Average Error: 29.1 → 0.6
Time: 10.3s
Precision: binary64
Cost: 6976
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
↓
\[\frac{2 \cdot \frac{x + 1}{e^{x}}}{2} \]
(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 (/ (+ x 1.0) (exp x))) 2.0))
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 * ((x + 1.0) / exp(x))) / 2.0;
}

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 * ((x + 1.0d0) / exp(x))) / 2.0d0
end function

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 * ((x + 1.0) / Math.exp(x))) / 2.0;
}

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 * ((x + 1.0) / math.exp(x))) / 2.0

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(x + 1.0) / exp(x))) / 2.0)
end

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 * ((x + 1.0) / exp(x))) / 2.0;
end

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[(x + 1.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

\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{x + 1}{e^{x}}}{2}

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 29.1

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

  2. 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. Simplified0.6

    \[\leadsto \frac{\color{blue}{2 \cdot \left(e^{-x} \cdot \left(x - -1\right)\right)}}{2} \]
    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)))): 1 points increase in error, 0 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. Applied egg-rr0.6

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

  5. Final simplification0.6

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

Alternatives

Alternative 1
Error1.1
Cost1220
\[\begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;0.5 \cdot \frac{4 - \left(x \cdot x\right) \cdot \left(x \cdot x\right)}{2 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
Alternative 2
Error1.1
Cost580
\[\begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;0.5 \cdot \left(2 - x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
Alternative 3
Error1.2
Cost196
\[\begin{array}{l} \mathbf{if}\;x \leq 1400:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
Alternative 4
Error16.9
Cost64
\[1 \]

Error

Reproduce

herbie shell --seed 2022297 
(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))