NMSE Section 6.1 mentioned, A

Average Error: 29.5 → 0.6

Time: 11.7s

Precision: binary64

Cost: 13504

Math

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

↓

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

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 (exp (- x)))) (/ (* 2.0 (+ t_0 (* x 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 = exp(-x);
	return (2.0 * (t_0 + (x * 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 = exp(-x)
    code = (2.0d0 * (t_0 + (x * 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 = Math.exp(-x);
	return (2.0 * (t_0 + (x * 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 = math.exp(-x)
	return (2.0 * (t_0 + (x * 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 = exp(Float64(-x))
	return Float64(Float64(2.0 * Float64(t_0 + Float64(x * 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 = exp(-x);
	tmp = (2.0 * (t_0 + (x * 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[Exp[(-x)], $MachinePrecision]}, N[(N[(2.0 * N[(t$95$0 + N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Initial program 29.5

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

2. Taylor expanded in eps around 0 0.6

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

3. Simplified 0.6

   \[\leadsto \frac{\color{blue}{2 \cdot \left(e^{-x} \cdot \left(x - -1\right)\right)}}{2} \]
   Proof
   (*.f64 2 (*.f64 (exp.f64 (neg.f64 x)) (-.f64 x -1))): 0 points increase in error, 0 points decrease in error
   (*.f64 (Rewrite<= metadata-eval (+.f64 1 1)) (*.f64 (exp.f64 (neg.f64 x)) (-.f64 x -1))): 0 points increase in error, 0 points decrease in error
   (*.f64 (+.f64 1 1) (Rewrite<= distribute-lft-out--_binary64 (-.f64 (*.f64 (exp.f64 (neg.f64 x)) x) (*.f64 (exp.f64 (neg.f64 x)) -1)))): 3 points increase in error, 2 points decrease in error
   (*.f64 (+.f64 1 1) (-.f64 (*.f64 (exp.f64 (neg.f64 x)) x) (Rewrite<= *-commutative_binary64 (*.f64 -1 (exp.f64 (neg.f64 x)))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (+.f64 1 1) (Rewrite=> sub-neg_binary64 (+.f64 (*.f64 (exp.f64 (neg.f64 x)) x) (neg.f64 (*.f64 -1 (exp.f64 (neg.f64 x))))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (+.f64 1 1) (+.f64 (*.f64 (exp.f64 (neg.f64 x)) x) (neg.f64 (Rewrite=> mul-1-neg_binary64 (neg.f64 (exp.f64 (neg.f64 x))))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (+.f64 1 1) (+.f64 (*.f64 (exp.f64 (neg.f64 x)) x) (Rewrite=> remove-double-neg_binary64 (exp.f64 (neg.f64 x))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (+.f64 1 1) (Rewrite<= +-commutative_binary64 (+.f64 (exp.f64 (neg.f64 x)) (*.f64 (exp.f64 (neg.f64 x)) x)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= distribute-rgt1-in_binary64 (+.f64 (+.f64 (exp.f64 (neg.f64 x)) (*.f64 (exp.f64 (neg.f64 x)) x)) (*.f64 1 (+.f64 (exp.f64 (neg.f64 x)) (*.f64 (exp.f64 (neg.f64 x)) x))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (+.f64 (exp.f64 (neg.f64 x)) (*.f64 (exp.f64 (neg.f64 x)) x)) (*.f64 (Rewrite<= metadata-eval (neg.f64 -1)) (+.f64 (exp.f64 (neg.f64 x)) (*.f64 (exp.f64 (neg.f64 x)) x)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 (+.f64 (exp.f64 (neg.f64 x)) (*.f64 (exp.f64 (neg.f64 x)) x)) (*.f64 -1 (+.f64 (exp.f64 (neg.f64 x)) (*.f64 (exp.f64 (neg.f64 x)) x))))): 0 points increase in error, 0 points decrease in error
   (-.f64 (+.f64 (exp.f64 (neg.f64 x)) (*.f64 (exp.f64 (neg.f64 x)) x)) (*.f64 -1 (Rewrite=> +-commutative_binary64 (+.f64 (*.f64 (exp.f64 (neg.f64 x)) x) (exp.f64 (neg.f64 x)))))): 0 points increase in error, 0 points decrease in error
   (-.f64 (+.f64 (exp.f64 (neg.f64 x)) (*.f64 (exp.f64 (neg.f64 x)) x)) (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 -1 (*.f64 (exp.f64 (neg.f64 x)) x)) (*.f64 -1 (exp.f64 (neg.f64 x)))))): 0 points increase in error, 0 points decrease in error

4. Taylor expanded in x around inf 0.6

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

5. Final simplification 0.6

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

Alternatives

Alternative 1
Error	0.8
Cost	13248

\[\frac{2 \cdot e^{\mathsf{log1p}\left(x\right) - x}}{2} \]

Alternative 2
Error	1.1
Cost	7108

\[\begin{array}{l} \mathbf{if}\;x \leq 1.42:\\ \;\;\;\;\frac{2 \cdot \mathsf{fma}\left(x, x \cdot -0.5, 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 3
Error	0.6
Cost	6976

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

Alternative 4
Error	1.3
Cost	196

\[\begin{array}{l} \mathbf{if}\;x \leq 4000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 5
Error	16.7
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022291 
(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))