NMSE Section 6.1 mentioned, A

Average Error: 29.7 → 1.0

Time: 14.2s

Precision: binary64

Cost: 13760

Math

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

↓

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

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

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

4. Simplified 1.0

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

   [Start] 1.0	\[ \frac{e^{\left(\varepsilon - 1\right) \cdot x} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}{2} \]
   rational.json-simplify-2 [=>] 1.0	\[ \frac{e^{\color{blue}{x \cdot \left(\varepsilon - 1\right)}} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}{2} \]
   rational.json-simplify-2 [=>] 1.0	\[ \frac{e^{x \cdot \left(\varepsilon - 1\right)} - \color{blue}{e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)} \cdot -1}}{2} \]
   rational.json-simplify-9 [=>] 1.0	\[ \frac{e^{x \cdot \left(\varepsilon - 1\right)} - \color{blue}{\left(-e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}\right)}}{2} \]
   rational.json-simplify-1 [=>] 1.0	\[ \frac{e^{x \cdot \left(\varepsilon - 1\right)} - \left(-e^{-1 \cdot \left(\color{blue}{\left(1 + \varepsilon\right)} \cdot x\right)}\right)}{2} \]
   rational.json-simplify-2 [=>] 1.0	\[ \frac{e^{x \cdot \left(\varepsilon - 1\right)} - \left(-e^{-1 \cdot \color{blue}{\left(x \cdot \left(1 + \varepsilon\right)\right)}}\right)}{2} \]
   rational.json-simplify-43 [<=] 1.0	\[ \frac{e^{x \cdot \left(\varepsilon - 1\right)} - \left(-e^{\color{blue}{\left(1 + \varepsilon\right) \cdot \left(-1 \cdot x\right)}}\right)}{2} \]
   rational.json-simplify-2 [=>] 1.0	\[ \frac{e^{x \cdot \left(\varepsilon - 1\right)} - \left(-e^{\color{blue}{\left(-1 \cdot x\right) \cdot \left(1 + \varepsilon\right)}}\right)}{2} \]
   rational.json-simplify-2 [=>] 1.0	\[ \frac{e^{x \cdot \left(\varepsilon - 1\right)} - \left(-e^{\color{blue}{\left(x \cdot -1\right)} \cdot \left(1 + \varepsilon\right)}\right)}{2} \]
   rational.json-simplify-9 [=>] 1.0	\[ \frac{e^{x \cdot \left(\varepsilon - 1\right)} - \left(-e^{\color{blue}{\left(-x\right)} \cdot \left(1 + \varepsilon\right)}\right)}{2} \]
   rational.json-simplify-1 [<=] 1.0	\[ \frac{e^{x \cdot \left(\varepsilon - 1\right)} - \left(-e^{\left(-x\right) \cdot \color{blue}{\left(\varepsilon + 1\right)}}\right)}{2} \]

5. Final simplification 1.0

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

Alternatives

Alternative 1
Error	0.7
Cost	7232

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

Alternative 2
Error	1.5
Cost	6784

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

Alternative 3
Error	1.5
Cost	6720

\[\frac{\frac{2}{e^{x}}}{2} \]

Alternative 4
Error	1.2
Cost	1092

\[\begin{array}{l} t_0 := \varepsilon + \frac{-1}{\varepsilon}\\ \mathbf{if}\;x \leq 360:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(t_0 - t_0\right)}{2}\\ \end{array} \]

Alternative 5
Error	16.5
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023074 
(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))