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

↓

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

\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{e^{-\mathsf{fma}\left(x, \varepsilon, x\right)} + e^{x \cdot \varepsilon - 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
 (/ (+ (exp (- (fma x eps x))) (exp (- (* x eps) 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 (exp(-fma(x, eps, x)) + exp((x * eps) - x)) / 2.0;
}

Derivation

Initial program 28.8
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
Taylor expanded in eps around inf 0.9
\[\leadsto \frac{\color{blue}{e^{-\left(\varepsilon \cdot x + x\right)} + e^{\varepsilon \cdot x - x}}}{2} \]
Simplified0.9
\[\leadsto \frac{\color{blue}{e^{-\mathsf{fma}\left(x, \varepsilon, x\right)} + e^{x \cdot \varepsilon - x}}}{2} \]
Final simplification0.9
\[\leadsto \frac{e^{-\mathsf{fma}\left(x, \varepsilon, x\right)} + e^{x \cdot \varepsilon - x}}{2} \]

Reproduce

herbie shell --seed 2021275 
(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