NMSE Section 6.1 mentioned, A

Average Accuracy: 54.1% → 98.6%

Time: 17.3s

Precision: binary64

Cost: 26496

Math

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

↓

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

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
 (/
  (+ (exp (* x (- -1.0 eps))) (pow (sqrt (exp (* x (+ -1.0 eps)))) 2.0))
  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 (exp((x * (-1.0 - eps))) + pow(sqrt(exp((x * (-1.0 + eps)))), 2.0)) / 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 = (exp((x * ((-1.0d0) - eps))) + (sqrt(exp((x * ((-1.0d0) + eps)))) ** 2.0d0)) / 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 (Math.exp((x * (-1.0 - eps))) + Math.pow(Math.sqrt(Math.exp((x * (-1.0 + eps)))), 2.0)) / 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 (math.exp((x * (-1.0 - eps))) + math.pow(math.sqrt(math.exp((x * (-1.0 + eps)))), 2.0)) / 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(exp(Float64(x * Float64(-1.0 - eps))) + (sqrt(exp(Float64(x * Float64(-1.0 + eps)))) ^ 2.0)) / 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 = (exp((x * (-1.0 - eps))) + (sqrt(exp((x * (-1.0 + eps)))) ^ 2.0)) / 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[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Power[N[Sqrt[N[Exp[N[(x * N[(-1.0 + eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Derivation

1. Initial program 54.1%

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

2. Simplified 54.1%

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

   [Start] 54.1	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   distribute-rgt-neg-in [=>] 54.1	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\color{blue}{\left(1 - \varepsilon\right) \cdot \left(-x\right)}} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   sub-neg [=>] 54.1	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \color{blue}{\left(\frac{1}{\varepsilon} + \left(-1\right)\right)} \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   metadata-eval [=>] 54.1	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + \color{blue}{-1}\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   distribute-rgt-neg-in [=>] 54.1	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\color{blue}{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}}{2} \]

3. Taylor expanded in eps around inf 98.6%

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

4. Simplified 98.6%

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

   [Start] 98.6	\[ \frac{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}{2} \]
   mul-1-neg [=>] 98.6	\[ \frac{e^{\color{blue}{-\left(1 - \varepsilon\right) \cdot x}} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}{2} \]
   *-commutative [=>] 98.6	\[ \frac{e^{-\color{blue}{x \cdot \left(1 - \varepsilon\right)}} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}{2} \]
   mul-1-neg [=>] 98.6	\[ \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \color{blue}{\left(-e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}\right)}}{2} \]
   mul-1-neg [=>] 98.6	\[ \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{\color{blue}{-\left(\varepsilon + 1\right) \cdot x}}\right)}{2} \]
   +-commutative [<=] 98.6	\[ \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-\color{blue}{\left(1 + \varepsilon\right)} \cdot x}\right)}{2} \]
   *-commutative [=>] 98.6	\[ \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-\color{blue}{x \cdot \left(1 + \varepsilon\right)}}\right)}{2} \]
   +-commutative [=>] 98.6	\[ \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \color{blue}{\left(\varepsilon + 1\right)}}\right)}{2} \]

5. Taylor expanded in x around inf 98.6%

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

6. Simplified 98.6%

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

   [Start] 98.6	\[ \frac{e^{-\left(\varepsilon \cdot x + x\right)} + e^{\varepsilon \cdot x - x}}{2} \]
   distribute-neg-in [=>] 98.6	\[ \frac{e^{\color{blue}{\left(-\varepsilon \cdot x\right) + \left(-x\right)}} + e^{\varepsilon \cdot x - x}}{2} \]
   distribute-lft-neg-out [<=] 98.6	\[ \frac{e^{\color{blue}{\left(-\varepsilon\right) \cdot x} + \left(-x\right)} + e^{\varepsilon \cdot x - x}}{2} \]
   neg-mul-1 [=>] 98.6	\[ \frac{e^{\left(-\varepsilon\right) \cdot x + \color{blue}{-1 \cdot x}} + e^{\varepsilon \cdot x - x}}{2} \]
   distribute-rgt-out [=>] 98.6	\[ \frac{e^{\color{blue}{x \cdot \left(\left(-\varepsilon\right) + -1\right)}} + e^{\varepsilon \cdot x - x}}{2} \]
   +-commutative [=>] 98.6	\[ \frac{e^{x \cdot \color{blue}{\left(-1 + \left(-\varepsilon\right)\right)}} + e^{\varepsilon \cdot x - x}}{2} \]
   sub-neg [<=] 98.6	\[ \frac{e^{x \cdot \color{blue}{\left(-1 - \varepsilon\right)}} + e^{\varepsilon \cdot x - x}}{2} \]

7. Applied egg-rr 98.6%

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

   [Start] 98.6	\[ \frac{e^{x \cdot \left(-1 - \varepsilon\right)} + e^{\varepsilon \cdot x - x}}{2} \]
   add-sqr-sqrt [=>] 98.6	\[ \frac{e^{x \cdot \left(-1 - \varepsilon\right)} + \color{blue}{\sqrt{e^{\varepsilon \cdot x - x}} \cdot \sqrt{e^{\varepsilon \cdot x - x}}}}{2} \]
   pow2 [=>] 98.6	\[ \frac{e^{x \cdot \left(-1 - \varepsilon\right)} + \color{blue}{{\left(\sqrt{e^{\varepsilon \cdot x - x}}\right)}^{2}}}{2} \]
   *-un-lft-identity [=>] 98.6	\[ \frac{e^{x \cdot \left(-1 - \varepsilon\right)} + {\left(\sqrt{e^{\varepsilon \cdot x - \color{blue}{1 \cdot x}}}\right)}^{2}}{2} \]
   distribute-rgt-out-- [=>] 98.6	\[ \frac{e^{x \cdot \left(-1 - \varepsilon\right)} + {\left(\sqrt{e^{\color{blue}{x \cdot \left(\varepsilon - 1\right)}}}\right)}^{2}}{2} \]
   expm1-log1p-u [=>] 86.6	\[ \frac{e^{x \cdot \left(-1 - \varepsilon\right)} + {\left(\sqrt{e^{x \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\varepsilon\right)\right)} - 1\right)}}\right)}^{2}}{2} \]
   expm1-udef [=>] 86.6	\[ \frac{e^{x \cdot \left(-1 - \varepsilon\right)} + {\left(\sqrt{e^{x \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\varepsilon\right)} - 1\right)} - 1\right)}}\right)}^{2}}{2} \]
   associate--l- [=>] 86.6	\[ \frac{e^{x \cdot \left(-1 - \varepsilon\right)} + {\left(\sqrt{e^{x \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\varepsilon\right)} - \left(1 + 1\right)\right)}}}\right)}^{2}}{2} \]
   metadata-eval [=>] 86.6	\[ \frac{e^{x \cdot \left(-1 - \varepsilon\right)} + {\left(\sqrt{e^{x \cdot \left(e^{\mathsf{log1p}\left(\varepsilon\right)} - \color{blue}{2}\right)}}\right)}^{2}}{2} \]
   metadata-eval [<=] 86.6	\[ \frac{e^{x \cdot \left(-1 - \varepsilon\right)} + {\left(\sqrt{e^{x \cdot \left(e^{\mathsf{log1p}\left(\varepsilon\right)} - \color{blue}{\left(1 - -1\right)}\right)}}\right)}^{2}}{2} \]
   associate-+l- [<=] 86.6	\[ \frac{e^{x \cdot \left(-1 - \varepsilon\right)} + {\left(\sqrt{e^{x \cdot \color{blue}{\left(\left(e^{\mathsf{log1p}\left(\varepsilon\right)} - 1\right) + -1\right)}}}\right)}^{2}}{2} \]
   expm1-udef [<=] 86.6	\[ \frac{e^{x \cdot \left(-1 - \varepsilon\right)} + {\left(\sqrt{e^{x \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\varepsilon\right)\right)} + -1\right)}}\right)}^{2}}{2} \]
   expm1-log1p-u [<=] 98.6	\[ \frac{e^{x \cdot \left(-1 - \varepsilon\right)} + {\left(\sqrt{e^{x \cdot \left(\color{blue}{\varepsilon} + -1\right)}}\right)}^{2}}{2} \]
   +-commutative [<=] 98.6	\[ \frac{e^{x \cdot \left(-1 - \varepsilon\right)} + {\left(\sqrt{e^{x \cdot \color{blue}{\left(-1 + \varepsilon\right)}}}\right)}^{2}}{2} \]

8. Final simplification 98.6%

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

Alternatives

Alternative 1
Accuracy	98.6%
Cost	13632

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

Alternative 2
Accuracy	99.1%
Cost	6976

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

Alternative 3
Accuracy	97.8%
Cost	6720

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

Alternative 4
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 5
Accuracy	98.3%
Cost	196

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

Alternative 6
Accuracy	26.9%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023143 
(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))