NMSE Section 6.1 mentioned, A

Average Error: 29.7 → 0.6

Time: 21.6s

Precision: binary64

Cost: 13952

Math

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

↓

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

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

   \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(\varepsilon + -1\right) \cdot x} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(-1 - \varepsilon\right) \cdot x}}{2}} \]
   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 0 0.5

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

4. Simplified 0.6

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

   [Start] 0.5	\[ \frac{\left(e^{-1 \cdot x} \cdot x + e^{-1 \cdot x}\right) - \left(-1 \cdot \left(e^{-1 \cdot x} \cdot x\right) + -1 \cdot e^{-1 \cdot x}\right)}{2} \]
   rational.json-simplify-1 [=>] 0.5	\[ \frac{\color{blue}{\left(e^{-1 \cdot x} + e^{-1 \cdot x} \cdot x\right)} - \left(-1 \cdot \left(e^{-1 \cdot x} \cdot x\right) + -1 \cdot e^{-1 \cdot x}\right)}{2} \]
   rational.json-simplify-2 [=>] 0.5	\[ \frac{\left(e^{\color{blue}{x \cdot -1}} + e^{-1 \cdot x} \cdot x\right) - \left(-1 \cdot \left(e^{-1 \cdot x} \cdot x\right) + -1 \cdot e^{-1 \cdot x}\right)}{2} \]
   rational.json-simplify-9 [=>] 0.5	\[ \frac{\left(e^{\color{blue}{-x}} + e^{-1 \cdot x} \cdot x\right) - \left(-1 \cdot \left(e^{-1 \cdot x} \cdot x\right) + -1 \cdot e^{-1 \cdot x}\right)}{2} \]
   rational.json-simplify-2 [=>] 0.5	\[ \frac{\left(e^{-x} + \color{blue}{x \cdot e^{-1 \cdot x}}\right) - \left(-1 \cdot \left(e^{-1 \cdot x} \cdot x\right) + -1 \cdot e^{-1 \cdot x}\right)}{2} \]
   rational.json-simplify-2 [=>] 0.5	\[ \frac{\left(e^{-x} + x \cdot e^{\color{blue}{x \cdot -1}}\right) - \left(-1 \cdot \left(e^{-1 \cdot x} \cdot x\right) + -1 \cdot e^{-1 \cdot x}\right)}{2} \]
   rational.json-simplify-9 [=>] 0.5	\[ \frac{\left(e^{-x} + x \cdot e^{\color{blue}{-x}}\right) - \left(-1 \cdot \left(e^{-1 \cdot x} \cdot x\right) + -1 \cdot e^{-1 \cdot x}\right)}{2} \]
   rational.json-simplify-43 [=>] 0.5	\[ \frac{\left(e^{-x} + x \cdot e^{-x}\right) - \left(\color{blue}{e^{-1 \cdot x} \cdot \left(x \cdot -1\right)} + -1 \cdot e^{-1 \cdot x}\right)}{2} \]
   rational.json-simplify-2 [<=] 0.5	\[ \frac{\left(e^{-x} + x \cdot e^{-x}\right) - \left(e^{-1 \cdot x} \cdot \color{blue}{\left(-1 \cdot x\right)} + -1 \cdot e^{-1 \cdot x}\right)}{2} \]
   rational.json-simplify-47 [=>] 0.6	\[ \frac{\left(e^{-x} + x \cdot e^{-x}\right) - \color{blue}{e^{-1 \cdot x} \cdot \left(-1 + -1 \cdot x\right)}}{2} \]
   rational.json-simplify-2 [=>] 0.6	\[ \frac{\left(e^{-x} + x \cdot e^{-x}\right) - e^{\color{blue}{x \cdot -1}} \cdot \left(-1 + -1 \cdot x\right)}{2} \]
   rational.json-simplify-9 [=>] 0.6	\[ \frac{\left(e^{-x} + x \cdot e^{-x}\right) - e^{\color{blue}{-x}} \cdot \left(-1 + -1 \cdot x\right)}{2} \]
   rational.json-simplify-2 [=>] 0.6	\[ \frac{\left(e^{-x} + x \cdot e^{-x}\right) - e^{-x} \cdot \left(-1 + \color{blue}{x \cdot -1}\right)}{2} \]
   rational.json-simplify-9 [=>] 0.6	\[ \frac{\left(e^{-x} + x \cdot e^{-x}\right) - e^{-x} \cdot \left(-1 + \color{blue}{\left(-x\right)}\right)}{2} \]

5. Applied egg-rr 0.6

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

6. Simplified 0.6

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

   [Start] 0.6	\[ \frac{e^{-1 - x} \cdot \left(\left(x + 1\right) \cdot e^{1}\right) - e^{-x} \cdot \left(-1 + \left(-x\right)\right)}{2} \]
   rational.json-simplify-43 [=>] 0.6	\[ \frac{\color{blue}{\left(x + 1\right) \cdot \left(e^{1} \cdot e^{-1 - x}\right)} - e^{-x} \cdot \left(-1 + \left(-x\right)\right)}{2} \]
   exponential.json-simplify-3 [=>] 0.6	\[ \frac{\left(x + 1\right) \cdot \color{blue}{e^{1 + \left(-1 - x\right)}} - e^{-x} \cdot \left(-1 + \left(-x\right)\right)}{2} \]

7. Taylor expanded in x around inf 0.5

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

8. Simplified 0.6

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

   [Start] 0.5	\[ \frac{\left(x + 1\right) \cdot e^{1 + \left(-1 - x\right)} - \left(-1 \cdot \left(e^{-x} \cdot x\right) + -1 \cdot e^{-x}\right)}{2} \]
   rational.json-simplify-43 [=>] 0.5	\[ \frac{\left(x + 1\right) \cdot e^{1 + \left(-1 - x\right)} - \left(\color{blue}{e^{-x} \cdot \left(x \cdot -1\right)} + -1 \cdot e^{-x}\right)}{2} \]
   rational.json-simplify-8 [<=] 0.5	\[ \frac{\left(x + 1\right) \cdot e^{1 + \left(-1 - x\right)} - \left(e^{-x} \cdot \color{blue}{\left(-x\right)} + -1 \cdot e^{-x}\right)}{2} \]
   rational.json-simplify-47 [=>] 0.6	\[ \frac{\left(x + 1\right) \cdot e^{1 + \left(-1 - x\right)} - \color{blue}{e^{-x} \cdot \left(-1 + \left(-x\right)\right)}}{2} \]
   rational.json-simplify-1 [=>] 0.6	\[ \frac{\left(x + 1\right) \cdot e^{1 + \left(-1 - x\right)} - e^{-x} \cdot \color{blue}{\left(\left(-x\right) + -1\right)}}{2} \]
   rational.json-simplify-15 [=>] 0.6	\[ \frac{\left(x + 1\right) \cdot e^{1 + \left(-1 - x\right)} - e^{-x} \cdot \color{blue}{\left(\left(-x\right) - 1\right)}}{2} \]
   rational.json-simplify-12 [=>] 0.6	\[ \frac{\left(x + 1\right) \cdot e^{1 + \left(-1 - x\right)} - e^{-x} \cdot \left(\color{blue}{\left(0 - x\right)} - 1\right)}{2} \]
   rational.json-simplify-46 [<=] 0.6	\[ \frac{\left(x + 1\right) \cdot e^{1 + \left(-1 - x\right)} - e^{-x} \cdot \color{blue}{\left(0 - \left(x + 1\right)\right)}}{2} \]
   rational.json-simplify-1 [=>] 0.6	\[ \frac{\left(x + 1\right) \cdot e^{1 + \left(-1 - x\right)} - e^{-x} \cdot \left(0 - \color{blue}{\left(1 + x\right)}\right)}{2} \]
   rational.json-simplify-46 [=>] 0.6	\[ \frac{\left(x + 1\right) \cdot e^{1 + \left(-1 - x\right)} - e^{-x} \cdot \color{blue}{\left(\left(0 - 1\right) - x\right)}}{2} \]
   metadata-eval [=>] 0.6	\[ \frac{\left(x + 1\right) \cdot e^{1 + \left(-1 - x\right)} - e^{-x} \cdot \left(\color{blue}{-1} - x\right)}{2} \]

9. Final simplification 0.6

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

Alternatives

Alternative 1
Error	0.6
Cost	13760

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

Alternative 2
Error	1.0
Cost	13700

\[\begin{array}{l} \mathbf{if}\;x \leq 360:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 - \varepsilon\right)} - \left(-e^{\varepsilon \cdot x}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0}{\varepsilon}}{2}\\ \end{array} \]

Alternative 3
Error	1.0
Cost	13696

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

Alternative 4
Error	1.1
Cost	7172

\[\begin{array}{l} \mathbf{if}\;x \leq 360:\\ \;\;\;\;\frac{\left(x + 1\right) \cdot e^{-x} - -1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0}{\varepsilon}}{2}\\ \end{array} \]

Alternative 5
Error	1.1
Cost	7044

\[\begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;\frac{2 + -0.5 \cdot {x}^{2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0}{\varepsilon}}{2}\\ \end{array} \]

Alternative 6
Error	1.1
Cost	452

\[\begin{array}{l} \mathbf{if}\;x \leq 360:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0}{\varepsilon}}{2}\\ \end{array} \]

Alternative 7
Error	54.6
Cost	64

\[0.5 \]

Alternative 8
Error	16.7
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023077 
(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))