NMSE Section 6.1 mentioned, A

Average Error: 29.8 → 0.6

Time: 13.7s

Precision: binary64

Cost: 20160

Math

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

↓

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

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
 (let* ((t_0 (exp (- x))))
   (/ (+ (+ (* x t_0) t_0) (/ (+ x 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 t_0 = exp(-x);
	return (((x * t_0) + t_0) + ((x + 1.0) / 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
    real(8) :: t_0
    t_0 = exp(-x)
    code = (((x * t_0) + t_0) + ((x + 1.0d0) / 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) {
	double t_0 = Math.exp(-x);
	return (((x * t_0) + t_0) + ((x + 1.0) / 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):
	t_0 = math.exp(-x)
	return (((x * t_0) + t_0) + ((x + 1.0) / 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)
	t_0 = exp(Float64(-x))
	return Float64(Float64(Float64(Float64(x * t_0) + t_0) + Float64(Float64(x + 1.0) / 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)
	t_0 = exp(-x);
	tmp = (((x * t_0) + t_0) + ((x + 1.0) / 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_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, N[(N[(N[(N[(x * t$95$0), $MachinePrecision] + t$95$0), $MachinePrecision] + N[(N[(x + 1.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Initial program 29.8

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

2. Simplified 29.8

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

   [Start] 29.8	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   distribute-rgt-neg-in [=>] 29.8	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\color{blue}{\left(1 - \varepsilon\right) \cdot \left(-x\right)}} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   sub-neg [=>] 29.8	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \color{blue}{\left(\frac{1}{\varepsilon} + \left(-1\right)\right)} \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   metadata-eval [=>] 29.8	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + \color{blue}{-1}\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   distribute-rgt-neg-in [=>] 29.8	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\color{blue}{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}}{2} \]

3. Taylor expanded in eps around 0 0.6

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

4. Applied egg-rr 0.6

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

5. Simplified 0.6

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

   [Start] 0.6	\[ \frac{\left(e^{-1 \cdot x} \cdot x + e^{-1 \cdot x}\right) - -1 \cdot \left(e^{-1 \cdot x} \cdot \left(1 + x\right)\right)}{2} \]
   +-commutative [<=] 0.6	\[ \frac{\left(e^{-1 \cdot x} \cdot x + e^{-1 \cdot x}\right) - -1 \cdot \left(e^{-1 \cdot x} \cdot \color{blue}{\left(x + 1\right)}\right)}{2} \]
   *-commutative [<=] 0.6	\[ \frac{\left(e^{-1 \cdot x} \cdot x + e^{-1 \cdot x}\right) - -1 \cdot \color{blue}{\left(\left(x + 1\right) \cdot e^{-1 \cdot x}\right)}}{2} \]
   associate-*r* [=>] 0.6	\[ \frac{\left(e^{-1 \cdot x} \cdot x + e^{-1 \cdot x}\right) - \color{blue}{\left(-1 \cdot \left(x + 1\right)\right) \cdot e^{-1 \cdot x}}}{2} \]
   neg-mul-1 [<=] 0.6	\[ \frac{\left(e^{-1 \cdot x} \cdot x + e^{-1 \cdot x}\right) - \color{blue}{\left(-\left(x + 1\right)\right)} \cdot e^{-1 \cdot x}}{2} \]
   mul-1-neg [=>] 0.6	\[ \frac{\left(e^{-1 \cdot x} \cdot x + e^{-1 \cdot x}\right) - \left(-\left(x + 1\right)\right) \cdot e^{\color{blue}{-x}}}{2} \]
   exp-neg [=>] 0.6	\[ \frac{\left(e^{-1 \cdot x} \cdot x + e^{-1 \cdot x}\right) - \left(-\left(x + 1\right)\right) \cdot \color{blue}{\frac{1}{e^{x}}}}{2} \]
   associate-*r/ [=>] 0.6	\[ \frac{\left(e^{-1 \cdot x} \cdot x + e^{-1 \cdot x}\right) - \color{blue}{\frac{\left(-\left(x + 1\right)\right) \cdot 1}{e^{x}}}}{2} \]
   distribute-lft-neg-in [<=] 0.6	\[ \frac{\left(e^{-1 \cdot x} \cdot x + e^{-1 \cdot x}\right) - \frac{\color{blue}{-\left(x + 1\right) \cdot 1}}{e^{x}}}{2} \]
   *-rgt-identity [=>] 0.6	\[ \frac{\left(e^{-1 \cdot x} \cdot x + e^{-1 \cdot x}\right) - \frac{-\color{blue}{\left(x + 1\right)}}{e^{x}}}{2} \]
   neg-sub0 [=>] 0.6	\[ \frac{\left(e^{-1 \cdot x} \cdot x + e^{-1 \cdot x}\right) - \frac{\color{blue}{0 - \left(x + 1\right)}}{e^{x}}}{2} \]
   +-commutative [=>] 0.6	\[ \frac{\left(e^{-1 \cdot x} \cdot x + e^{-1 \cdot x}\right) - \frac{0 - \color{blue}{\left(1 + x\right)}}{e^{x}}}{2} \]
   associate--r+ [=>] 0.6	\[ \frac{\left(e^{-1 \cdot x} \cdot x + e^{-1 \cdot x}\right) - \frac{\color{blue}{\left(0 - 1\right) - x}}{e^{x}}}{2} \]
   metadata-eval [=>] 0.6	\[ \frac{\left(e^{-1 \cdot x} \cdot x + e^{-1 \cdot x}\right) - \frac{\color{blue}{-1} - x}{e^{x}}}{2} \]

6. Final simplification 0.6

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

Alternatives

Alternative 1
Error	0.6
Cost	13440

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

Alternative 2
Error	1.6
Cost	6784

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

Alternative 3
Error	1.1
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;\frac{2 - x \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 4
Error	1.2
Cost	196

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

Alternative 5
Error	16.7
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023031 
(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))