NMSE Section 6.1 mentioned, A

Average Error: 29.5 → 0.6

Time: 23.9s

Precision: binary64

Cost: 13184

Math

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

↓

\[\frac{x}{e^{x}} + e^{-x} \]

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 (+ (/ x (exp x)) (exp (- x))))

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 (x / exp(x)) + exp(-x);
}

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 = (x / exp(x)) + exp(-x)
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 (x / Math.exp(x)) + Math.exp(-x);
}

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 (x / math.exp(x)) + math.exp(-x)

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(x / exp(x)) + exp(Float64(-x)))
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 = (x / exp(x)) + exp(-x);
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[(x / N[Exp[x], $MachinePrecision]), $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 29.5

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

2. Simplified 29.5

   \[\leadsto \color{blue}{e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{-0.5}{\varepsilon} - -0.5\right) - e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(\frac{-0.5}{\varepsilon} - 0.5\right)} \]
   Proof

   [Start] 29.5	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   metadata-eval [<=] 29.5	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{\color{blue}{--2}} \]
   metadata-eval [<=] 29.5	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{-\color{blue}{\left(-2\right)}} \]
   rational_best-simplify-54 [<=] 29.5	\[ \color{blue}{\frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{-2} - \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{-2}} \]

3. Taylor expanded in eps around 0 0.6

   \[\leadsto \color{blue}{\left(0.5 \cdot e^{-1 \cdot x} + 0.5 \cdot \left(e^{-1 \cdot x} \cdot x\right)\right) - -1 \cdot \left(0.5 \cdot e^{-1 \cdot x} + 0.5 \cdot \left(e^{-1 \cdot x} \cdot x\right)\right)} \]

4. Simplified 0.6

   \[\leadsto \color{blue}{e^{-x} \cdot \left(\left(0.5 + 0.5 \cdot x\right) - -1 \cdot \left(0.5 + 0.5 \cdot x\right)\right)} \]
   Proof

   [Start] 0.6	\[ \left(0.5 \cdot e^{-1 \cdot x} + 0.5 \cdot \left(e^{-1 \cdot x} \cdot x\right)\right) - -1 \cdot \left(0.5 \cdot e^{-1 \cdot x} + 0.5 \cdot \left(e^{-1 \cdot x} \cdot x\right)\right) \]
   rational_best-simplify-1 [=>] 0.6	\[ \color{blue}{\left(0.5 \cdot \left(e^{-1 \cdot x} \cdot x\right) + 0.5 \cdot e^{-1 \cdot x}\right)} - -1 \cdot \left(0.5 \cdot e^{-1 \cdot x} + 0.5 \cdot \left(e^{-1 \cdot x} \cdot x\right)\right) \]
   rational_best-simplify-44 [=>] 0.6	\[ \left(\color{blue}{e^{-1 \cdot x} \cdot \left(0.5 \cdot x\right)} + 0.5 \cdot e^{-1 \cdot x}\right) - -1 \cdot \left(0.5 \cdot e^{-1 \cdot x} + 0.5 \cdot \left(e^{-1 \cdot x} \cdot x\right)\right) \]
   rational_best-simplify-51 [=>] 0.6	\[ \color{blue}{e^{-1 \cdot x} \cdot \left(0.5 + 0.5 \cdot x\right)} - -1 \cdot \left(0.5 \cdot e^{-1 \cdot x} + 0.5 \cdot \left(e^{-1 \cdot x} \cdot x\right)\right) \]
   rational_best-simplify-1 [=>] 0.6	\[ e^{-1 \cdot x} \cdot \left(0.5 + 0.5 \cdot x\right) - -1 \cdot \color{blue}{\left(0.5 \cdot \left(e^{-1 \cdot x} \cdot x\right) + 0.5 \cdot e^{-1 \cdot x}\right)} \]
   rational_best-simplify-44 [=>] 0.6	\[ e^{-1 \cdot x} \cdot \left(0.5 + 0.5 \cdot x\right) - -1 \cdot \left(\color{blue}{e^{-1 \cdot x} \cdot \left(0.5 \cdot x\right)} + 0.5 \cdot e^{-1 \cdot x}\right) \]
   rational_best-simplify-51 [=>] 0.6	\[ e^{-1 \cdot x} \cdot \left(0.5 + 0.5 \cdot x\right) - -1 \cdot \color{blue}{\left(e^{-1 \cdot x} \cdot \left(0.5 + 0.5 \cdot x\right)\right)} \]
   rational_best-simplify-44 [=>] 0.6	\[ e^{-1 \cdot x} \cdot \left(0.5 + 0.5 \cdot x\right) - \color{blue}{e^{-1 \cdot x} \cdot \left(-1 \cdot \left(0.5 + 0.5 \cdot x\right)\right)} \]
   rational_best-simplify-50 [=>] 0.6	\[ \color{blue}{e^{-1 \cdot x} \cdot \left(\left(0.5 + 0.5 \cdot x\right) - -1 \cdot \left(0.5 + 0.5 \cdot x\right)\right)} \]
   rational_best-simplify-2 [=>] 0.6	\[ e^{\color{blue}{x \cdot -1}} \cdot \left(\left(0.5 + 0.5 \cdot x\right) - -1 \cdot \left(0.5 + 0.5 \cdot x\right)\right) \]
   rational_best-simplify-12 [=>] 0.6	\[ e^{\color{blue}{-x}} \cdot \left(\left(0.5 + 0.5 \cdot x\right) - -1 \cdot \left(0.5 + 0.5 \cdot x\right)\right) \]

5. Taylor expanded in x around inf 0.6

   \[\leadsto \color{blue}{e^{-x} + e^{-x} \cdot x} \]

6. Simplified 0.6

   \[\leadsto \color{blue}{\frac{x + 1}{e^{x}}} \]
   Proof

   [Start] 0.6	\[ e^{-x} + e^{-x} \cdot x \]
   rational_best-simplify-5 [<=] 0.6	\[ \color{blue}{1 \cdot e^{-x}} + e^{-x} \cdot x \]
   rational_best-simplify-2 [=>] 0.6	\[ \color{blue}{e^{-x} \cdot 1} + e^{-x} \cdot x \]
   rational_best-simplify-2 [=>] 0.6	\[ e^{-x} \cdot 1 + \color{blue}{x \cdot e^{-x}} \]
   rational_best-simplify-51 [=>] 0.6	\[ \color{blue}{e^{-x} \cdot \left(x + 1\right)} \]
   rational_best-simplify-2 [=>] 0.6	\[ \color{blue}{\left(x + 1\right) \cdot e^{-x}} \]
   exponential-simplify-2 [=>] 0.6	\[ \left(x + 1\right) \cdot \color{blue}{\frac{1}{e^{x}}} \]
   rational_best-simplify-47 [<=] 0.6	\[ \color{blue}{\frac{1 \cdot \left(x + 1\right)}{e^{x}}} \]
   rational_best-simplify-5 [=>] 0.6	\[ \frac{\color{blue}{x + 1}}{e^{x}} \]

7. Taylor expanded in x around inf 0.6

   \[\leadsto \color{blue}{\frac{1}{e^{x}} + \frac{x}{e^{x}}} \]

8. Simplified 0.6

   \[\leadsto \color{blue}{\frac{x}{e^{x}} + e^{-x}} \]
   Proof

   [Start] 0.6	\[ \frac{1}{e^{x}} + \frac{x}{e^{x}} \]
   rational_best-simplify-1 [=>] 0.6	\[ \color{blue}{\frac{x}{e^{x}} + \frac{1}{e^{x}}} \]
   exponential-simplify-2 [<=] 0.6	\[ \frac{x}{e^{x}} + \color{blue}{e^{-x}} \]

9. Final simplification 0.6

   \[\leadsto \frac{x}{e^{x}} + e^{-x} \]

Alternatives

Alternative 1
Error	0.6
Cost	6720

\[\frac{x + 1}{e^{x}} \]

Alternative 2
Error	1.1
Cost	196

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

Alternative 3
Error	46.9
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023096 
(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))