NMSE Section 6.1 mentioned, A

Average Error: 28.9 → 1.0

Time: 21.5s

Precision: binary64

Cost: 13824

Math

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

↓

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

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 (* (- eps 1.0) x)) (- (exp (* (- 1.0 (- eps)) (- 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(((eps - 1.0) * x)) - -exp(((1.0 - -eps) * -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(((eps - 1.0d0) * x)) - -exp(((1.0d0 - -eps) * -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(((eps - 1.0) * x)) - -Math.exp(((1.0 - -eps) * -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(((eps - 1.0) * x)) - -math.exp(((1.0 - -eps) * -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(Float64(eps - 1.0) * x)) - Float64(-exp(Float64(Float64(1.0 - Float64(-eps)) * Float64(-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(((eps - 1.0) * x)) - -exp(((1.0 - -eps) * -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[(N[(eps - 1.0), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] - (-N[Exp[N[(N[(1.0 - (-eps)), $MachinePrecision] * (-x)), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / 2.0), $MachinePrecision]

Derivation

1. Initial program 28.9

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

2. Taylor expanded in eps around -inf 1.0

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

3. Simplified 1.0

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

Alternatives

Alternative 1
Error	1.6
Cost	6784

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

Alternative 2
Error	2.1
Cost	708

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

Alternative 3
Error	54.6
Cost	192

\[\frac{1}{2} \]

Alternative 4
Error	17.0
Cost	192

\[\frac{2}{2} \]

Reproduce

herbie shell --seed 2023010 
(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))