NMSE Section 6.1 mentioned, A

Average Error: 29.5 → 0.8

Time: 14.2s

Precision: binary64

Cost: 20160

Math

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

↓

\[\begin{array}{l} t_0 := e^{-x}\\ 0.5 \cdot \left(x \cdot \left(\frac{1}{e^{x}} + t_0\right) + t_0 \cdot 2\right) \end{array} \]

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
 (let* ((t_0 (exp (- x))))
   (* 0.5 (+ (* x (+ (/ 1.0 (exp x)) t_0)) (* t_0 2.0)))))

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) {
	double t_0 = exp(-x);
	return 0.5 * ((x * ((1.0 / exp(x)) + t_0)) + (t_0 * 2.0));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (((1.0d0 + (1.0d0 / eps)) * exp(-((1.0d0 - eps) * x))) - (((1.0d0 / eps) - 1.0d0) * exp(-((1.0d0 + eps) * x)))) / 2.0d0
end function

↓

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    t_0 = exp(-x)
    code = 0.5d0 * ((x * ((1.0d0 / exp(x)) + t_0)) + (t_0 * 2.0d0))
end function

Java

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

↓

public static double code(double x, double eps) {
	double t_0 = Math.exp(-x);
	return 0.5 * ((x * ((1.0 / Math.exp(x)) + t_0)) + (t_0 * 2.0));
}

Python

def code(x, eps):
	return (((1.0 + (1.0 / eps)) * math.exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * math.exp(-((1.0 + eps) * x)))) / 2.0

↓

def code(x, eps):
	t_0 = math.exp(-x)
	return 0.5 * ((x * ((1.0 / math.exp(x)) + t_0)) + (t_0 * 2.0))

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)
	t_0 = exp(Float64(-x))
	return Float64(0.5 * Float64(Float64(x * Float64(Float64(1.0 / exp(x)) + t_0)) + Float64(t_0 * 2.0)))
end

MATLAB

function tmp = code(x, eps)
	tmp = (((1.0 + (1.0 / eps)) * exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * exp(-((1.0 + eps) * x)))) / 2.0;
end

↓

function tmp = code(x, eps)
	t_0 = exp(-x);
	tmp = 0.5 * ((x * ((1.0 / exp(x)) + t_0)) + (t_0 * 2.0));
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_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, N[(0.5 * N[(N[(x * N[(N[(1.0 / N[Exp[x], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 29.5

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

2. Simplified 29.5

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

   [Start] 29.5	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   div-sub [=>] 29.5	\[ \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}} \]
   associate-/l* [=>] 29.5	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \color{blue}{\frac{\frac{1}{\varepsilon} - 1}{\frac{2}{e^{-\left(1 + \varepsilon\right) \cdot x}}}} \]
   *-lft-identity [<=] 29.5	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \frac{\color{blue}{1 \cdot \left(\frac{1}{\varepsilon} - 1\right)}}{\frac{2}{e^{-\left(1 + \varepsilon\right) \cdot x}}} \]
   associate-*l/ [<=] 29.5	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \color{blue}{\frac{1}{\frac{2}{e^{-\left(1 + \varepsilon\right) \cdot x}}} \cdot \left(\frac{1}{\varepsilon} - 1\right)} \]
   associate-/r/ [=>] 29.5	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \color{blue}{\left(\frac{1}{2} \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right) \]
   associate-*l* [=>] 29.5	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \color{blue}{\frac{1}{2} \cdot \left(e^{-\left(1 + \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)} \]
   *-commutative [<=] 29.5	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \frac{1}{2} \cdot \color{blue}{\left(\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)} \]

3. Taylor expanded in eps around 0 30.2

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

4. Simplified 0.8

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

   [Start] 30.2	\[ 0.5 \cdot \left(\left(\frac{1}{e^{x}} + \left(\frac{e^{-x}}{\varepsilon} + \left(e^{-x} + e^{-x} \cdot x\right)\right)\right) - \left(\frac{1}{\varepsilon \cdot e^{x}} + -1 \cdot \frac{x}{e^{x}}\right)\right) \]
   associate--l+ [=>] 25.6	\[ 0.5 \cdot \color{blue}{\left(\frac{1}{e^{x}} + \left(\left(\frac{e^{-x}}{\varepsilon} + \left(e^{-x} + e^{-x} \cdot x\right)\right) - \left(\frac{1}{\varepsilon \cdot e^{x}} + -1 \cdot \frac{x}{e^{x}}\right)\right)\right)} \]
   +-commutative [=>] 25.6	\[ 0.5 \cdot \left(\frac{1}{e^{x}} + \left(\color{blue}{\left(\left(e^{-x} + e^{-x} \cdot x\right) + \frac{e^{-x}}{\varepsilon}\right)} - \left(\frac{1}{\varepsilon \cdot e^{x}} + -1 \cdot \frac{x}{e^{x}}\right)\right)\right) \]
   associate--l+ [=>] 2.1	\[ 0.5 \cdot \left(\frac{1}{e^{x}} + \color{blue}{\left(\left(e^{-x} + e^{-x} \cdot x\right) + \left(\frac{e^{-x}}{\varepsilon} - \left(\frac{1}{\varepsilon \cdot e^{x}} + -1 \cdot \frac{x}{e^{x}}\right)\right)\right)}\right) \]

5. Taylor expanded in x around inf 0.8

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

6. Final simplification 0.8

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

Alternatives

Alternative 1
Error	0.8
Cost	13504

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

Alternative 2
Error	0.8
Cost	6976

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

Alternative 3
Error	1.2
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \leq 1.42:\\ \;\;\;\;0.5 \cdot \left(2 - x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 4
Error	1.4
Cost	196

\[\begin{array}{l} \mathbf{if}\;x \leq 365:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 5
Error	46.9
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023018 
(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))