NMSE Section 6.1 mentioned, A

Average Error: 29.2 → 0.7

Time: 15.6s

Precision: binary64

Cost: 13632

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x + 1}{e^{x}}\\ \frac{t_0 + t_0}{2} \end{array} \]

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
 (let* ((t_0 (/ (+ x 1.0) (exp x)))) (/ (+ t_0 t_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) {
	double t_0 = (x + 1.0) / exp(x);
	return (t_0 + t_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
    real(8) :: t_0
    t_0 = (x + 1.0d0) / exp(x)
    code = (t_0 + t_0) / 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) {
	double t_0 = (x + 1.0) / Math.exp(x);
	return (t_0 + t_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):
	t_0 = (x + 1.0) / math.exp(x)
	return (t_0 + t_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)
	t_0 = Float64(Float64(x + 1.0) / exp(x))
	return Float64(Float64(t_0 + t_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)
	t_0 = (x + 1.0) / exp(x);
	tmp = (t_0 + t_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_] := Block[{t$95$0 = N[(N[(x + 1.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]}, N[(N[(t$95$0 + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Initial program 29.2

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

2. Simplified 43.8

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

   [Start] 29.2	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   distribute-rgt-neg-in [=>] 29.2	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\color{blue}{\left(1 - \varepsilon\right) \cdot \left(-x\right)}} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   exp-prod [=>] 43.8	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{{\left(e^{1 - \varepsilon}\right)}^{\left(-x\right)}} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   sub-neg [=>] 43.8	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot {\left(e^{1 - \varepsilon}\right)}^{\left(-x\right)} - \color{blue}{\left(\frac{1}{\varepsilon} + \left(-1\right)\right)} \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   metadata-eval [=>] 43.8	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot {\left(e^{1 - \varepsilon}\right)}^{\left(-x\right)} - \left(\frac{1}{\varepsilon} + \color{blue}{-1}\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   distribute-lft-neg-in [=>] 43.8	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot {\left(e^{1 - \varepsilon}\right)}^{\left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\color{blue}{\left(-\left(1 + \varepsilon\right)\right) \cdot x}}}{2} \]
   *-commutative [=>] 43.8	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot {\left(e^{1 - \varepsilon}\right)}^{\left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\color{blue}{x \cdot \left(-\left(1 + \varepsilon\right)\right)}}}{2} \]
   +-commutative [=>] 43.8	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot {\left(e^{1 - \varepsilon}\right)}^{\left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{x \cdot \left(-\color{blue}{\left(\varepsilon + 1\right)}\right)}}{2} \]
   distribute-neg-in [=>] 43.8	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot {\left(e^{1 - \varepsilon}\right)}^{\left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{x \cdot \color{blue}{\left(\left(-\varepsilon\right) + \left(-1\right)\right)}}}{2} \]
   metadata-eval [=>] 43.8	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot {\left(e^{1 - \varepsilon}\right)}^{\left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{x \cdot \left(\left(-\varepsilon\right) + \color{blue}{-1}\right)}}{2} \]

3. Taylor expanded in eps around 0 0.7

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

4. Simplified 0.7

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

   [Start] 0.7	\[ \frac{\left(e^{-1 \cdot x} \cdot x + e^{-1 \cdot x}\right) - \left(-1 \cdot \left(e^{-1 \cdot x} \cdot x\right) + -1 \cdot e^{-1 \cdot x}\right)}{2} \]
   *-commutative [=>] 0.7	\[ \frac{\left(\color{blue}{x \cdot e^{-1 \cdot x}} + e^{-1 \cdot x}\right) - \left(-1 \cdot \left(e^{-1 \cdot x} \cdot x\right) + -1 \cdot e^{-1 \cdot x}\right)}{2} \]
   distribute-lft1-in [=>] 0.7	\[ \frac{\color{blue}{\left(x + 1\right) \cdot e^{-1 \cdot x}} - \left(-1 \cdot \left(e^{-1 \cdot x} \cdot x\right) + -1 \cdot e^{-1 \cdot x}\right)}{2} \]
   mul-1-neg [=>] 0.7	\[ \frac{\left(x + 1\right) \cdot e^{\color{blue}{-x}} - \left(-1 \cdot \left(e^{-1 \cdot x} \cdot x\right) + -1 \cdot e^{-1 \cdot x}\right)}{2} \]
   distribute-lft-out [=>] 0.7	\[ \frac{\left(x + 1\right) \cdot e^{-x} - \color{blue}{-1 \cdot \left(e^{-1 \cdot x} \cdot x + e^{-1 \cdot x}\right)}}{2} \]
   mul-1-neg [=>] 0.7	\[ \frac{\left(x + 1\right) \cdot e^{-x} - \color{blue}{\left(-\left(e^{-1 \cdot x} \cdot x + e^{-1 \cdot x}\right)\right)}}{2} \]
   *-commutative [=>] 0.7	\[ \frac{\left(x + 1\right) \cdot e^{-x} - \left(-\left(\color{blue}{x \cdot e^{-1 \cdot x}} + e^{-1 \cdot x}\right)\right)}{2} \]
   distribute-lft1-in [=>] 0.7	\[ \frac{\left(x + 1\right) \cdot e^{-x} - \left(-\color{blue}{\left(x + 1\right) \cdot e^{-1 \cdot x}}\right)}{2} \]
   mul-1-neg [=>] 0.7	\[ \frac{\left(x + 1\right) \cdot e^{-x} - \left(-\left(x + 1\right) \cdot e^{\color{blue}{-x}}\right)}{2} \]

5. Applied egg-rr 0.7

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

6. Applied egg-rr 0.7

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

7. Final simplification 0.7

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

Alternatives

Alternative 1
Error	1.1
Cost	8068

\[\begin{array}{l} \mathbf{if}\;x \leq 2.05:\\ \;\;\;\;\frac{\left(x + 1\right) \cdot e^{-x} - \left(-1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(-0.3333333333333333 - x \cdot -0.125\right)\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 2
Error	1.1
Cost	8004

\[\begin{array}{l} \mathbf{if}\;x \leq 2.05:\\ \;\;\;\;\frac{\frac{x + 1}{e^{x}} - \left(-1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(-0.3333333333333333 - x \cdot -0.125\right)\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 3
Error	1.1
Cost	7812

\[\begin{array}{l} \mathbf{if}\;x \leq 350:\\ \;\;\;\;\frac{\left(x + 1\right) \cdot e^{-x} + \left(1 - \left(x \cdot x\right) \cdot \left(0.5 + x \cdot -0.3333333333333333\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 4
Error	1.2
Cost	7556

\[\begin{array}{l} \mathbf{if}\;x \leq 1.75:\\ \;\;\;\;\frac{\left(x + 1\right) \cdot e^{-x} + \left(1 + \left(x \cdot x\right) \cdot -0.5\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 5
Error	1.2
Cost	7492

\[\begin{array}{l} \mathbf{if}\;x \leq 1.75:\\ \;\;\;\;\frac{\frac{x + 1}{e^{x}} + \left(1 + \left(x \cdot x\right) \cdot -0.5\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 6
Error	1.2
Cost	1732

\[\begin{array}{l} \mathbf{if}\;x \leq 1.56:\\ \;\;\;\;\frac{\left(1 + \left(x \cdot x\right) \cdot -0.5\right) - \left(-1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(-0.3333333333333333 - x \cdot -0.125\right)\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 7
Error	1.3
Cost	1348

\[\begin{array}{l} \mathbf{if}\;x \leq 2.3:\\ \;\;\;\;\frac{1 - \left(-1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(-0.3333333333333333 - x \cdot -0.125\right)\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 8
Error	1.3
Cost	196

\[\begin{array}{l} \mathbf{if}\;x \leq 370:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 9
Error	46.7
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2022364 
(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))