NMSE Section 6.1 mentioned, A

Average Error: 29.3 → 0.6

Time: 13.4s

Precision: binary64

Cost: 19840

Math

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

↓

\[0.5 \cdot \mathsf{fma}\left(2, \frac{x}{e^{x}}, 2 \cdot e^{-x}\right) \]

FPCore

(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
 (* 0.5 (fma 2.0 (/ x (exp x)) (* 2.0 (exp (- x))))))

C

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 0.5 * fma(2.0, (x / exp(x)), (2.0 * exp(-x)));
}

Julia

function code(x, eps)
	return Float64(Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) * exp(Float64(-Float64(Float64(1.0 - eps) * x)))) - Float64(Float64(Float64(1.0 / eps) - 1.0) * exp(Float64(-Float64(Float64(1.0 + eps) * x))))) / 2.0)
end

↓

function code(x, eps)
	return Float64(0.5 * fma(2.0, Float64(x / exp(x)), Float64(2.0 * exp(Float64(-x)))))
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * N[Exp[(-N[(N[(1.0 - eps), $MachinePrecision] * x), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] - N[(N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision] * N[Exp[(-N[(N[(1.0 + eps), $MachinePrecision] * x), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

↓

code[x_, eps_] := N[(0.5 * N[(2.0 * N[(x / N[Exp[x], $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 29.3

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

2. Simplified 29.4

   \[\leadsto \color{blue}{0.5 \cdot \mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{\mathsf{fma}\left(\varepsilon, x, -x\right)}, \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)} \]
   Proof

   [Start] 29.3	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   div-sub [=>] 29.3	\[ \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}} \]
   associate-/l* [=>] 29.4	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \color{blue}{\frac{\frac{1}{\varepsilon} - 1}{\frac{2}{e^{-\left(1 + \varepsilon\right) \cdot x}}}} \]
   *-lft-identity [<=] 29.4	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \frac{\color{blue}{1 \cdot \left(\frac{1}{\varepsilon} - 1\right)}}{\frac{2}{e^{-\left(1 + \varepsilon\right) \cdot x}}} \]
   associate-*l/ [<=] 29.4	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \color{blue}{\frac{1}{\frac{2}{e^{-\left(1 + \varepsilon\right) \cdot x}}} \cdot \left(\frac{1}{\varepsilon} - 1\right)} \]
   associate-/r/ [=>] 29.3	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \color{blue}{\left(\frac{1}{2} \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right) \]
   associate-*l* [=>] 29.3	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \color{blue}{\frac{1}{2} \cdot \left(e^{-\left(1 + \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)} \]
   *-commutative [<=] 29.3	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \frac{1}{2} \cdot \color{blue}{\left(\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)} \]

3. Taylor expanded in eps around 0 29.9

   \[\leadsto 0.5 \cdot \color{blue}{\left(\left(\frac{1}{e^{x}} + \left(\frac{e^{-x}}{\varepsilon} + \left(e^{-x} + e^{-x} \cdot x\right)\right)\right) - \left(\frac{1}{\varepsilon \cdot e^{x}} + -1 \cdot \frac{x}{e^{x}}\right)\right)} \]

4. Simplified 0.6

   \[\leadsto 0.5 \cdot \color{blue}{\left(e^{-x} + \left(\frac{x}{e^{x}} + \left(x + 1\right) \cdot e^{-x}\right)\right)} \]
   Proof

   [Start] 29.9	\[ 0.5 \cdot \left(\left(\frac{1}{e^{x}} + \left(\frac{e^{-x}}{\varepsilon} + \left(e^{-x} + e^{-x} \cdot x\right)\right)\right) - \left(\frac{1}{\varepsilon \cdot e^{x}} + -1 \cdot \frac{x}{e^{x}}\right)\right) \]
   associate--l+ [=>] 25.3	\[ 0.5 \cdot \color{blue}{\left(\frac{1}{e^{x}} + \left(\left(\frac{e^{-x}}{\varepsilon} + \left(e^{-x} + e^{-x} \cdot x\right)\right) - \left(\frac{1}{\varepsilon \cdot e^{x}} + -1 \cdot \frac{x}{e^{x}}\right)\right)\right)} \]
   exp-neg [<=] 25.3	\[ 0.5 \cdot \left(\color{blue}{e^{-x}} + \left(\left(\frac{e^{-x}}{\varepsilon} + \left(e^{-x} + e^{-x} \cdot x\right)\right) - \left(\frac{1}{\varepsilon \cdot e^{x}} + -1 \cdot \frac{x}{e^{x}}\right)\right)\right) \]
   +-commutative [=>] 25.3	\[ 0.5 \cdot \left(e^{-x} + \left(\color{blue}{\left(\left(e^{-x} + e^{-x} \cdot x\right) + \frac{e^{-x}}{\varepsilon}\right)} - \left(\frac{1}{\varepsilon \cdot e^{x}} + -1 \cdot \frac{x}{e^{x}}\right)\right)\right) \]
   associate--l+ [=>] 1.8	\[ 0.5 \cdot \left(e^{-x} + \color{blue}{\left(\left(e^{-x} + e^{-x} \cdot x\right) + \left(\frac{e^{-x}}{\varepsilon} - \left(\frac{1}{\varepsilon \cdot e^{x}} + -1 \cdot \frac{x}{e^{x}}\right)\right)\right)}\right) \]

5. Applied egg-rr 0.6

   \[\leadsto 0.5 \cdot \left(e^{-x} + \left(\frac{x}{e^{x}} + \color{blue}{\frac{x + 1}{e^{x}}}\right)\right) \]

6. Taylor expanded in x around inf 0.6

   \[\leadsto 0.5 \cdot \color{blue}{\left(2 \cdot \frac{x}{e^{x}} + \left(\frac{1}{e^{x}} + e^{-x}\right)\right)} \]

7. Simplified 0.6

   \[\leadsto 0.5 \cdot \color{blue}{\mathsf{fma}\left(2, \frac{x}{e^{x}}, 2 \cdot e^{-x}\right)} \]
   Proof

   [Start] 0.6	\[ 0.5 \cdot \left(2 \cdot \frac{x}{e^{x}} + \left(\frac{1}{e^{x}} + e^{-x}\right)\right) \]
   fma-def [=>] 0.6	\[ 0.5 \cdot \color{blue}{\mathsf{fma}\left(2, \frac{x}{e^{x}}, \frac{1}{e^{x}} + e^{-x}\right)} \]
   exp-neg [=>] 0.6	\[ 0.5 \cdot \mathsf{fma}\left(2, \frac{x}{e^{x}}, \frac{1}{e^{x}} + \color{blue}{\frac{1}{e^{x}}}\right) \]
   count-2 [=>] 0.6	\[ 0.5 \cdot \mathsf{fma}\left(2, \frac{x}{e^{x}}, \color{blue}{2 \cdot \frac{1}{e^{x}}}\right) \]
   exp-neg [<=] 0.6	\[ 0.5 \cdot \mathsf{fma}\left(2, \frac{x}{e^{x}}, 2 \cdot \color{blue}{e^{-x}}\right) \]

8. Final simplification 0.6

   \[\leadsto 0.5 \cdot \mathsf{fma}\left(2, \frac{x}{e^{x}}, 2 \cdot e^{-x}\right) \]

Alternatives

Alternative 1
Error	0.6
Cost	19776

\[\begin{array}{l} t_0 := \frac{2}{e^{x}}\\ 0.5 \cdot \mathsf{fma}\left(t_0, x, t_0\right) \end{array} \]

Alternative 2
Error	0.9
Cost	13632

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

Alternative 3
Error	0.6
Cost	13572

\[\begin{array}{l} \mathbf{if}\;x \leq 360:\\ \;\;\;\;\frac{e^{x \cdot \varepsilon} + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 4
Error	0.9
Cost	7300

\[\begin{array}{l} \mathbf{if}\;x \leq 350:\\ \;\;\;\;0.5 \cdot \left(\left(2 - x \cdot x\right) + {x}^{3} \cdot 0.6666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 5
Error	1.0
Cost	964

\[\begin{array}{l} \mathbf{if}\;x \leq 350:\\ \;\;\;\;\frac{2 - \left(x \cdot x\right) \cdot \left(1 + x \cdot -0.3333333333333333\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 6
Error	1.0
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \leq 1.45:\\ \;\;\;\;\frac{2 - x \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 7
Error	1.1
Cost	196

\[\begin{array}{l} \mathbf{if}\;x \leq 360:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 8
Error	46.8
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023027 
(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))