NMSE Section 6.1 mentioned, A

Average Accuracy: 53.4% → 99.1%

Time: 13.9s

Precision: binary64

Cost: 6976

Math

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

↓

\[\frac{2 \cdot \frac{1 + x}{e^{x}}}{2} \]

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 (/ (* 2.0 (/ (+ 1.0 x) (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) {
	return (2.0 * ((1.0 + x) / 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
    code = (2.0d0 * ((1.0d0 + x) / 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) {
	return (2.0 * ((1.0 + x) / 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):
	return (2.0 * ((1.0 + x) / 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)
	return Float64(Float64(2.0 * Float64(Float64(1.0 + x) / 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)
	tmp = (2.0 * ((1.0 + x) / 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_] := N[(N[(2.0 * N[(N[(1.0 + x), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Derivation

1. Initial program 53.4%

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

2. Simplified 31.3%

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

   [Start] 53.4	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

3. Taylor expanded in eps around 0 52.6%

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

4. Simplified 99.1%

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

   [Start] 52.6	\[ \frac{\left(\frac{e^{-1 \cdot x}}{\varepsilon} + \left(e^{-1 \cdot x} \cdot x + \left(\frac{1}{e^{x}} + e^{-1 \cdot x}\right)\right)\right) - \left(\frac{1}{\varepsilon \cdot e^{x}} + -1 \cdot \frac{x}{e^{x}}\right)}{2} \]
   +-commutative [=>] 52.6	\[ \frac{\color{blue}{\left(\left(e^{-1 \cdot x} \cdot x + \left(\frac{1}{e^{x}} + e^{-1 \cdot x}\right)\right) + \frac{e^{-1 \cdot x}}{\varepsilon}\right)} - \left(\frac{1}{\varepsilon \cdot e^{x}} + -1 \cdot \frac{x}{e^{x}}\right)}{2} \]
   associate--l+ [=>] 97.2	\[ \frac{\color{blue}{\left(e^{-1 \cdot x} \cdot x + \left(\frac{1}{e^{x}} + e^{-1 \cdot x}\right)\right) + \left(\frac{e^{-1 \cdot x}}{\varepsilon} - \left(\frac{1}{\varepsilon \cdot e^{x}} + -1 \cdot \frac{x}{e^{x}}\right)\right)}}{2} \]
   mul-1-neg [=>] 97.2	\[ \frac{\left(e^{\color{blue}{-x}} \cdot x + \left(\frac{1}{e^{x}} + e^{-1 \cdot x}\right)\right) + \left(\frac{e^{-1 \cdot x}}{\varepsilon} - \left(\frac{1}{\varepsilon \cdot e^{x}} + -1 \cdot \frac{x}{e^{x}}\right)\right)}{2} \]
   rec-exp [<=] 97.2	\[ \frac{\left(\color{blue}{\frac{1}{e^{x}}} \cdot x + \left(\frac{1}{e^{x}} + e^{-1 \cdot x}\right)\right) + \left(\frac{e^{-1 \cdot x}}{\varepsilon} - \left(\frac{1}{\varepsilon \cdot e^{x}} + -1 \cdot \frac{x}{e^{x}}\right)\right)}{2} \]
   associate-*l/ [=>] 97.2	\[ \frac{\left(\color{blue}{\frac{1 \cdot x}{e^{x}}} + \left(\frac{1}{e^{x}} + e^{-1 \cdot x}\right)\right) + \left(\frac{e^{-1 \cdot x}}{\varepsilon} - \left(\frac{1}{\varepsilon \cdot e^{x}} + -1 \cdot \frac{x}{e^{x}}\right)\right)}{2} \]
   *-commutative [<=] 97.2	\[ \frac{\left(\frac{\color{blue}{x \cdot 1}}{e^{x}} + \left(\frac{1}{e^{x}} + e^{-1 \cdot x}\right)\right) + \left(\frac{e^{-1 \cdot x}}{\varepsilon} - \left(\frac{1}{\varepsilon \cdot e^{x}} + -1 \cdot \frac{x}{e^{x}}\right)\right)}{2} \]
   *-rgt-identity [=>] 97.2	\[ \frac{\left(\frac{\color{blue}{x}}{e^{x}} + \left(\frac{1}{e^{x}} + e^{-1 \cdot x}\right)\right) + \left(\frac{e^{-1 \cdot x}}{\varepsilon} - \left(\frac{1}{\varepsilon \cdot e^{x}} + -1 \cdot \frac{x}{e^{x}}\right)\right)}{2} \]
   rec-exp [=>] 97.2	\[ \frac{\left(\frac{x}{e^{x}} + \left(\color{blue}{e^{-x}} + e^{-1 \cdot x}\right)\right) + \left(\frac{e^{-1 \cdot x}}{\varepsilon} - \left(\frac{1}{\varepsilon \cdot e^{x}} + -1 \cdot \frac{x}{e^{x}}\right)\right)}{2} \]
   mul-1-neg [<=] 97.2	\[ \frac{\left(\frac{x}{e^{x}} + \left(e^{\color{blue}{-1 \cdot x}} + e^{-1 \cdot x}\right)\right) + \left(\frac{e^{-1 \cdot x}}{\varepsilon} - \left(\frac{1}{\varepsilon \cdot e^{x}} + -1 \cdot \frac{x}{e^{x}}\right)\right)}{2} \]
   count-2 [=>] 97.2	\[ \frac{\left(\frac{x}{e^{x}} + \color{blue}{2 \cdot e^{-1 \cdot x}}\right) + \left(\frac{e^{-1 \cdot x}}{\varepsilon} - \left(\frac{1}{\varepsilon \cdot e^{x}} + -1 \cdot \frac{x}{e^{x}}\right)\right)}{2} \]
   mul-1-neg [=>] 97.2	\[ \frac{\left(\frac{x}{e^{x}} + 2 \cdot e^{\color{blue}{-x}}\right) + \left(\frac{e^{-1 \cdot x}}{\varepsilon} - \left(\frac{1}{\varepsilon \cdot e^{x}} + -1 \cdot \frac{x}{e^{x}}\right)\right)}{2} \]

5. Taylor expanded in x around inf 99.1%

   \[\leadsto \frac{\color{blue}{2 \cdot \frac{x}{e^{x}} + 2 \cdot e^{-x}}}{2} \]

6. Simplified 99.1%

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

   [Start] 99.1	\[ \frac{2 \cdot \frac{x}{e^{x}} + 2 \cdot e^{-x}}{2} \]
   distribute-lft-out [=>] 99.1	\[ \frac{\color{blue}{2 \cdot \left(\frac{x}{e^{x}} + e^{-x}\right)}}{2} \]

7. Applied egg-rr 74.3%

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

   [Start] 99.1	\[ \frac{2 \cdot \left(\frac{x}{e^{x}} + e^{-x}\right)}{2} \]
   +-commutative [=>] 99.1	\[ \frac{2 \cdot \color{blue}{\left(e^{-x} + \frac{x}{e^{x}}\right)}}{2} \]
   exp-neg [=>] 99.1	\[ \frac{2 \cdot \left(\color{blue}{\frac{1}{e^{x}}} + \frac{x}{e^{x}}\right)}{2} \]
   frac-add [=>] 74.3	\[ \frac{2 \cdot \color{blue}{\frac{1 \cdot e^{x} + e^{x} \cdot x}{e^{x} \cdot e^{x}}}}{2} \]

8. Simplified 99.1%

   \[\leadsto \frac{2 \cdot \color{blue}{\frac{1 + x}{e^{x}}}}{2} \]
   Proof

   [Start] 74.3	\[ \frac{2 \cdot \frac{\mathsf{fma}\left(x, e^{x}, e^{x}\right)}{e^{x + x}}}{2} \]
   fma-udef [=>] 74.3	\[ \frac{2 \cdot \frac{\color{blue}{x \cdot e^{x} + e^{x}}}{e^{x + x}}}{2} \]
   distribute-lft1-in [=>] 74.3	\[ \frac{2 \cdot \frac{\color{blue}{\left(x + 1\right) \cdot e^{x}}}{e^{x + x}}}{2} \]
   associate-/l* [=>] 74.3	\[ \frac{2 \cdot \color{blue}{\frac{x + 1}{\frac{e^{x + x}}{e^{x}}}}}{2} \]
   +-commutative [=>] 74.3	\[ \frac{2 \cdot \frac{\color{blue}{1 + x}}{\frac{e^{x + x}}{e^{x}}}}{2} \]
   exp-sum [=>] 74.3	\[ \frac{2 \cdot \frac{1 + x}{\frac{\color{blue}{e^{x} \cdot e^{x}}}{e^{x}}}}{2} \]
   associate-/l* [=>] 74.4	\[ \frac{2 \cdot \frac{1 + x}{\color{blue}{\frac{e^{x}}{\frac{e^{x}}{e^{x}}}}}}{2} \]
   associate-/r/ [=>] 74.4	\[ \frac{2 \cdot \frac{1 + x}{\color{blue}{\frac{e^{x}}{e^{x}} \cdot e^{x}}}}{2} \]
   *-inverses [=>] 99.1	\[ \frac{2 \cdot \frac{1 + x}{\color{blue}{1} \cdot e^{x}}}{2} \]
   *-lft-identity [=>] 99.1	\[ \frac{2 \cdot \frac{1 + x}{\color{blue}{e^{x}}}}{2} \]

9. Final simplification 99.1%

   \[\leadsto \frac{2 \cdot \frac{1 + x}{e^{x}}}{2} \]

Alternatives

Alternative 1
Accuracy	98.5%
Cost	580

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

Alternative 2
Accuracy	98.3%
Cost	196

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

Alternative 3
Accuracy	27.1%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023126 
(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))