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

↓

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

\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^{\log \left(\mathsf{fma}\left(2, x, 2\right)\right) - 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 (- (log (fma 2.0 x 2.0)) 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(log(fma(2.0, x, 2.0)) - x) / 2.0;
}

Derivation

Initial program 29.5
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
Taylor expanded in eps around 0 0.7
\[\leadsto \frac{\color{blue}{2 \cdot \left(x \cdot e^{-x}\right) + 2 \cdot e^{-x}}}{2} \]
Applied add-exp-log_binary640.8
\[\leadsto \frac{\color{blue}{e^{\log \left(2 \cdot \left(x \cdot e^{-x}\right) + 2 \cdot e^{-x}\right)}}}{2} \]
Simplified0.8
\[\leadsto \frac{e^{\color{blue}{\log \left(\mathsf{fma}\left(2, x, 2\right)\right) - x}}}{2} \]
Final simplification0.8
\[\leadsto \frac{e^{\log \left(\mathsf{fma}\left(2, x, 2\right)\right) - x}}{2} \]

Reproduce

herbie shell --seed 2022067 
(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