eps) x)))) (* (- (/ 1.0 eps) 1.0) (exp (- (* (+ 1.0 eps) x))))) 2.0))

Error

Derivation

Reproduce