NMSE Section 6.1 mentioned, A

Average Error: 28.8 → 1.0

Time: 15.3s

Precision: binary64

Cost: 13636

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq 3.4:\\ \;\;\;\;\frac{e^{\varepsilon \cdot x} + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon - 1\right)} + e^{-x}}{2}\\ \end{array} \]

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
 (if (<= x 3.4)
   (/ (+ (exp (* eps x)) (exp (* x (- -1.0 eps)))) 2.0)
   (/ (+ (exp (* x (- eps 1.0))) (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) {
	double tmp;
	if (x <= 3.4) {
		tmp = (exp((eps * x)) + exp((x * (-1.0 - eps)))) / 2.0;
	} else {
		tmp = (exp((x * (eps - 1.0))) + exp(-x)) / 2.0;
	}
	return tmp;
}

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) :: tmp
    if (x <= 3.4d0) then
        tmp = (exp((eps * x)) + exp((x * ((-1.0d0) - eps)))) / 2.0d0
    else
        tmp = (exp((x * (eps - 1.0d0))) + exp(-x)) / 2.0d0
    end if
    code = tmp
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 tmp;
	if (x <= 3.4) {
		tmp = (Math.exp((eps * x)) + Math.exp((x * (-1.0 - eps)))) / 2.0;
	} else {
		tmp = (Math.exp((x * (eps - 1.0))) + Math.exp(-x)) / 2.0;
	}
	return tmp;
}

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):
	tmp = 0
	if x <= 3.4:
		tmp = (math.exp((eps * x)) + math.exp((x * (-1.0 - eps)))) / 2.0
	else:
		tmp = (math.exp((x * (eps - 1.0))) + math.exp(-x)) / 2.0
	return tmp

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)
	tmp = 0.0
	if (x <= 3.4)
		tmp = Float64(Float64(exp(Float64(eps * x)) + exp(Float64(x * Float64(-1.0 - eps)))) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(eps - 1.0))) + exp(Float64(-x))) / 2.0);
	end
	return tmp
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_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 3.4)
		tmp = (exp((eps * x)) + exp((x * (-1.0 - eps)))) / 2.0;
	else
		tmp = (exp((x * (eps - 1.0))) + exp(-x)) / 2.0;
	end
	tmp_2 = tmp;
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_] := If[LessEqual[x, 3.4], N[(N[(N[Exp[N[(eps * x), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(eps - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.39999999999999991

   1. Initial program 38.3

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

   2. Simplified 38.3

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

      [Start] 38.3	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   3. Taylor expanded in eps around inf 1.2

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

   4. Simplified 1.2

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

      [Start] 1.2	\[ \frac{e^{\left(\varepsilon - 1\right) \cdot x} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}{2} \]
      rational.json-simplify-15 [<=] 1.2	\[ \frac{e^{\color{blue}{\left(\varepsilon + -1\right)} \cdot x} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}{2} \]
      rational.json-simplify-2 [<=] 1.2	\[ \frac{e^{\color{blue}{x \cdot \left(\varepsilon + -1\right)}} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}{2} \]
      rational.json-simplify-4 [<=] 1.2	\[ \frac{\color{blue}{\left(e^{x \cdot \left(\varepsilon + -1\right)} + 0\right)} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}{2} \]
      rational.json-simplify-1 [=>] 1.2	\[ \frac{\color{blue}{\left(0 + e^{x \cdot \left(\varepsilon + -1\right)}\right)} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}{2} \]
      rational.json-simplify-48 [=>] 1.2	\[ \frac{\color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} + \left(0 - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}\right)}}{2} \]
      rational.json-simplify-2 [=>] 1.2	\[ \frac{e^{\color{blue}{\left(\varepsilon + -1\right) \cdot x}} + \left(0 - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}\right)}{2} \]
      rational.json-simplify-15 [=>] 1.2	\[ \frac{e^{\color{blue}{\left(\varepsilon - 1\right)} \cdot x} + \left(0 - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}\right)}{2} \]
      rational.json-simplify-2 [=>] 1.2	\[ \frac{e^{\color{blue}{x \cdot \left(\varepsilon - 1\right)}} + \left(0 - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}\right)}{2} \]
      rational.json-simplify-2 [=>] 1.2	\[ \frac{e^{x \cdot \left(\varepsilon - 1\right)} + \left(0 - \color{blue}{e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)} \cdot -1}\right)}{2} \]
      rational.json-simplify-9 [=>] 1.2	\[ \frac{e^{x \cdot \left(\varepsilon - 1\right)} + \left(0 - \color{blue}{\left(-e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}\right)}\right)}{2} \]
      rational.json-simplify-12 [=>] 1.2	\[ \frac{e^{x \cdot \left(\varepsilon - 1\right)} + \left(0 - \color{blue}{\left(0 - e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}\right)}\right)}{2} \]
      rational.json-simplify-45 [=>] 1.2	\[ \frac{e^{x \cdot \left(\varepsilon - 1\right)} + \color{blue}{\left(e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)} - \left(0 - 0\right)\right)}}{2} \]

   5. Taylor expanded in eps around inf 1.1

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

   if 3.39999999999999991 < x

   1. Initial program 0.5

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

   2. Simplified 0.5

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

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

   3. Taylor expanded in eps around inf 0.4

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

   4. Simplified 0.4

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

      [Start] 0.4	\[ \frac{e^{\left(\varepsilon - 1\right) \cdot x} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}{2} \]
      rational.json-simplify-15 [<=] 0.4	\[ \frac{e^{\color{blue}{\left(\varepsilon + -1\right)} \cdot x} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}{2} \]
      rational.json-simplify-2 [<=] 0.4	\[ \frac{e^{\color{blue}{x \cdot \left(\varepsilon + -1\right)}} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}{2} \]
      rational.json-simplify-4 [<=] 0.4	\[ \frac{\color{blue}{\left(e^{x \cdot \left(\varepsilon + -1\right)} + 0\right)} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}{2} \]
      rational.json-simplify-1 [=>] 0.4	\[ \frac{\color{blue}{\left(0 + e^{x \cdot \left(\varepsilon + -1\right)}\right)} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}{2} \]
      rational.json-simplify-48 [=>] 0.4	\[ \frac{\color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} + \left(0 - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}\right)}}{2} \]
      rational.json-simplify-2 [=>] 0.4	\[ \frac{e^{\color{blue}{\left(\varepsilon + -1\right) \cdot x}} + \left(0 - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}\right)}{2} \]
      rational.json-simplify-15 [=>] 0.4	\[ \frac{e^{\color{blue}{\left(\varepsilon - 1\right)} \cdot x} + \left(0 - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}\right)}{2} \]
      rational.json-simplify-2 [=>] 0.4	\[ \frac{e^{\color{blue}{x \cdot \left(\varepsilon - 1\right)}} + \left(0 - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}\right)}{2} \]
      rational.json-simplify-2 [=>] 0.4	\[ \frac{e^{x \cdot \left(\varepsilon - 1\right)} + \left(0 - \color{blue}{e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)} \cdot -1}\right)}{2} \]
      rational.json-simplify-9 [=>] 0.4	\[ \frac{e^{x \cdot \left(\varepsilon - 1\right)} + \left(0 - \color{blue}{\left(-e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}\right)}\right)}{2} \]
      rational.json-simplify-12 [=>] 0.4	\[ \frac{e^{x \cdot \left(\varepsilon - 1\right)} + \left(0 - \color{blue}{\left(0 - e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}\right)}\right)}{2} \]
      rational.json-simplify-45 [=>] 0.4	\[ \frac{e^{x \cdot \left(\varepsilon - 1\right)} + \color{blue}{\left(e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)} - \left(0 - 0\right)\right)}}{2} \]

   5. Taylor expanded in eps around 0 0.4

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

   6. Simplified 0.4

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

      [Start] 0.4	\[ \frac{e^{x \cdot \left(\varepsilon - 1\right)} + e^{-1 \cdot x}}{2} \]
      rational.json-simplify-2 [=>] 0.4	\[ \frac{e^{x \cdot \left(\varepsilon - 1\right)} + e^{\color{blue}{x \cdot -1}}}{2} \]
      rational.json-simplify-9 [=>] 0.4	\[ \frac{e^{x \cdot \left(\varepsilon - 1\right)} + e^{\color{blue}{-x}}}{2} \]

3. Recombined 2 regimes into one program.

4. Final simplification 1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.4:\\ \;\;\;\;\frac{e^{\varepsilon \cdot x} + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon - 1\right)} + e^{-x}}{2}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.0
Cost	13632

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

Alternative 2
Error	1.8
Cost	13440

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

Alternative 3
Error	1.1
Cost	7236

\[\begin{array}{l} \mathbf{if}\;x \leq 3.9:\\ \;\;\;\;\frac{e^{\varepsilon \cdot x} + \left(1 - x \cdot \varepsilon\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(\varepsilon - 1\right) \cdot x}}{2}\\ \end{array} \]

Alternative 4
Error	1.2
Cost	6980

\[\begin{array}{l} \mathbf{if}\;x \leq 3.7:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(\varepsilon - 1\right) \cdot x}}{2}\\ \end{array} \]

Alternative 5
Error	3.0
Cost	2244

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq 3.4 \cdot 10^{-23}:\\ \;\;\;\;\frac{x + \left(2 + x \cdot \left(-1 + \varepsilon\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(1 + \left(\varepsilon - 1\right) \cdot x\right) - \left(\frac{1}{\varepsilon} + \left(-1 - x \cdot \left(\left(-1 + \frac{1}{\varepsilon}\right) \cdot \left(1 + \varepsilon\right)\right)\right)\right)}{2}\\ \end{array} \]

Alternative 6
Error	9.9
Cost	708

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

Alternative 7
Error	3.1
Cost	704

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

Alternative 8
Error	54.5
Cost	64

\[0.5 \]

Alternative 9
Error	16.7
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023075 
(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))