?

Average Error: 0.0 → 0.0
Time: 2.3min
Precision: binary64
Cost: 39424

?

\[0 \leq s \land s \leq 1.0651631\]
\[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
↓
\[\begin{array}{l} t_0 := \frac{\left|x\right|}{-s}\\ \frac{\frac{1}{{\left(e^{-1}\right)}^{t_0} + 1}}{\mathsf{fma}\left(e^{t_0}, s, s\right)} \end{array} \]
(FPCore (x s)
 :precision binary64
 (/
  (exp (/ (- (fabs x)) s))
  (* (* s (+ 1.0 (exp (/ (- (fabs x)) s)))) (+ 1.0 (exp (/ (- (fabs x)) s))))))
↓
(FPCore (x s)
 :precision binary64
 (let* ((t_0 (/ (fabs x) (- s))))
   (/ (/ 1.0 (+ (pow (exp -1.0) t_0) 1.0)) (fma (exp t_0) s s))))
double code(double x, double s) {
	return exp((-fabs(x) / s)) / ((s * (1.0 + exp((-fabs(x) / s)))) * (1.0 + exp((-fabs(x) / s))));
}
↓
double code(double x, double s) {
	double t_0 = fabs(x) / -s;
	return (1.0 / (pow(exp(-1.0), t_0) + 1.0)) / fma(exp(t_0), s, s);
}

function code(x, s)
	return Float64(exp(Float64(Float64(-abs(x)) / s)) / Float64(Float64(s * Float64(1.0 + exp(Float64(Float64(-abs(x)) / s)))) * Float64(1.0 + exp(Float64(Float64(-abs(x)) / s)))))
end
↓
function code(x, s)
	t_0 = Float64(abs(x) / Float64(-s))
	return Float64(Float64(1.0 / Float64((exp(-1.0) ^ t_0) + 1.0)) / fma(exp(t_0), s, s))
end

code[x_, s_] := N[(N[Exp[N[((-N[Abs[x], $MachinePrecision]) / s), $MachinePrecision]], $MachinePrecision] / N[(N[(s * N[(1.0 + N[Exp[N[((-N[Abs[x], $MachinePrecision]) / s), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[Exp[N[((-N[Abs[x], $MachinePrecision]) / s), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[x_, s_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] / (-s)), $MachinePrecision]}, N[(N[(1.0 / N[(N[Power[N[Exp[-1.0], $MachinePrecision], t$95$0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[Exp[t$95$0], $MachinePrecision] * s + s), $MachinePrecision]), $MachinePrecision]]

\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}
↓
\begin{array}{l}
t_0 := \frac{\left|x\right|}{-s}\\
\frac{\frac{1}{{\left(e^{-1}\right)}^{t_0} + 1}}{\mathsf{fma}\left(e^{t_0}, s, s\right)}
\end{array}

Error?

Derivation?

  1. Initial program 0.0

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

  2. Applied egg-rr0.0

    \[\leadsto \color{blue}{\frac{\frac{1}{e^{\frac{\left|x\right|}{s}} + 1}}{\mathsf{fma}\left(e^{\frac{\left|x\right|}{-s}}, s, s\right)}} \]

  3. Applied egg-rr0.0

    \[\leadsto \frac{\frac{1}{\color{blue}{{\left(e^{-1}\right)}^{\left(\frac{\left|x\right|}{-s}\right)}} + 1}}{\mathsf{fma}\left(e^{\frac{\left|x\right|}{-s}}, s, s\right)} \]

Alternatives

Alternative 1
Error0.1
Cost32960
\[\frac{\frac{-1}{s}}{-e^{\frac{\left|x\right|}{s} - \mathsf{log1p}\left(e^{\frac{\left|x\right|}{-s}}\right) \cdot -2}} \]
Alternative 2
Error0.0
Cost32768
\[\frac{e^{\mathsf{log1p}\left(e^{\frac{\left|x\right|}{-s}}\right) \cdot -2 - \frac{\left|x\right|}{s}}}{s} \]
Alternative 3
Error0.9
Cost26688
\[\begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ \frac{t_0}{\left(s \cdot \left(1 + t_0\right)\right) \cdot \left(1 + 1\right)} \end{array} \]
Alternative 4
Error7.3
Cost19716
\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 10^{-172}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{-\left|x\right|}{s}}}{s}\\ \end{array} \]
Alternative 5
Error1.0
Cost13248
\[\frac{\frac{0.25}{s}}{e^{\frac{\left|x\right|}{s}}} \]
Alternative 6
Error40.7
Cost7884
\[\begin{array}{l} t_0 := \begin{array}{l} \mathbf{if}\;\frac{-0.25}{s} \ne 0:\\ \;\;\;\;\frac{\frac{0.0625}{s \cdot s}}{\sqrt[3]{\frac{\frac{-0.0625}{s}}{s} \cdot \frac{-0.25}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array}\\ \mathbf{if}\;x \leq -1.25 \cdot 10^{-108}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3.85 \cdot 10^{-112}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 7
Error40.7
Cost7884
\[\begin{array}{l} t_0 := \frac{\frac{-0.0625}{s}}{s}\\ t_1 := \frac{-0.25}{s} \ne 0\\ t_2 := \frac{0.0625}{s \cdot s}\\ \mathbf{if}\;x \leq -1.8 \cdot 10^{-106}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;t_1:\\ \;\;\;\;\frac{t_2}{\sqrt[3]{t_0 \cdot \frac{-0.25}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array}\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-111}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{elif}\;t_1:\\ \;\;\;\;\frac{t_0}{\sqrt[3]{\frac{-0.25}{s} \cdot t_2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]
Alternative 8
Error47.3
Cost192
\[\frac{0.25}{s} \]

Error

Reproduce?

herbie shell --seed 2023033 
(FPCore (x s)
  :name "Logistic distribution"
  :precision binary64
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ (exp (/ (- (fabs x)) s)) (* (* s (+ 1.0 (exp (/ (- (fabs x)) s)))) (+ 1.0 (exp (/ (- (fabs x)) s))))))