Average Error: 29.4 → 0.7
Time: 4.8s
Precision: binary64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
↓
\[\frac{\mathsf{log1p}\left(\mathsf{expm1}\left(e^{\mathsf{log1p}\left(x\right) - x} \cdot 2\right)\right)}{2} \]
(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
 (/ (log1p (expm1 (* (exp (- (log1p x) x)) 2.0))) 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 log1p(expm1((exp((log1p(x) - x)) * 2.0))) / 2.0;
}

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.log1p(Math.expm1((Math.exp((Math.log1p(x) - x)) * 2.0))) / 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 math.log1p(math.expm1((math.exp((math.log1p(x) - x)) * 2.0))) / 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(log1p(expm1(Float64(exp(Float64(log1p(x) - x)) * 2.0))) / 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[Log[1 + N[(Exp[N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]] - 1), $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{\mathsf{log1p}\left(\mathsf{expm1}\left(e^{\mathsf{log1p}\left(x\right) - x} \cdot 2\right)\right)}{2}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 29.4

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

  2. Taylor expanded in eps around 0 0.6

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

  3. Simplified0.6

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

  4. Applied egg-rr0.6

    \[\leadsto \frac{\color{blue}{\frac{x + 1}{e^{x} \cdot 0.5}}}{2} \]

  5. Applied egg-rr0.7

    \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(e^{\mathsf{log1p}\left(x\right) - x} \cdot 2\right)\right)}}{2} \]

  6. Final simplification0.7

    \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(e^{\mathsf{log1p}\left(x\right) - x} \cdot 2\right)\right)}{2} \]

Reproduce

herbie shell --seed 2022150 
(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))