NMSE Section 6.1 mentioned, A

Average Accuracy: 53.6% → 98.8%

Time: 17.8s

Precision: binary64

Cost: 13568

Math

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

↓

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

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
 (/ (+ (/ 2.0 (exp x)) (* x (* 2.0 (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) {
	return ((2.0 / exp(x)) + (x * (2.0 * 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
    code = ((2.0d0 / exp(x)) + (x * (2.0d0 * 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) {
	return ((2.0 / Math.exp(x)) + (x * (2.0 * 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):
	return ((2.0 / math.exp(x)) + (x * (2.0 * 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)
	return Float64(Float64(Float64(2.0 / exp(x)) + Float64(x * Float64(2.0 * exp(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 = ((2.0 / exp(x)) + (x * (2.0 * 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_] := N[(N[(N[(2.0 / N[Exp[x], $MachinePrecision]), $MachinePrecision] + N[(x * N[(2.0 * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Derivation

1. Initial program 53.6%

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

2. Simplified 31.4%

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

   [Start] 53.6	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

3. Taylor expanded in eps around inf 53.6%

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

4. Taylor expanded in eps around 0 52.4%

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

5. Simplified 98.8%

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

   [Start] 52.4	\[ \frac{\left(\frac{e^{-1 \cdot x}}{\varepsilon} + \left(e^{-1 \cdot x} \cdot x + \left(\frac{1}{e^{x}} + e^{-1 \cdot x}\right)\right)\right) - \left(\frac{1}{\varepsilon \cdot e^{x}} + -1 \cdot \frac{x}{e^{x}}\right)}{2} \]
   +-commutative [=>] 52.4	\[ \frac{\color{blue}{\left(\left(e^{-1 \cdot x} \cdot x + \left(\frac{1}{e^{x}} + e^{-1 \cdot x}\right)\right) + \frac{e^{-1 \cdot x}}{\varepsilon}\right)} - \left(\frac{1}{\varepsilon \cdot e^{x}} + -1 \cdot \frac{x}{e^{x}}\right)}{2} \]
   associate--l+ [=>] 96.5	\[ \frac{\color{blue}{\left(e^{-1 \cdot x} \cdot x + \left(\frac{1}{e^{x}} + e^{-1 \cdot x}\right)\right) + \left(\frac{e^{-1 \cdot x}}{\varepsilon} - \left(\frac{1}{\varepsilon \cdot e^{x}} + -1 \cdot \frac{x}{e^{x}}\right)\right)}}{2} \]
   +-commutative [=>] 96.5	\[ \frac{\color{blue}{\left(\frac{e^{-1 \cdot x}}{\varepsilon} - \left(\frac{1}{\varepsilon \cdot e^{x}} + -1 \cdot \frac{x}{e^{x}}\right)\right) + \left(e^{-1 \cdot x} \cdot x + \left(\frac{1}{e^{x}} + e^{-1 \cdot x}\right)\right)}}{2} \]

6. Taylor expanded in x around -inf 98.8%

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

7. Simplified 98.8%

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

   [Start] 98.8	\[ \frac{-1 \cdot \left(\left(-1 \cdot e^{-1 \cdot x} - \frac{1}{e^{x}}\right) \cdot x\right) + 2 \cdot e^{-1 \cdot x}}{2} \]
   +-commutative [=>] 98.8	\[ \frac{\color{blue}{2 \cdot e^{-1 \cdot x} + -1 \cdot \left(\left(-1 \cdot e^{-1 \cdot x} - \frac{1}{e^{x}}\right) \cdot x\right)}}{2} \]
   mul-1-neg [=>] 98.8	\[ \frac{2 \cdot e^{\color{blue}{-x}} + -1 \cdot \left(\left(-1 \cdot e^{-1 \cdot x} - \frac{1}{e^{x}}\right) \cdot x\right)}{2} \]
   mul-1-neg [=>] 98.8	\[ \frac{2 \cdot e^{-x} + \color{blue}{\left(-\left(-1 \cdot e^{-1 \cdot x} - \frac{1}{e^{x}}\right) \cdot x\right)}}{2} \]
   unsub-neg [=>] 98.8	\[ \frac{\color{blue}{2 \cdot e^{-x} - \left(-1 \cdot e^{-1 \cdot x} - \frac{1}{e^{x}}\right) \cdot x}}{2} \]
   exp-neg [=>] 98.8	\[ \frac{2 \cdot \color{blue}{\frac{1}{e^{x}}} - \left(-1 \cdot e^{-1 \cdot x} - \frac{1}{e^{x}}\right) \cdot x}{2} \]
   associate-*r/ [=>] 98.8	\[ \frac{\color{blue}{\frac{2 \cdot 1}{e^{x}}} - \left(-1 \cdot e^{-1 \cdot x} - \frac{1}{e^{x}}\right) \cdot x}{2} \]
   metadata-eval [=>] 98.8	\[ \frac{\frac{\color{blue}{2}}{e^{x}} - \left(-1 \cdot e^{-1 \cdot x} - \frac{1}{e^{x}}\right) \cdot x}{2} \]
   *-commutative [=>] 98.8	\[ \frac{\frac{2}{e^{x}} - \color{blue}{x \cdot \left(-1 \cdot e^{-1 \cdot x} - \frac{1}{e^{x}}\right)}}{2} \]
   sub-neg [=>] 98.8	\[ \frac{\frac{2}{e^{x}} - x \cdot \color{blue}{\left(-1 \cdot e^{-1 \cdot x} + \left(-\frac{1}{e^{x}}\right)\right)}}{2} \]
   mul-1-neg [=>] 98.8	\[ \frac{\frac{2}{e^{x}} - x \cdot \left(-1 \cdot e^{\color{blue}{-x}} + \left(-\frac{1}{e^{x}}\right)\right)}{2} \]
   exp-neg [=>] 98.8	\[ \frac{\frac{2}{e^{x}} - x \cdot \left(-1 \cdot \color{blue}{\frac{1}{e^{x}}} + \left(-\frac{1}{e^{x}}\right)\right)}{2} \]
   associate-*r/ [=>] 98.8	\[ \frac{\frac{2}{e^{x}} - x \cdot \left(\color{blue}{\frac{-1 \cdot 1}{e^{x}}} + \left(-\frac{1}{e^{x}}\right)\right)}{2} \]
   metadata-eval [=>] 98.8	\[ \frac{\frac{2}{e^{x}} - x \cdot \left(\frac{\color{blue}{-1}}{e^{x}} + \left(-\frac{1}{e^{x}}\right)\right)}{2} \]
   distribute-neg-frac [=>] 98.8	\[ \frac{\frac{2}{e^{x}} - x \cdot \left(\frac{-1}{e^{x}} + \color{blue}{\frac{-1}{e^{x}}}\right)}{2} \]
   metadata-eval [=>] 98.8	\[ \frac{\frac{2}{e^{x}} - x \cdot \left(\frac{-1}{e^{x}} + \frac{\color{blue}{-1}}{e^{x}}\right)}{2} \]

8. Final simplification 98.8%

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

Alternatives

Alternative 1
Accuracy	98.0%
Cost	7108

\[\begin{array}{l} t_0 := \frac{x}{e^{x}}\\ \mathbf{if}\;x \leq 1.7:\\ \;\;\;\;\frac{t_0 + \left(2 - x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot t_0}{2}\\ \end{array} \]

Alternative 2
Accuracy	98.8%
Cost	7040

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

Alternative 3
Accuracy	98.0%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;x \leq 1.15:\\ \;\;\;\;\frac{2 - x \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{x}{e^{x}}}{2}\\ \end{array} \]

Alternative 4
Accuracy	97.9%
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \leq 1.42:\\ \;\;\;\;\frac{2 - x \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 5
Accuracy	97.7%
Cost	196

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

Alternative 6
Accuracy	26.5%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023135 
(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))