NMSE Section 6.1 mentioned, A

Average Error: 29.7 → 0.5

Time: 14.8s

Precision: binary64

Cost: 13888

Math

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

↓

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

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
 (/ (+ (exp (* x (+ eps -1.0))) (/ (+ 1.0 (* x (- 2.0 eps))) (exp x))) 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) {
	return (exp((x * (eps + -1.0))) + ((1.0 + (x * (2.0 - eps))) / exp(x))) / 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
    code = (exp((x * (eps + (-1.0d0)))) + ((1.0d0 + (x * (2.0d0 - eps))) / exp(x))) / 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) {
	return (Math.exp((x * (eps + -1.0))) + ((1.0 + (x * (2.0 - eps))) / Math.exp(x))) / 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):
	return (math.exp((x * (eps + -1.0))) + ((1.0 + (x * (2.0 - eps))) / math.exp(x))) / 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)
	return Float64(Float64(exp(Float64(x * Float64(eps + -1.0))) + Float64(Float64(1.0 + Float64(x * Float64(2.0 - eps))) / exp(x))) / 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)
	tmp = (exp((x * (eps + -1.0))) + ((1.0 + (x * (2.0 - eps))) / exp(x))) / 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_] := N[(N[(N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[(1.0 + N[(x * N[(2.0 - eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Derivation

1. Initial program 29.7

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

2. Simplified 29.7

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

   [Start] 29.7	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

3. Applied egg-rr 25.0

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

4. Taylor expanded in eps around 0 0.5

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

5. Simplified 0.5

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

   [Start] 0.5	\[ \frac{e^{x \cdot \left(\varepsilon + -1\right)} + \left(\left(-1 \cdot \left(\varepsilon \cdot \left(e^{-1 \cdot x} \cdot x\right)\right) + e^{-1 \cdot x} \cdot x\right) - \left(-1 \cdot \left(e^{-1 \cdot x} \cdot x\right) + -1 \cdot e^{-1 \cdot x}\right)\right)}{2} \]
   rational.json-simplify-48 [=>] 0.5	\[ \frac{e^{x \cdot \left(\varepsilon + -1\right)} + \color{blue}{\left(e^{-1 \cdot x} \cdot x + \left(-1 \cdot \left(\varepsilon \cdot \left(e^{-1 \cdot x} \cdot x\right)\right) - \left(-1 \cdot \left(e^{-1 \cdot x} \cdot x\right) + -1 \cdot e^{-1 \cdot x}\right)\right)\right)}}{2} \]
   rational.json-simplify-1 [=>] 0.5	\[ \frac{e^{x \cdot \left(\varepsilon + -1\right)} + \color{blue}{\left(\left(-1 \cdot \left(\varepsilon \cdot \left(e^{-1 \cdot x} \cdot x\right)\right) - \left(-1 \cdot \left(e^{-1 \cdot x} \cdot x\right) + -1 \cdot e^{-1 \cdot x}\right)\right) + e^{-1 \cdot x} \cdot x\right)}}{2} \]
   rational.json-simplify-43 [=>] 0.5	\[ \frac{e^{x \cdot \left(\varepsilon + -1\right)} + \left(\left(-1 \cdot \color{blue}{\left(e^{-1 \cdot x} \cdot \left(x \cdot \varepsilon\right)\right)} - \left(-1 \cdot \left(e^{-1 \cdot x} \cdot x\right) + -1 \cdot e^{-1 \cdot x}\right)\right) + e^{-1 \cdot x} \cdot x\right)}{2} \]
   rational.json-simplify-43 [=>] 0.5	\[ \frac{e^{x \cdot \left(\varepsilon + -1\right)} + \left(\left(\color{blue}{e^{-1 \cdot x} \cdot \left(\left(x \cdot \varepsilon\right) \cdot -1\right)} - \left(-1 \cdot \left(e^{-1 \cdot x} \cdot x\right) + -1 \cdot e^{-1 \cdot x}\right)\right) + e^{-1 \cdot x} \cdot x\right)}{2} \]
   rational.json-simplify-43 [=>] 0.5	\[ \frac{e^{x \cdot \left(\varepsilon + -1\right)} + \left(\left(e^{-1 \cdot x} \cdot \left(\left(x \cdot \varepsilon\right) \cdot -1\right) - \left(\color{blue}{e^{-1 \cdot x} \cdot \left(x \cdot -1\right)} + -1 \cdot e^{-1 \cdot x}\right)\right) + e^{-1 \cdot x} \cdot x\right)}{2} \]
   rational.json-simplify-2 [<=] 0.5	\[ \frac{e^{x \cdot \left(\varepsilon + -1\right)} + \left(\left(e^{-1 \cdot x} \cdot \left(\left(x \cdot \varepsilon\right) \cdot -1\right) - \left(e^{-1 \cdot x} \cdot \color{blue}{\left(-1 \cdot x\right)} + -1 \cdot e^{-1 \cdot x}\right)\right) + e^{-1 \cdot x} \cdot x\right)}{2} \]
   rational.json-simplify-51 [=>] 0.5	\[ \frac{e^{x \cdot \left(\varepsilon + -1\right)} + \left(\left(e^{-1 \cdot x} \cdot \left(\left(x \cdot \varepsilon\right) \cdot -1\right) - \color{blue}{e^{-1 \cdot x} \cdot \left(-1 + -1 \cdot x\right)}\right) + e^{-1 \cdot x} \cdot x\right)}{2} \]
   rational.json-simplify-2 [=>] 0.5	\[ \frac{e^{x \cdot \left(\varepsilon + -1\right)} + \left(\left(e^{-1 \cdot x} \cdot \left(\left(x \cdot \varepsilon\right) \cdot -1\right) - \color{blue}{\left(-1 + -1 \cdot x\right) \cdot e^{-1 \cdot x}}\right) + e^{-1 \cdot x} \cdot x\right)}{2} \]
   rational.json-simplify-52 [=>] 0.5	\[ \frac{e^{x \cdot \left(\varepsilon + -1\right)} + \left(\color{blue}{e^{-1 \cdot x} \cdot \left(\left(x \cdot \varepsilon\right) \cdot -1 - \left(-1 + -1 \cdot x\right)\right)} + e^{-1 \cdot x} \cdot x\right)}{2} \]
   rational.json-simplify-2 [=>] 0.5	\[ \frac{e^{x \cdot \left(\varepsilon + -1\right)} + \left(e^{-1 \cdot x} \cdot \left(\left(x \cdot \varepsilon\right) \cdot -1 - \left(-1 + -1 \cdot x\right)\right) + \color{blue}{x \cdot e^{-1 \cdot x}}\right)}{2} \]
   rational.json-simplify-51 [=>] 0.5	\[ \frac{e^{x \cdot \left(\varepsilon + -1\right)} + \color{blue}{e^{-1 \cdot x} \cdot \left(x + \left(\left(x \cdot \varepsilon\right) \cdot -1 - \left(-1 + -1 \cdot x\right)\right)\right)}}{2} \]

6. Taylor expanded in x around 0 0.5

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

7. Applied egg-rr 0.5

   \[\leadsto \frac{e^{x \cdot \left(\varepsilon + -1\right)} + \color{blue}{\left(\frac{1 + x \cdot \left(2 - \varepsilon\right)}{e^{x}} + 0\right)}}{2} \]

8. Simplified 0.5

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

   [Start] 0.5	\[ \frac{e^{x \cdot \left(\varepsilon + -1\right)} + \left(\frac{1 + x \cdot \left(2 - \varepsilon\right)}{e^{x}} + 0\right)}{2} \]
   rational.json-simplify-4 [=>] 0.5	\[ \frac{e^{x \cdot \left(\varepsilon + -1\right)} + \color{blue}{\frac{1 + x \cdot \left(2 - \varepsilon\right)}{e^{x}}}}{2} \]

9. Final simplification 0.5

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

Alternatives

Alternative 1
Error	1.1
Cost	13572

\[\begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{\left(x \cdot \varepsilon + \left(1 - x\right)\right) + t_0 \cdot \left(1 + \left(2 - \varepsilon\right) \cdot x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 + 2 \cdot \frac{x}{e^{x}}}{2}\\ \end{array} \]

Alternative 2
Error	0.5
Cost	13568

\[\frac{e^{-x} + \frac{1 - x \cdot -2}{e^{x}}}{2} \]

Alternative 3
Error	1.1
Cost	7812

\[\begin{array}{l} \mathbf{if}\;x \leq 1.3:\\ \;\;\;\;\frac{\left(x \cdot \varepsilon + \left(1 - x\right)\right) + e^{-x} \cdot \left(1 + \left(2 - \varepsilon\right) \cdot x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{2 - \varepsilon}{e^{x}}}{2}\\ \end{array} \]

Alternative 4
Error	1.1
Cost	7108

\[\begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(-1 + \varepsilon\right) - \frac{1 + \varepsilon}{\frac{\varepsilon}{\varepsilon + -1}}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{2 - \varepsilon}{e^{x}}}{2}\\ \end{array} \]

Alternative 5
Error	1.2
Cost	1732

\[\begin{array}{l} \mathbf{if}\;x \leq 360:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(-1 + \varepsilon\right) - \frac{1 + \varepsilon}{\frac{\varepsilon}{\varepsilon + -1}}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0}{\varepsilon}}{2}\\ \end{array} \]

Alternative 6
Error	1.1
Cost	452

\[\begin{array}{l} \mathbf{if}\;x \leq 360:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0}{\varepsilon}}{2}\\ \end{array} \]

Alternative 7
Error	16.7
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023073 
(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))